Patent Publication Number: US-2011070962-A1

Title: Principle-based device and method for using an asymmetrical target zone to improve golf-putting skill

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to devices and methods for improving golf-putting skills, such as golf-putting practice greens, targets, games, and game procedures. Specifically, the present invention relates to a vastly improved golf-putting target and golf-putting green which, by virtue of its principled design, reinforces good putting habits. Also disclosed herein are one or more methods for using the device. The present invention provides, among other things, an improvement on a standard golf-practice putting green and golf-practice putting target that is designed around a principled method of analyzing different putting situations based on the topology of the putting surface. 
     Obviously, in the game of golf the primary objective of putting is to get the ball into the cup. That said, most first putts taken from the green actually miss. What is needed then is some principled way of teaching putters how to putt toward the hole so that the ball stands the best possible chance of going into the hole and, perhaps more importantly, that, if it misses the hole, then it goes in on no more than the second putt taken from the green. That is to say, to the extent there is a principled way to view the art of putting so that one can avoid three-putting (i.e., taking three putts from the green to sink the ball), what is needed, then, is a device and method for reinforcing such principles. 
     Fortunately, such principles do exist and can be stated relatively easily. The first of such principles reads:
         Secondary Objective of Putting: A “good miss” preferably comes within “two-putt” range of the cup, which for most golfers is approximately 18 inches. Beyond this distance, most players run a non-trivial risk of three-putting.
 
Quite intuitive in its meaning, this objective merely states that a “good miss” comes within no more than 18 inches of the hole. Experience teaches that putts landing within this range have a very high likelihood of being sunk on the next attempt. Obviously, different playing populations (e.g., juniors, children, professionals, seniors, etc.) may have different needs, but fixing the two-putt range at roughly 18 inches seems appropriate for the largest segment of the golfing world. Modifications in the present invention can be made as needed, however, since this distance is not etched in stone.
       

     Central to the present invention is another basic though less intuitive principle of good putting, which reads:
         Percentage-Based Principle of Good Putting: For every putt taken, there is a high-percentage side of the hole and a low-percentage side of the hole that are determined by the topology of the putting surface as viewed from the putting location. Specifically, in most circumstances the high-percentage side is on the side of the hole with the highest elevation, and the low-percentage side is on the side of the hole with the lowest elevation.
 
This principle states merely that for every putting situation an imaginary line can be drawn through the cup that divides the area near the cup into a low-percentage side and a high-percentage side. Generally speaking, putts that miss the hole but land within a short distance of the cup on the high-percentage side should be viewed as at least slightly preferable to putts that land an equal distance from the cup but on the low-percentage side. This is so because, first, as the name suggests putts that are directed toward (or that land within) the high-percentage side of the hole stand a greater chance of going into the hole than those putts that are directed toward (or land within) the low-percentage side of the hole. Secondly, putts that are directed toward or land within the high-percentage side of the hole stand a greater chance of landing within eighteen inches of the hole than putts that are directed toward or land within the low-percentage side of the hole, because on the side of the hole with the higher elevation gravity tends to move the ball back toward the hole, whereas on the low side of the hole gravity tends to move the ball away from the hole. As such, it is possible to rank the approximate quality of a “good miss” both in terms of its distance from the hole and in terms of which side of the hole it lands on.
       

     The Percentage-Based Principle of Good Putting can best be explained by a simple example that will form the basis upon which more complex situations can be clearly analyzed. On a flat putting surface without any right-to-left or left-to-right pitch, a ball that does not make it at least all the way to the hole stands absolutely no chance of going in. Such a ball simply did not travel far enough and is said to have been “underputt.” Conversely, if a ball makes it all the way to the hole but misses and goes a little bit farther than the hole (i.e., is “overputt”), while it still may have missed, it had at least some opportunity of going in. It did, after all, travel at least far enough to have had some possibility of falling in. Hence, for flat pitchless putting surfaces we can think of the area in front of the cup (as viewed from the putting location) as the low-percentage side of the cup—i.e., because underputted balls never go in. Similarly, we can think of the area behind the cup as the high-percentage side of the cup—i.e., because overputted balls at least stand some chance of going in. All things being equal (which is a vastly encompassing and greatly oversimplifying assumption indeed) on flat, pitchless surfaces overputted balls are at least slightly preferable to underputted balls. 
     Consequently, the line that divides the cup area on a flat pitchless putting surface into a high-percentage side and a low-percentage side is simply a straight line through the cup that is perpendicular to the balls&#39; (straight) trajectory from the putting location to the cup. Similarly, as detailed more thoroughly in the discussion that follows, we can construct a similar dividing line for all common putting-surface topologies—e.g., uphill slope, downhill slope, flat or no slope, left-to-right pitch, right-to-left pitch, and reasonable combinations of these slopes and pitches. Although the following discussion will elaborate the point in fine detail, it can be generalized that the high side of the cup (i.e., the side of the cup at a higher elevation) is in fact also the high-percentage side. 
     What is needed, then, is a putting device and method that encourages players to putt to within two-putt range (or 18 inches) of the hole on the high-percentage side and that discourages putting to the low-percentage side. As will be illustrated throughout the following discussion, among the many aims of the present invention, it fulfills these needs. Specifically, the present invention is directed toward encouraging users to practice good putting habits by establishing a practice putting green and target region that encourage putts directed toward the high-percentage side of the hole and that reward “good misses” both that stop within 18 inches on the high-percentage side of the hole and that stop within 12 inches on the low-percentage side of the hole. 
     The present invention accomplishes this and other objectives by providing an asymmetrical putting target with a large region (or “long side”) and a small region (or “short side”). By placing the large region of the target on the high-percentage side of the hole and the small region on the low-percentage side, the present invention encourages players always to aim for the high-percentage side of the hole. Furthermore, although several sizes and shapes can be used as the target zone in accordance with the present invention, one set of embodiments encourages users to putt the ball to within no more than 12 inches on the low-percentage side and to within no more than 18 inches on the high-percentage side. Within this set of embodiments, one embodiment also uses a target zone that is circular in shape. 
     While choosing a boundary 12 inches from the cup on the low-percentage side is somewhat arbitrary, it is a distance that is generally agreed to be the outer boundary of what is referred to as “tap-in range,” i.e., that distance from which a putting attempt is almost always successful. It was furthermore selected with two things in mind—i.e., i) that such distance had to be a shorter than 18 inches (the boundary on the high-percentage side), so as to discourage putting to the low-percentage side, and ii) that the short distance had to be realistic in conforming to widely held views about the meaning of a “good put,” inasmuch as putts that come close to the hole even on the low-percentage side are often widely considered to be quite good. Under such considerations 15 inches seems too generous, and 6 inches seems too stingy, although for some circumstances and some player populations, they too might be appropriate. While strict determination for the low-percentage side distance is not possible with exacting precision, 12 inches seems appropriate for the widest range of circumstances and players—although modifications are possible. 
     SUMMARY OF THE PRIOR ART 
     While several creative arrangements for a golf-putting target region have been offered through the years, to date easily the most common arrangement contains a set of one or more concentric circles forming a “bull&#39;s-eye” pattern with the cup placed at dead center. U.S. Pat. No. 1,738,265 to Scanlon; U.S. Pat. No. 1,979,584 to Thompson; U.S. Pat. No. 3,490,769 to Torbett; U.S. Pat. No. 3,649,027 to Vallas; U.S. Pat. No. 3,695,619 to Brobston; U.S. Pat. No. 5,383,665 to Schultz et al.; U.S. Pat. No. 5,830,076 to Borys; U.S. Pat. No. 6,176,789 to Kluttz et al.; U.S. Pat. No. 6,419,590 to Criger; U.S. Pat. No. 6,875,121 to McKeen, Jr.; and U.S. Pat. No. 7,192,360 to Tamulweicz, as examples, all teach golf-putting games or devices that include placing a golf cup at dead center to one or more concentric areas (typically circular in shape) to form a target region. Scoring methods are then devised so as to give a preferential score to putts that land closer to the cup (as measured by circular areas of smaller radius) than to those that land farther away, but without giving consideration to which side of the cup the ball eventually lands on. Justifications for such arrangements tend to focus on their simplicity or perhaps the visual aesthetics of a certain degree of symmetry within a bull&#39;s-eye. 
     These arrangements, however, are not justified by the aforementioned principles of good putting (i.e., getting the ball to within two-putt range on the high-percentage side of the hole). The problem with placing the cup at the exact center of a circular target region is that doing so effectively reinforces bad putting habits. For example, if a putt is made from a flat putting surface with no breaking slant to the green and then ends 14 inches in front of the cup, it is scored by each of the aforementioned prior-art methods to be equal in caliber to a putt made from the same location that lands 14 inches behind the cup. As such, this scoring method, and hence the target regions themselves, does not follow the percentage-based approach to good putting—i.e., that the high-percentage side of the hole is to be preferred over the low-percentage side. Other things being equal (again, a widely encompassing assumption), the ball that was overputt by 14 inches should be seen as somewhat superior to the ball that was underputt by 14 inches and should be scored more preferentially. 
     Consequently, any scores arising from putting games played with such target regions do not accurately reflect relative putting quality. Even more importantly, by placing a highly visible set of rings concentrically surrounding the cup—upon which the putter must focus intense mental concentration while lining up her stroke—the aforementioned devices are subconsciously reinforcing the notion that all putts that land an equal distance from the cup are of equal quality. The player will then begin to visualize a set of concentric circles around the cup on all of her putts, including those putts taken during real golf play. This is a powerful but unintentional subconscious message that these devices are sending to unsuspecting golfers looking to improve their putting game. 
     Other arrangements for a putting target have been proposed as well, including a shuffleboard pattern (U.S. Pat. No. 3,490,769 to Florian), a plurality of cups arranged like bumpers in a bumper-pool table (U.S. Pat. No. 4,877,250 to Torbett), and even a Cartesian grid with the cup placed at the origin (U.S. Pat. No. 5,607,360 to Shiffman). While these arrangements may make for entertaining and novel game play, they do not do enough to reinforce good putting habits so as to maximize the value of a user&#39;s valuable practice time in the same manner as the presently disclosed invention does. 
     At best some prior-art putting games have taught the placement of a cup inside one or more concentric areas (whether circular or otherwise) at an unspecified location instead of at dead center. While it may be possible for random cup placement to reinforce good putting habits in accordance with the principles of the present invention, such a state of affairs comes about only by pure accident, since the references do not teach or claim, either explicitly or implicitly, such principles. In fact at least one such device indirectly teaches placement of the cup at a location that exaggerates the impact of reinforcing bad putting habits. U.S. Pat. No. 5,401,027 to Surbeck discloses a set of thin plastic tubes of varying length that can be made to form enclosed irregular regions of differently sized areas. The enclosed regions thereby formed are then placed around the cup to form a series of telescoping boundaries surrounding the cup. No specific methodology for placing the tubes around the cup so that proper putting skills are reinforced is taught explicitly by Surbeck. The main illustration from the Surbeck patent, however, (Surbeck FIG. 5) shows an exemplary arrangement in which the cup is placed at or near the very farthest point within each of the enclosed regions, which are presumed to be on a flat surface with no noticeable pitch. 
     While other embodiments without this drawback are indeed contemplated by Surbeck, this arrangement strongly encourages underputting on flat surfaces by rewarding putts that fall short of the cup by several feet while simultaneously punishing heavily those putts that go beyond the cup by only a few inches. Though it is very likely unintentional, this arrangement encourages users to putt in such a manner that the ball stands a relatively lower chance of going in than even the center-placed cup within concentric circular regions. Namely, the user is encouraged to underputt on flat surfaces for fear of being penalized by going beyond the marked boundary by even a small distance and thereby securing a lower score for the putt. 
     What is needed, then, is a target region that rewards putting toward the high-percentage side of the hole, that encourages the player to come within no more than two-putt range of the cup (i.e., no more than 18 inches or so, as modified by player necessity), and that provides visual reinforcement of a properly constructed target region surrounding the cup. Of its many objectives the present invention fulfills each of these needs. 
     BRIEF SUMMARY OF THE INVENTION 
     Unlike prior-art putting greens, the present invention places a golf cup slightly off center within a target region in a precise manner that encourages adherence to the two aforementioned putting principles. Traditionally, putting targets were designed by placing the cup at dead center to one or more concentric circles, but as will be demonstrated throughout the present discussion, this old way of thinking does not conform to the principles of good putting. As described more thoroughly below, an improved target region is formed by identifying a proper, preferred, or even ideal line of approach to the cup in view of the putting location&#39;s spatial relationship to the hole and in light of the slope (uphill or downhill) and pitch (right-to-left or left-to-right break) of the putting surface. The target region is then formed either: a) by establishing a boundary of the region approximately 18 inches away from the hole on the high-percentage side of the hole and 12 inches away from the hole on the low-percentage side of the hole, and then completing the boundary in any desired shape; or b) by assuring that a pronounced majority—in the neighborhood of sixty percent (60%) to seventy percent (70%)—of a boundary&#39;s area is located on the high-percentage side of the hole in light of the aforementioned putting-surface topology (e.g., behind the hole for uphill and flat surfaces and in front of the hole for downhill surfaces). As will be demonstrated below in connection with  FIG. 1B , these two alternative ways of constructing the target zone (by distance or by area) are closely related in a geometric sense. 
     Since one preferred embodiment of the present invention comprises the use of a circular target region, and since a standard golf cup (another preferred embodiment) is four and one-quarter inches (4.25″) wide, taking into account the two aforementioned distances to the target region&#39;s boundaries results in a circular target region with a diameter of exactly thirty-four and one quarter inches (34.25″), wherein the cup is located twelve inches (12″) away from one side of the target region and eighteen inches (18″) away from the opposite side of the target region as measured along a shared diameter of the circle. The target region is then situated on the target green with the short (twelve inch) side in one direction and the long (eighteen inch) side in the other direction according the principles discussed herein that address the slant and slope of the putting surface. This particular embodiment has been demonstrated to produce good results in the improvement of a player&#39;s putting game. 
     Alternative embodiments to the circular region can also be constructed as well. As will also be demonstrated below in connection with  FIG. 1B , placing a standard golf cup twelve inches from one side of a circle and eighteen inches from the opposite side along a single diameter results in the cup dividing the circle (by constructing a line through the cup that is also perpendicular to the diameter in question) into two regions of approximately 40% and 60% of the circle&#39;s total area. This 40:60 (or 2:3) ratio of areas on each side of the cup can then be applied to target shapes that are not strictly circular. As such, a variety of target shapes can be produced in accordance with the present invention as discussed below in connection with  FIGS. 2A through 2E . 
     Alternatively, some applications of the present invention require a deviation from strict adherence to there being twelve inches and eighteen inches on either side of the cup so as to accommodate the needs of different player groups—e.g., longer for juniors and children, and shorter for professionals. Consequently, additional embodiments of the present invention are constructed by modifying these two distances but maintaining the same 12:18 ratio of distances (which coincidentally also equals 2:3) to the hole from either side of the target boundary. Several variations (although not a comprehensive set) on this and the prior theme are explained at length below in connection with a detailed description of the invention. 
     Furthermore, another objective of the present invention is to provide a playing method that simulates the scoring of regulation golf play and that at least roughly correlates to the actual putting performance of a true round of golf in which putts of the same quality are made. 
     Therefore in accordance with another embodiment of the invention, a method for playing a game with the disclosed apparatus is claimed in which the aforementioned scoring objectives are achieved. Specifically, one stroke is made from each of one or more fixed locations at varying distances from a cup that is enclosed within a target region designed in accordance with the principles disclosed herein. For each putt that makes its way into the hole with only this one putt, a preferred score (typically negative one point) is assigned to this putt; for each putt that makes its way into the target zone but not into the cup, a slightly less preferred score (typically zero points) is assigned; for every putt that stays on the putting surface but does not make its way into the target zone, an even slightly less preferred score (typically positive one point) is assigned; and for every putt that does not remain on the official putting surface, a still yet slightly less preferred score (typically positive two points) is assigned. (It is worth noting that in the typical scoring scheme, a higher score, like in actual golf play, is considered less preferred than a lower score.) 
     Most significantly, in accordance with another embodiment of the present invention, a final putt is made from a location that is particularly far away from the cup in relative comparison to the aforementioned one or more putting locations. This final putt (called a “superputt”) is scored in a fashion similar to the regular putts described above, but the point values assigned to the different destination regions are adjusted to reflect the increased difficulty of the superputt. (Typically, each of the point values provided above is lowered by one point—i.e., minus two for the cup, minus one for the target zone, zero for the surface, and positive one for out of bounds.) In an alternative embodiment of the invention, the putt made from the extra-long putting distance is not a compulsory component of the putting game, but rather is made only if the score achieved from the original set of putts achieves some threshold score. (In such cases, the superputt is considered a bonus putt because it is optional.) 
     The score thus assigned correlates nicely to actual golf play, encourages good putting habits, and provides a rough estimate of a player&#39;s actual putting skill as would be reflected in a true round of golf. To wit, if the suggested values are used, a net score of zero would indicate that the golfer is putting on par for the “course” reflected in the one or more initial putting locations. Conversely, a positive score would reflect strokes over par, and a negative score would reflect strokes under par, for the putting component of a hypothetical round of golf. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The foregoing summary, as well as the following detailed description of preferred embodiments of the invention, will be better understood when read in conjunction with the appended drawings. For the purpose of illustrating the invention, there is shown in the drawings embodiments that are presently preferred. It should be understood, however, that the invention is not limited to the precise arrangements and instrumentalities shown, in which: 
       The multiple views of  FIG. 1  ( FIGS. 1A  through  FIG. 1C ) illustrate a preferred embodiment of the present invention in which a golf cup is placed inside an asymmetrical target zone.  FIG. 1A . illustrates a circular golf-putting target zone wherein the hole is placed off center on a given diameter.  FIG. 1B  provides important geometric details to the aforementioned asymmetrical circular target zone so as to derive important proportionality information for subsequent analogous application to other, non-circular shapes and non-preferred-embodiment sizes.  FIG. 1C  illustrates in clear detail an important trigonometric relationship between the center-to-hole distance and the circular target zone&#39;s radius. 
       The multiple views of  FIG. 2  ( FIGS. 2A through 2E ) illustrate additional embodiments of the present invention in which a golf cup is placed inside asymmetrical target zones of various, non-circular shapes.  FIG. 2A  illustrates a triangular target zone.  FIG. 2B  illustrates a square target zone.  FIG. 2C  illustrates a rectangular target zone.  FIG. 2D  illustrates an oval-shaped target zone.  FIG. 2E  illustrates an irregular (or “amoeboid-shaped”) target zone. 
       The multiple views of  FIG. 3  ( FIGS. 3A through 3E ) illustrate the concept of a “line of ideal approach,” which is a critical component for understanding the present invention.  FIG. 3A  illustrates the construction of a line of ideal approach for a putt taken on a surface without any right-to-left or left-to-right pitch.  FIG. 3B  illustrates the construction of a line of ideal approach for a putt taken on a surface with left-to-right pitch.  FIG. 3C  illustrates the construction of a line of ideal approach for a putt taken on a surface with right-to-left pitch.  FIG. 3D  illustrates how slight perturbations may occur within the construction of a line of ideal approach on a given surface with pitch (in this case, right-to-left pitch) and that such perturbations are contained within some reasonable boundaries.  FIG. 3E  illustrates the construction of a line of ideal approach for a given hole in light of a plurality of putting locations. 
       The multiple views of  FIG. 4  ( FIGS. 4A through 4G ) illustrate how the target zone is situated on putting surfaces of differing topologies (i.e., slopes and pitches) in accordance with the presently described invention.  FIG. 4A  illustrates how the target zone is situated on a flat or uphill surface that has no pitch.  FIG. 4B  illustrates how the target zone is situated on a flat or uphill surface that has left-to-right pitch.  FIG. 4C  illustrates how the target zone is situated on a flat or uphill surface that has right-to-left pitch.  FIG. 4D  illustrates how the target zone is situated on a downhill surface that has no pitch.  FIG. 4E  illustrates how the target zone is situated on a downhill surface that left-to-right pitch.  FIG. 4F  illustrates how the target zone is situated on a downhill surface that has right-to-left pitch.  FIG. 4G  illustrates the manner in which the various combinations of slope and pitch, as depicted in  FIGS. 4A through 4F , can be conceptually joined as all lying on a hypothetical, perfectly planar surface and oriented with respect to the cup from different directions. 
       The multiple views of  FIG. 5  ( FIGS. 5A and 5B ) illustrate methods for using the presently described invention in accordance with two preferred embodiments of a putting-practice game.  FIG. 5A  illustrates a method for using the presently described invention, such that a superputt is taken from a particularly long putting location so as to provide an adjustment means to a putting-game&#39;s score.  FIG. 5B  illustrates a method for using the presently described invention, such that a bonus putt is taken from a particularly long putting location so as to provide an adjustment means to a putting-game&#39;s score, provided that the player&#39;s interim score meets a minimum, threshold quality. 
       The multiple views of  FIG. 6  ( FIGS. 6A through 6D ) illustrate an additional four (4) preferred embodiments in which a plurality of putting locations are placed in relation to a golf cup enclosed by an asymmetrical target zone on different topological variations of a putting surface in accordance with the presently described invention.  FIG. 6A  illustrates the placement of a plurality of putting locations on a putting surface that slopes downhill without pitch.  FIG. 6B  illustrates the placement of a plurality of putting locations on a putting surface that slopes uphill without pitch.  FIG. 6C  illustrates the placement of a plurality of putting locations on a putting surface that is flat but pitches from right to left.  FIG. 6C  illustrates the placement of a plurality of putting locations on a putting surface that slopes uphill and that pitches from left to right. 
         FIG. 7  illustrates an additional embodiment in which both uphill and downhill putting can be practiced on the same surface, containing a plurality of putting locations, with use of a single target zone designed in accordance with the presently described invention. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Certain terminology is used herein for convenience only and is not to be taken as a limitation on the present invention. In the drawings, the same reference numerals and reference letters are employed for designating the same elements throughout the several figures. The following discussion describes in detail several embodiments of the invention and several variations on those embodiments. This discussion should not be construed as limiting the invention to those particular embodiments or variations. Persons skilled in the art will recognize numerous other embodiments and variations as well. For definition of the complete scope of the invention, reference is made to the appended claims. 
     The words “right”, “left”, “lower” and “upper” designate directions in the drawings to which reference is made. The words “two-o&#39;clock,” “four-o&#39;clock,” “six-o&#39;clock,” “eight-o&#39;clock,” “ten-o&#39;clock” and “twelve o&#39;clock” (and the identically meaning “2-o&#39;clock,” “4-o&#39;clock,” “6-o&#39;clock,” “8-o&#39;clock,” “10-o&#39;clock,” and “12-o&#39;clock”) refer to the relative off-center placement of a golf cup within a circular target region. If the circular region is analogized to a watch or clock face, the golf cup would then be placed at or near the tip of a hypothetical hour hand placed in the appropriate position at the described time, where the clock face is viewed from a fixed putting location (or plurality thereof) facing the circular region and where the “twelve-o&#39;clock” position is imaged to lie directly ahead. When such orientation terms are used in connection with the included drawings, a legend or key is provided so that clarity can be secured. 
     The terms “pitch,” “slope,” “break,” and “slant” also have specific non-synonymous meanings in connection with the topology of a putting surface, as explained in the following discussion. Similarly, the term “line of ideal approach” is a term constructed for the purpose of describing the disclosed invention. It too is defined in great detail in the following discussion in connection with the multiple views of  FIG. 3 . 
     At the core of the presently disclosed invention lies a simple but thoughtfully engineered golf target zone that encourages good putting habits. While the present invention encompasses much more than just a target region, understanding the other facets of the present invention first requires understanding how the target region is designed. 
       FIG. 1A  illustrates the basic golf-putting target contemplated by the present invention. A golf cup  101  is placed slightly off center within a circular target region  102  on a given diameter of the circle  108 . To assure that the cup  101  is off center within the circular region  102 , the distance  103  from the cup  101  to one side of the circle along the diameter  108  is shorter than the distance  104  from the cup to the other side of the circle along the diameter  108 . A preferred embodiment has the shorter of the two distances, in this case distance  103 , equal to twelve inches (12″), and the longer distance, in this case distance  104 , equal to eighteen inches (18″). Though it is not required in order to constitute an embodiment of the present invention, if cup  101  is a standard, regulation-sized golf cup of diameter four and one quarter inches (4.25″), then circular region  102  will have a total diameter  108  of thirty-four and one-quarter inches (34.25″). 
     It should also be noted that although the present discussion proceeds largely on the assumption that target zone  102  is incorporated into the surface of a putting green (e.g., a target zone painted onto the surface), that need not always be the case. According to one set of embodiments of the present invention, the target zone can be made of a thin, flexible material, such as plastic, vinyl, cloth, or the equivalent, which can be fixably removed from the putting surface. In such cases, the “cup”  101  of these embodiments corresponds to an aperture in the flexible material that would be placed over the actual cup on a putting surface, thereby permitting the golf ball to flow through the material into the actual golf cup. In other words these embodiments represent stand-alone golf putting targets that are portable from one putting surface and golf cup to another—not exclusively golf-putting greens permanently attached to the ground (although the present discussion will treat these cases as equivalent and focus on the latter). For the sake of convenience, such apertures will be treated as the equivalent to a golf cup in the present discussion unless otherwise specified. 
     It should be noted that the long-side distance  104  was selected as the “two-putt” range, which in a preferred embodiment equals 18 inches. The short-side distance  103  was chosen so as to be shorter than long-side distance  104  but still reasonably accommodating for putts that land close to the hole but on the low-percentage side. If the long-side distance equals 18 inches, in accordance with one preferred embodiment, then the short-side distance equals 12 inches, in accordance with the same preferred embodiment. Naturally, variations can be introduced into both of these distances in accordance with the foregoing discussion. 
     This particular 12 and 18 inch embodiment has been demonstrated to be quite effective, when used in accordance with the other principles and teachings of the present invention, in reinforcing good golf-putting habits. From time to time, however, modifications must be made in order to accommodate players with different needs or otherwise to introduce some degree of variety into the playing qualities of the present invention. In order to make such modifications to circular target region  102 , two design features can be extracted from the aforementioned preferred embodiment in a generalized manner. 
     First, it is to be noted that the ratio of shorter length  103  to longer length  104  is 12:18, which reduces to the equivalent ratio of 2:3. Circles of different sizes and even other non-circular shapes of different sizes can be constructed, (as will be demonstrated below in connection with  FIGS. 2A through 2E ) by more or less observing this ratio between different distances from the cup  101  to the outer boundary of the target region  102 . 
     Secondly, other modifications can be made to circular target region  102 , inasmuch as a comparison of the relative magnitude of the two sub-areas  106  (the larger, or “high-percentage” sub-area) and  107  (the smaller or “low-percentage” sub-area) within the target region  102  (one on either side of the cup  101 ) can be made and then approximately generalized to other shapes and sizes. To address these other shapes and sizes, a detailed geometric understanding of the preferred embodiment shown in  FIG. 1A  is first required. The appropriate geometric principles can then be analogized accordingly. 
     In  FIG. 1A  a diving line  105  is drawn through the center of the cup  101  that is perpendicular to the aforementioned reference diameter  108  through the cup  101 . As such, circular region  102  is divided into a region, small region  107 , on one side of perpendicular dividing line  105  and another region, large region  106 , on the other side of perpendicular dividing line  105 . (For convenience of reference these two subregions,  106  and  107 , will be described as lying on one or the other side of cup  101 , not just as lying on one or the other sides of perpendicular dividing line  105 .) Perpendicular dividing line  105  is provided for reference purposes only. Actual embodiments of the present invention do not necessarily need to have the line physically represented on the actual target region  102  (although they may). 
       FIG. 1B  elaborates this idea by providing sufficient geometric detail to permit relatively accurate calculation of the relative sizes of small region  107  and large region  106  for the embodiment in which the diameter of circular region  102  is a preferred thirty-four and one-quarter inches (34.25″) as described above. Within circular region  102 , dividing line  105  is redrawn through the cup  101  perpendicular to the diameter  108  on which the cup  101  sits. In addition, radii  109   a  and  109   b  are drawn connecting the cup  101  to the two points of intersection,  110   a  and  110   b,  respectively, between perpendicular dividing line  105  and circular region  102 . An equilateral triangle is thereby formed with sides comprising the aforementioned radii  109   a  and  109   b  along with the segment from the perpendicular dividing line  105  that lies within circular region  102 . These lines connect the center  112  of the circular region  102  with the aforementioned points of intersection  110   a  and  110   b  between the perpendicular dividing line  105  and the circular region  102 . 
     The angle  111  formed by the intersection of the two radii  109   a  and  109   b  at the center  112  of circular region  102  is denoted Θ (the capital Greek letter theta) for convenience. From a standard table of formulas, one learns that the size of small area  107 , denoted A for convenience, is given by: 
         A= ½ R   2 (Θ−sin Θ).
 
     If what one is concerned about, however, is the ratio of this area  107  to the area of the circular region  102 , (which equals π R 2 ), then one needs: 
     
       
         
           
             
               
                 
                   
                     A 
                     / 
                     
                       A 
                       ⊙ 
                     
                   
                   = 
                   
                     
                       1 
                       / 
                       2 
                     
                      
                     
                       
                         
                           R 
                           2 
                         
                          
                         
                           ( 
                           
                             Θ 
                             - 
                             
                               sin 
                                
                               
                                   
                               
                                
                               Θ 
                             
                           
                           ) 
                         
                       
                       / 
                       π 
                     
                      
                     
                         
                     
                      
                     
                       R 
                       2 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                     
                       
                         
                           ( 
                           
                             Θ 
                             - 
                             
                               sin 
                                
                               
                                   
                               
                                
                               Θ 
                             
                           
                           ) 
                         
                         / 
                         2 
                       
                        
                       π 
                     
                   
                   , 
                 
               
             
           
         
       
     
     where R denotes the length of radius (or one half of the diameter  108 ) of the circular region, which, in a preferred embodiment, is 34.25″. If we let w denote the width of the cup  101  (which, in a preferred but not mandatory embodiment, is the standard 4.25″), and if we let 2a and 3a equal the lengths of the subject diameter  108  on either side of the cup  101 , respectively, (where, in a preferred embodiment a=6″, so that 2a=12″ and 3a=18″ thereby giving the precise lengths listed in connection with  FIG. 1A  describing one preferred embodiment), we can calculate the distance  113  between the center of the cup  101  and the center  112  of the circular region  102 . Notably, this cup-to-center distance  113 , denoted L for convenience, is given by the formula: 
         L=R −(2 a+ ½ w ).
 
     Since w, R, and a are all known quantities for a preferred embodiment and follow the relation w+2a+3a=2R (because each side of this equation equals the diameter  108  of the circular region  102 ), we see that 2a+½w=2R−3a−½w. Therefore: 
     
       
         
           
             
               
                 
                   L 
                   = 
                   
                     R 
                     - 
                     
                       ( 
                       
                         
                           2 
                            
                           R 
                         
                         - 
                         
                           3 
                            
                           a 
                         
                         - 
                         
                           
                             1 
                             / 
                             2 
                           
                            
                           w 
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                     
                       
                         3 
                          
                         a 
                       
                       + 
                       
                         
                           1 
                           / 
                           2 
                         
                          
                         w 
                       
                       - 
                       R 
                     
                   
                   , 
                 
               
             
           
         
       
     
     which for the aforementioned preferred embodiment is equal to: 
     
       
         
           
             
               
                 
                   L 
                   = 
                   
                     
                       3 
                        
                       
                         ( 
                         
                           6 
                           ″ 
                         
                         ) 
                       
                     
                     + 
                     
                       
                         1 
                         / 
                         2 
                       
                        
                       
                         ( 
                         
                           4.25 
                           ″ 
                         
                         ) 
                       
                     
                     - 
                     
                       
                         1 
                         / 
                         2 
                       
                        
                       
                         ( 
                         
                           34.25 
                           ″ 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       18 
                       ″ 
                     
                     + 
                     
                       2.125 
                       ″ 
                     
                     - 
                     
                       17.125 
                       ″ 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       3 
                       ″ 
                     
                     . 
                   
                 
               
             
           
         
       
     
       FIG. 1C  aids in the calculation of the magnitude of Θ, and thereby the size of small sub-area  107  by the aforementioned equation. It brings into clear focus the right triangle formed by the center  112  of the circular region  102 , the cup  101 , and the point of intersection  110  between the perpendicular dividing line  105  and the circular region  102 . By definition of its construction, perpendicular dividing line  105  forms a right angle between the segment joining the cup  101  and the center  112  of the circular region  102 . As discussed in connection with  FIG. 1B , the cup-to-center line segment  113 , is length L. Symmetry dictates, furthermore, that the angle formed between the radius  109  (of length R) and the cup-to-center segment  113  equals Θ/2. Applying the definition of the cosine function to this triangle, we note: 
     
       
         
           
             
               
                 
                   
                     cos 
                      
                     
                       ( 
                       
                         Θ 
                         / 
                         2 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         L 
                         + 
                         
                           
                             1 
                             / 
                             2 
                           
                            
                           w 
                         
                       
                       ) 
                     
                     / 
                     R 
                   
                 
               
             
             
               
                 
                   Θ 
                   = 
                   
                     2 
                      
                     
                         
                     
                      
                     
                       
                         cos 
                         
                           - 
                           1 
                         
                       
                        
                       
                         ( 
                         
                           
                             ( 
                             
                               L 
                               + 
                               
                                 
                                   1 
                                   / 
                                   2 
                                 
                                  
                                 w 
                               
                             
                             ) 
                           
                           / 
                           R 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     2 
                      
                     
                         
                     
                      
                     
                       
                         cos 
                         
                           - 
                           1 
                         
                       
                        
                       
                         ( 
                         
                           
                             ( 
                             
                               R 
                               - 
                               
                                 ( 
                                 
                                   
                                     2 
                                      
                                     a 
                                   
                                   + 
                                   
                                     
                                       1 
                                       / 
                                       2 
                                     
                                      
                                     r 
                                   
                                 
                                 ) 
                               
                               + 
                               
                                 
                                   1 
                                   / 
                                   2 
                                 
                                  
                                 w 
                               
                             
                             ) 
                           
                           / 
                           R 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     2 
                      
                     
                         
                     
                      
                     
                       
                         
                           cos 
                           
                             - 
                             1 
                           
                         
                          
                         
                           ( 
                           
                             
                               ( 
                               
                                 R 
                                 - 
                                 
                                   2 
                                    
                                   a 
                                 
                               
                               ) 
                             
                             / 
                             R 
                           
                           ) 
                         
                       
                       . 
                     
                   
                 
               
             
           
         
       
     
     For target regions whose diameter is considerably larger than the diameter of the cup, as is the case in most preferred embodiments of the present invention, 2R≈5a, and therefore R−2a≈.2R. Hence, we can approximate Θ as: 
     
       
         
           
             
               
                 
                   
                     Θ 
                     ≈ 
                       
                      
                     
                       2 
                        
                       
                           
                       
                        
                       
                         
                           cos 
                           
                             - 
                             1 
                           
                         
                          
                         
                           ( 
                           .2 
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
                       for 
                        
                       
                           
                       
                        
                       w 
                     
                      
                     R 
                   
                 
               
             
             
               
                 
                   ≈ 
                     
                    
                   
                     
                       2.7389 
                       radians 
                     
                     . 
                   
                 
               
             
           
         
       
     
     Θ therefore equals approximately 2.7389 radians (or 157°), as is confirmed empirically by  FIGS. 1A and 1B , which are both drawn roughly (though not perfectly) to scale. ( FIGS. 1A and 1B  are not drawn precisely to scale, because center-to-hole distance  113  would be so short as to cause crowding of element reference numerals and other important details, and consequently the ratio of long-side distance  104  to short-side distance  103  is skewed, as is intimated in  FIG. 1B , where the length of long-side  104  is drawn as to be larger than the preferred 3a lengths.) 
     With a good approximation for Θ in hand, we can return to a calculation of the ratio of small area  107  compared to the totality of the circular region  102 , as follows: 
     
       
         
           
             
               
                 
                   
                     A 
                     / 
                     
                       A 
                       ⊙ 
                     
                   
                   = 
                     
                    
                   
                     
                       
                         ( 
                         
                           Θ 
                           - 
                           
                             sin 
                              
                             
                                 
                             
                              
                             Θ 
                           
                         
                         ) 
                       
                       / 
                       2 
                     
                      
                     π 
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         ( 
                         
                           
                             2 
                              
                             
                                 
                             
                              
                             
                               
                                 cos 
                                 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 ( 
                                 .2 
                                 ) 
                               
                             
                           
                           - 
                           
                             sin 
                              
                             
                               ( 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     cos 
                                     
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                     ( 
                                     .2 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                       / 
                       2 
                     
                      
                     π 
                   
                 
               
             
             
               
                 
                   ≈ 
                     
                    
                   
                     0.3735 
                     . 
                   
                 
               
             
           
         
       
     
     Or in other words, the small subregion  107  comprises approximately 37.35% of the area of circular region  102 . By implication then, the larger subregion  106  comprises approximately 62.65% of the area of the circular region  102 . Rounded off, this is roughly a 40-60 split, which amounts to a ratio of approximately 2:3. (This, coincidentally, is the same value as the ratio between the short  103  and long  104  lengths between the cup  101  and the opposite sides of the circular region  102 .) 
     These ratios, however, represent only an ideal embodiment of the present invention, not an immutable feature. Experimentation and experience with the art of instructing players on how to improve their putting game reveals that these ratios can be adjusted from anywhere between 1:3 and 3:1 for the length ratios and, likewise, anywhere between 1:3 and 3:1 for the area ratios. 
     Experimentation and experience also indicate that the target region  102  need not always be a preferred circular shape.  FIGS. 2A through 2E  illustrate alternative embodiments of the present invention in which the target region  102  is triangular  201 , square  202 , rectangular  203 , oblong ovular  204 , and irregular (or “amoeboid”)  205 , respectively. For each of  FIGS. 2A through 2E , the distance from the cup  101  to the edge of the target region is denoted by “b” for the short distance and “c” for the long distance. Similarly, the small subregion within each target zone is denoted as “d,” and the large subregion within each target zone is denoted as “e.” Constructing a target region in accordance with the present invention for each shape requires only that either the ratio between lengths b and c lies within the above-stated length-ratio ranges (i.e., 1:3≦b:c≦3:1) or the ratio between sub-areas d and e lie within the above stated area-ratio ranges (i.e., also 1:3≦d:e≦3:1). 
     The shapes illustrated in  FIGS. 2A through 2E  are not exhaustive, however, of the many embodiments of the invention, but the target region may instead assume any desired shape as long as the restrictions on either the length ratio or the area ratio are roughly followed. Within such a construction the present invention is designed to reinforce good putting habits in accordance with the principles discussed herein. As will be discussed in great detail below, the smaller area or shorter length is placed on the less desirable or “low-percentage” side of a hole, and the larger area or longer length is placed on the more desirable or “high-percentage” side of the hole. The golfer is thereby encouraged to putt within two-putt range on the high-percentage side of the hole. 
     The exact placement and orientation of the target region, however, requires a fair amount of discussion, since properly reinforcing the desired good putting habits requires that the target region accurately reflect the totality of a particular putting situation. Specifically, the slope (i.e., upward or downward elevation from the putting location to the cup) and the pitch (or “slant” or “break”—i.e., transverse changes in elevation from one side of the cup to the other as viewed from the putting location) of a given putting surface will impact the orientation and placement of the target zone. 
     It should be noted that as used herein the terms “slope” and “pitch” (or “slant” or “break” in place of “pitch”) are not synonymous as might be typically understood in most non-technical situations. As used herein “slope” refers to the change in elevation of the putting surface as viewed from the perspective at the putting location. Generally speaking if the golf cup is at a higher elevation than the putting location, the surface “slopes” uphill. Conversely if the cup is at a lower elevation than the putting location, the surface “slopes” downhill. And, furthermore, if the cup has the same elevation as the hole, we consider this surface to have “no slope” and possibly even to be “flat.” (Of course, there may be intermediating changes in elevation for some surfaces that erode the meaning of these terms as applied thereto, but we will not be concerned with such complex surfaces within the present discussion.) 
     As used herein the terms “slant,” “pitch,” or “break” refer to the quality of some surfaces to change elevation in a direction that is perpendicular to the direction from the putting location to the cup. Such surfaces leave one side of the cup, either the left or the right (as viewed from the putting location), at a higher elevation than the other. As the term is used herein, a surface with “right-to-left pitch” has a higher elevation on the right side of the cup than on the left side of the cup, as viewed from the putting location, such that a ball putted from the putting location will be inclined to veer off of a straight trajectory and move. (or “break”) in a leftward direction (i.e., from right to left) in accordance with gravity. Conversely, a surface with “left-to-right pitch” has a higher elevation on the left side than on the right, and a ball putted from the putting location will break to the right instead. If the surface has right and left sides of the cup with equal elevation, we refer to such a situation as “pitchless” (as distinct from “flat,” which refers to surfaces with no slope). Of course, some putting scenarios can involve both slope and pitch in the same putt, and the present discussion will address basic combinations of those scenarios as needed. 
       FIGS. 3A through 3E  illustrate different aspects of an important conceptual tool that must be firmly understood not only to use the present invention but also to master the art of accurate putting—i.e., the concept of a “line of ideal approach.” This term will be used frequently throughout the present discussion and, most importantly, within the appended claims that follow, in order to accurately and precisely define the present invention. As discussed below, this term is a very useful concept for analyzing putting situations and accurately reflects the intuitive feel for a particular putting surface that an experienced golfer eventually obtains. As further discussed below, however, it is not absolute in its nature, and its intuitive application by different golfers to a given putting situation may result in disagreements as to the proper way to approach a given putt—even if both of those golfers are experienced players or even highly decorated professionals. The concept, however, does provide enough certainty so as to properly disclose the present invention and particularly point out and claim its features. 
     Generally speaking, a line of ideal approach is the line formed by the trajectory of a hypothetical perfectly putted ball right at the moment it falls into the cup. It is specifically defined in a proper geometric sense by referencing the point lying at the center of the cup (i.e., a mathematical point) and the direction of the hypothetical perfectly putted ball right as it drops into the cup (i.e., with said mathematical point coupled to a mathematical direction, thus defining a line). 
       FIG. 3A  illustrates that for putting surfaces that have no pitch, the line of ideal approach  301  is simply a straight line between the putting location  303  and the cup  101  and, under perfect conditions, coincides with the ball&#39;s trajectory  302 . This is true irrespective of the putting surface&#39;s slope. That is to say, the line of, ideal approach is the same straight line whether the surface is flat, slopes uphill, or slopes downhill, as long as there is no left-to-right or right-to-left change in elevation near the cup  101  or anywhere along the trajectory  302 . 
     The line of ideal approach, as the term is used herein (and particularly in the following claims), must not be confused with the ball&#39;s actual trajectory however, even though for perfectly pitchless surfaces the two coincide. To wit, when pitch is introduced into the putting surface, the ball&#39;s trajectory is no longer a straight line to the cup from the putting location. The force of gravity on a pitched surface will force the ball to veer off of the direction in which it was putted and eventually roll in a “downwardly” direction along the pitched surface. (For uphill or downhill putts made from a surface without pitch, on the other hand, gravity may accelerate or decelerate the ball&#39;s movement, respectively, in its putted direction, but it will not force the ball to veer off a straight course.) The direction that the ball is traveling just as it enters the cup, however, is the defining feature for the “line of ideal approach” as used throughout the present discussion. The ball&#39;s direction is not constant throughout its travels along a pitched slope, and the line formed by its final direction is not the same as either a straight line from the putting location or the curved line of the balls&#39; trajectory. 
       FIGS. 3B and 3C  illustrate how the introduction of pitch into a putting surface affects the line of ideal approach. In  FIG. 3B , which offers a putting surface that slants from left to right (i.e., has left-to-right pitch), the golfer must compensate for gravity&#39;s impact on the ball by putting the ball in a direction that is leftward of the straight line  304  to the cup  101  by some relatively small offset angle  305 , denoted as φ. As the ball travels, gravity corrects its trajectory  302  such that just prior dropping into the cup  101 , the ball is traveling “down” the pitched surface in a generally rightward direction (on top of whatever remains of its forward motion, of course). The line of ideal approach  301  for this particular putting situation, then, is ascertained by fixing a point at the center of the cup  101  and constructing the line  301  through said point whose direction is the same as the ball&#39;s direction at the very last moment before the ball drops in.  FIG. 3C  reflects the precisely opposite but symmetrical situation when the putting surface is pitched from right to left instead. In either case, the line of ideal approach  301  is illustrated with a thick, dark line that coincides with the ball&#39;s trajectory  302  at the cup  101  and lies in the same direction thereat. The “line of ideal approach” should therefore not be confused with the actual trajectory followed by the ball. It is only on perfectly pitchless surfaces (whether flat, uphill or downhill) that the two coincide. 
     As mentioned previously, however, pitched surfaces may also slope upward or downward to the cup as viewed from the putting location. This means golfers may experience the combined effect of pitch and slope in any given putting scenario. The present invention, therefore, is designed to accommodate these scenarios by adjusting to them in the manner discussed below. Such matters will be discussed in reference to the precise placement and orientation of the target zone for each of the commonly encountered surface variations in connection with  FIGS. 4A through 4F  and in their totality in connection with  FIG. 4G . 
       FIG. 3D  illustrates how different players may have slight disagreements over the precise geometric construction of a line of ideal approach for a given putting situation. Experience dictates that, largely for reasons of how hard the ball is initially struck, a ball may follow any one of several trajectories into the cup.  FIG. 3D  illustrates that these trajectories  302  lie within a reasonably constrained region  306  (shown somewhat exaggerated in  FIG. 3D ). Different trajectories within the region  306  will result in slightly different lines of ideal approach  301  being constructed for the given hole. (Furthermore, while only the situation involving right-to-left pitch is illustrated in  FIG. 3D , there is no such limitation in reality; a mirror image reflecting the same situation for left-to-right pitch could also be shown, but is omitted for convenience.) In addition to these minor ambiguities, the hole has a specific diameter (instead of being a mathematical point), and in reality putting surfaces out on actual golf courses present all sorts of irregularities (e.g., gradual changes in pitch or slope, depressions, mounded areas, obstructions, changes in turf conditions, etc.). There is also wind, which, for reasons of simplicity, is entirely ignored within the present discussion. Each of these elements can introduce further small perturbations within the range of trajectories that will result in a sunken or well-putted putt. Hence, highly qualified and greatly experienced golfers may come to reasonable disagreements as to which precise line within these perturbations will qualify as the official or precise line of ideal approach that a given putting situation presents. These differences, however, appear only within a relatively small deviation of one another and do not render the concept of “a line of ideal approach” as overly vague or meaningless. Instead, the term refers to an asymptotic, hypothetical ideal over which there may be some reasonable bounded disagreement as to its precise construction under specific circumstances. 
     Furthermore, as  FIG. 3E  illustrates, the line of ideal approach will vary depending upon the putting location. While it is irrelevant to ordinary golf play out on a real course that different putting locations will result in different lines of ideal approach (since a player is presented with only one putting location per putt), these differences must be taken into account in certain golf-putting practice games, such as several embodiments of the present invention discussed in connection with  FIGS. 6A through 6D  and  FIG. 7 , below, where a plurality of putting locations are marked out on a practice putting green in relation to a single golf cup marked with a single target zone. In the case of a flat putting surface, like the one presented in  FIG. 3E , the lines of ideal approach for each of the marked putting locations,  303   a  through  303   e  will form different orientation angles,  308   a  through  308   e,  respectively (denoted with lower-case Greek letters alpha “α” through epsilon “ε,” respectively), with respect to an arbitrary line  307  drawn through the cup  101 . (For present purposes, line  307  is truly arbitrary and does not necessarily coincide with perpendicular dividing line  105  separating the low-percentage side of the cup from the high-percentage side, although it may be the same if circumstances warrant.) 
     As will be discussed in great detail below, the placement and orientation of the target zone used in connection with the present invention is dependent not just upon the slope and pitch of the surface, but is also dependent upon how the putting location is situated with respect to the cup. The proper placement and orientation of the target zone therefore requires choosing a single, specific line of ideal approach. To arrive at a line of ideal approach that can be used in connection with a plurality of putting locations, some type of averaging system must be used. 
     In the case of those putting locations illustrated in  FIG. 3E , the orientation angles α, β, γ, δ, and ε can be either simply averaged or combined into a weighted average, where the weight is equivalent to the distance d α , d β , d γ , d δ , and d ε  between the corresponding putting location  303   a  through  303   e  and the cup  101 , respectively. As such, any of the following equations can be used to determine an appropriate orientation angle  309  (denoted by lower-case Greek letter phi “ 104  ”) for the aggregate line of ideal approach  301  with respect to the same arbitrary line  307  from which orientation angles α, β, γ, δ, and ε are measured (although each equation provides a slightly different answer): 
     Simple Average: 
       ψ=(α+β+γ+δ+ε)/5
 
     Weighted Average (directly proportional to putt distance): 
       ψ=( d   α   α+d   β   β+d   γ   γ+d   δ   δ+d   ε ε)/( d   α   +d   β   +d   γ   +d   δ   +d   ε )
 
     Weighted Average (indirectly proportional to putt distance): 
       ψ=(α/ d   α   +β/d   β   +γ/d   γ   +δ/d   δ   +εd   ε )/(1/ d   α +1/ d   β +1/ d   γ +1/ d   δ +1/ d   ε )
 
     Of course, these formulas can also be extended indefinitely to account for a very large set of putting locations. For a set of N putting locations (not shown), instead of just five, the formulas generalize as follows, if the distances are denoted d i  through d N  (not shown) and orientation angles are denoted α 1  through α N  (not shown): 
     
       
         
           
             
               
                 Generalized 
                  
                 
                     
                 
                  
                 Simple 
                  
                 
                     
                 
                  
                 Average 
               
               _ 
             
              
             
               : 
             
           
         
       
       
         
           
             ψ 
             = 
             
               
                 ∑ 
                 
                   i 
                   = 
                   1 
                 
                 N 
               
                
               
                 
                   α 
                   i 
                 
                 / 
                 N 
               
             
           
         
       
       
         
           
             
               
                 Generalized 
                  
                 
                     
                 
                  
                 Weighted 
                  
                 
                     
                 
                  
                 Average 
                  
                 
                     
                 
                  
                 
                   ( 
                   
                     directly 
                      
                     
                         
                     
                      
                     proportional 
                      
                     
                         
                     
                      
                     to 
                      
                     
                         
                     
                      
                     putt 
                      
                     
                         
                     
                      
                     distance 
                   
                   ) 
                 
               
               _ 
             
              
             
               : 
             
           
         
       
       
         
           
             ψ 
             = 
             
               
                 ∑ 
                 
                   i 
                   = 
                   1 
                 
                 N 
               
                
               
                 
                   d 
                   i 
                 
                  
                 
                   
                     α 
                     i 
                   
                   / 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       N 
                     
                      
                     
                       d 
                       i 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   Generalized 
                    
                   
                       
                   
                    
                   Weighted 
                    
                   
                       
                   
                    
                   Average 
                    
                   
                       
                   
                    
                   
                     ( 
                     
                       indirectly 
                        
                       
                           
                       
                        
                       proportional 
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     Other averaging or aggregating functions or methods may be used to estimate an approximate line of ideal approach for a plurality of putting locations surrounding a given hole (e.g., using another common distribution function, such as a Gaussian, to further adjust the distance weights), and should be readily apparent to one of ordinary skill in the appropriate mathematical arts in light of the teachings of the present discussion. Such methods should therefore be seen as reasonably equivalent to those provided here and are thereby incorporated herein. 
     Except where specifically noted to the contrary, it will be assumed throughout the present discussion that the putting locations in reference to which a line of ideal approach is being determined for a given golf cup and putting surface are all grouped together in a sufficiently close proximity to one another and all lie to a single side of the golf cup so that a line of ideal approach can be reasonably constructed for them in the aggregate according to the aforementioned methods. ( FIG. 7 , as one example where this generalization is not strictly adhered to, illustrates an embodiment in which putting locations are placed on several sides of the target zone, both above and below the cup on an inclined surface, but the specific topology involved allows for the construction of a single, aggregate line of ideal approach because of a particular geometric symmetry explained at length in connection with  FIG. 4G  between pitchless uphill and pitchless downhill putting scenarios.) In actual practice, however, the ascertainment of line of ideal approach is not an exacting geometric task, but rather an exercise of experienced judgment. Aggregating a line of ideal approach for a cluster of putting locations is likewise that much more imprecise and approximate, but remains conceptually viable and can be carried out in exacting detail if required and if appropriate means for accurate measurement are used. 
     Now that the peculiar qualities of a target region have been specified and that the concept of a line of ideal approach has been articulated in accordance with the present invention, it is possible to discuss the placement and orientation of a target zone around a given golf hole in light of the surrounding putting conditions. Specifically, the target region will be placed around the golf cup slightly differently depending upon whether the surrounding putting surface is flat, slopes upward, slopes downward, pitches from right to left, pitches from left to right, or has some combination of slope and pitch. In each instance, however, the target zone is situated so as to reinforce good putting habits and to provide the player with proper visual reinforcement of a target zone that encourages good putting habits. The following discussion will address the placement and orientation of the target zone with respect to each of the above conditions. 
     The basic concept to keep in mind is that the perpendicular dividing line  105  between the low-percentage side  107  and the high-percentage side  106  of the hole  101  will be placed so that it is either approximately coincident with or approximately perpendicular to the line of ideal approach  301  depending upon the surface topology (subject to caveats discussed below). Then a determination must be made as to which sides of the hole  101  to place the short side (or small subregion)  107  and long side (or large subregion)  106  of the target region  102 . As will be illustrated in connection with  FIGS. 4A through 4F , the answer depends upon the slope and pitch of the surrounding putting surface. Generally speaking, however, the high-percentage area (or large area)  106  is placed on the side of the hole  101  with the highest elevation, where flat putting surfaces are treated equivalently to uphill putting surfaces for this purpose. 
       FIG. 4A  illustrates the proper placement and orientation of a target zone  102  on a putting surface  401   a  that is either flat (i.e., has no slope) or slopes uphill and that has relatively no pitch. For this topology the dividing line  105  between the low-percentage side  107  and the high-percentage side  106  of the hole  101  is placed so as to be approximately perpendicular to the line of ideal approach  301  (defined in relationship to trajectory  302 ), and the long distance  104  or, alternatively, the large subregion  106 , will be placed on the side of the cup  101  away from the putting location  303  (or plurality of putting locations, if a plurality is contemplated). If the surface slopes uphill, the side of the cup  101  that lies away from the putting location  303  will, in fact, be the high side of the cup  101 . (For ease of understanding, we will therefore treat flat surfaces as equivalent to uphill surfaces in this limited respect.) For ease of comprehension, it should be noted that under perfect conditions this placement of the cup coincides with the “6 o&#39;clock” position if the target region is seen as a watch or clock face as viewed from above the putting location. (A key is provided in connection with  FIG. 4  for the easy determination of clock positions.) 
       FIG. 4B  illustrates the proper placement and orientation of a target zone  102  on a putting surface  401   b  that is either flat (i.e., has no uphill or downhill slope) or slopes uphill and, in either case, that pitches from left to right. For this topology the dividing line  105  between the low-percentage side  107  and the high percentage-side  106  of the hole  101  is placed approximately so as to coincide with the line of ideal approach  301  (defined in relationship to trajectory  302 ), and the long distance  104 , or alternatively the large subregion  106 , will be placed on the high side of the cup  101  (or, in the case of flat putting surfaces, on the side of the cup that would be the high side if, in fact, the surface were to slope upward from the putting location). 
     As  FIG. 4B  illustrates, if the target zone  102  is seen as a watch or clock face, the hole  101  is placed at roughly somewhere between the 4 o&#39;clock and 5 o&#39;clock positions, as viewed from above the putting location. As is the case with all putting surfaces that have either left-to-right or right-to-left pitch, constructing the precise line of ideal approach  301 , and therefore the proper placement of the line dividing line  105  between the low-percentage side  107  and the high-percentage side  106  of the cup  101 , is a matter of experienced judgment, and not typically an exercise of geometric precision. Consequently, achieving a perfect coincidence of the dividing line  105  with the line of ideal approach  301  is not always possible. A preferred embodiment of the present invention, therefore, simply places the cup in the 4 o&#39;clock position for all flat or upward sloping surfaces that have a left-to-right pitch. While this generalization may not provide an exacting geometrical construction of the target zone  102 , the approximation achieved thereby is sufficient to reinforce good putting habits nonetheless. 
       FIG. 4C  illustrates the proper placement and orientation of a target zone  102  on a putting surface  401   c  that is flat or uphill and that also pitches from right to left. For this topology the dividing line  105  between the low-percentage side  107  and the high-percentage side  106  of the hole  101  is placed approximately so as to coincide with the line of ideal approach  301  (defined in relationship to trajectory  302 ), and the large subregion  106 , will be placed on the high side of the cup  101  (or on what would be the high side of the cup if a flat surface were in fact treated as an uphill surface when viewed from the putting location). 
     As  FIG. 4C  illustrates, if the target zone  102  is viewed as a watch or clock face, the hole  101  is placed at roughly somewhere between the 7 o&#39;clock or 8 o&#39;clock positions. For simplicity a preferred embodiment of the present invention places the cup in the 8 o&#39;clock position for all flat or upward sloping surfaces that have a right-to-left pitch. 
     When the target region is placed in this fashion for flat and uphill putts, the player is encouraged to putt the ball at least to the hole (or within a short distance thereof, which equals twelve inches in a preferred embodiment) but not to let the ball go too far beyond the hole (or eighteen inches in a preferred embodiment) so as to have an “ideal miss” if, in fact, the putt misses the hole at all. Furthermore, such placement also conforms to the well-known golfing maxim that on breaking putts one should “putt above the break,” inasmuch as in  FIGS. 4B and 4C , the high-percentage side  106  of the target zone  102  is placed in such a manner as to encourage putting to the high side of the cup  101 . Placement of the target zone  102  in this fashion also provides visual reinforcement that the “sweet spot” of the particular putt lies on the opposite side of the hole  101  from the putting location  303 , since that is where the majority of the target zone  102  (and therefore the majority of its visual impact) is situated. Proper selection of trajectory and swing effort is thereby reinforced with this arrangement for flat and uphill putts both with and without pitch. 
     It should be noted that for putts on flat or uphill surfaces, the high-percentage side of the hole always occurs on the side farthest away from the putting location (which is the “high side” of the hole). This is because, as alluded to throughout the present discussion, all things being equal it is best to make it at least all the way to the hole if not a little bit beyond. For downhill putts, however, this logic does not hold. (In fact, it can lead to very bad putts indeed.) Because there is a tendency to overputt on downhill surfaces, and because once a ball rolls past the cup on a downhill slope it is likely to continue on for quite some distance, good putting habits focus on getting the ball to “die into the cup” on downhill putts. That is, for downhill putting it is best to have the ball run out of momentum just as it drops into the cup. This prevents what might appear to be near misses from overshooting the cup and going a very long distance beyond, thereby creating a very bad putt. Consequently, the high-percentage side of the cup on a downhill putt is, generally speaking, the near side of the cup, which conveniently is still of higher elevation. It therefore still makes sense to say that the present invention teaches placing the high-percentage side of the target zone on the side of the cup with the higher elevation. This has repercussions for how the target region is situated on downhill surfaces, for which the following discussion addresses the typical scenarios. 
       FIG. 4D  illustrates the proper placement and orientation of a target zone  102  on a putting surface  401   d  that slopes downhill without any pitch. For this topology the dividing line  105  between the low-percentage side  107  and the high percentage-side  106  of the hole  101  is placed so as to be approximately perpendicular to the line of ideal approach  301  (defined in relationship to trajectory  302 ), and the long distance  104  or, alternatively, the large subregion  106 , is placed on the high side of the cup  101  toward the putting location  303  or plurality of putting locations. This placement of the cup coincides with the “12 o&#39;clock” position if the target region is viewed as a watch face as seen from above the putting location. 
       FIG. 4E  illustrates the proper placement and orientation of a target zone  102  on a putting surface  401   e  that slopes downhill and that pitches from left to right. For this topology the dividing line  105  between the low-percentage side  107  and the high percentage-side  106  of the hole  101  is placed so as to be approximately perpendicular to the line of ideal approach  301  (defined in relationship to trajectory  302 ), and the large distance  104  or, alternatively, the large subregion  106 , will be placed on the high side of the cup  101 , toward the putting location  303 . 
     As  FIG. 4E  illustrates, if the target zone  102  is viewed as a watch or clock face, the hole  101  is placed at roughly somewhere between the 1 o&#39;clock and 2 o&#39;clock positions. For simplicity a preferred embodiment of the present invention places the cup in the 2 o&#39;clock position for all downward sloping surfaces that have a left-to-right pitch. 
       FIG. 4F  illustrates the proper placement and orientation of a target zone  102  on a putting surface  401   f  that slopes downhill and that pitches from right to left. For this topology the dividing line  105  between the low-percentage side  107  and the high percentage-side  106  of the hole  101  is placed so as to be approximately perpendicular to the line of ideal approach  301  (defined in relationship to trajectory  302 ), and the large distance  104  or, alternatively, the large subregion  106 , will be placed on the high side of the cup  101 , near the putting location  303 . 
     As  FIG. 4F  illustrates, if the target zone  102  is viewed as a watch or clock face, the hole  101  is placed at roughly somewhere between the 10 o&#39;clock and 11 o&#39;clock positions. For simplicity a preferred embodiment of the present invention places the cup in the 10 o&#39;clock position for all downward sloping surfaces that have a right-to-left pitch. 
     When the target region  102  is placed in this fashion for downhill sloping surfaces, the player is encouraged not to putt the ball beyond the hole  101 . Placement of the target zone  102  in this fashion also provides visual reinforcement that the “sweet spot” of the particular putt lies on the close side of the hole  101  from the putting location  303 , since that is where the majority of the target zone  102  (and therefore the majority of its visual impact) is situated. Proper selection of trajectory and swing effort is thereby reinforced with this arrangement for downhill putts with or without pitch. 
     Table 1 summarizes the “clock-position” placement of the cup  101  within the target zone  102  and the “axis orientation” (either perpendicular or coincidental) between the line of ideal approach  301  and the dividing line  105  separating the high-percentage side  106  of the cup  101  from the low-percentage side  107 , for the surface topologies considered herein for a preferred embodiment of the invention. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Summary of Hole Placements and Axis Orientations for 
               
               
                 Different Putting Surface Topologies 
               
            
           
           
               
               
               
               
               
            
               
                   
                   
                   
                 HOLE 
                 AXIS 
               
               
                 FIG. 
                 SLOPE 
                 PITCH 
                 PLACEMENT 
                 ORIENTATION 
               
               
                   
               
               
                 4A 
                 flat/uphill 
                 none 
                 6 o&#39;clock 
                 perpendicular 
               
               
                 4B 
                 flat/uphill 
                 left-to-right 
                 4 o&#39;clock 
                 coincident 
               
               
                 4C 
                 flat/uphill 
                 right-to-left 
                 8 o&#39;clock 
                 coincident 
               
               
                 4D 
                 downhill 
                 none 
                 12 o&#39;clock  
                 perpendicular 
               
               
                 4E 
                 downhill 
                 left-to-right 
                 2 o&#39;clock 
                 perpendicular 
               
               
                 4F 
                 downhill 
                 right-to-left 
                 10 o&#39;clock  
                 Perpendicular 
               
               
                   
               
            
           
         
       
     
     While these “clock-position” placements of the cup within the target region are good general-purpose approximations for those surfaces with right-to-left or left-to-right pitch, it should be noted that these are not the only placements contemplated by all embodiments of the present invention. Under ideal circumstances it would be possible to construct a precise and unique line of ideal approach  301  for all putting surfaces, whether they are pitched or not, and from there to orient the dividing line  105  as either perpendicular or coincident thereto. While such precision is not always possible, some putting surfaces lend themselves to easy adjustment of the aforementioned preferred-embodiment cup locations. Experience playing on any of such surfaces may indicate that an adjustment is needed, and anyone versed in the principles of the present invention would be able to make reasonable adjustments as their need becomes evident. 
       FIG. 4G  provides the conceptual tools necessary to understand how the different surface topologies (i.e., combinations of slope and pitch) depicted in  FIGS. 4A through 4F  and summarized in Table 1 are geometrically related to one another, so as to simplify the rules and procedures for placing the target zone  102  around a golf cup  101  on a given surface  401 . Although the rules for placement and orientation of the target zone  102  have been taught herein via rote procedures for the surfaces  401   a  through  401   f  respectively, they are all connected through simple geometric symmetry. In short, one can consider each of the surfaces  401   a  through  401   f  from  FIGS. 4A through 4F  as subsets of a hypothetical, perfectly planar surface,  401   g,  as illustrated in  FIG. 4G . It can then be shown that a single, unique placement of the target zone  102  around the cup  101  can be made that is acceptable—according to the underlying principles of the presently disclosed invention—for all of the putting locations  303   a  through  303   f  that are associated with the different surfaces  401   a  through  401   f.    
     Specifically, if a hypothetical, perfectly planar surface  401   g  is tilted with respect to the force of gravity so as to create an inclined plane, then putting locations  303   a  through  303   f  can be located on the plane  401   g,  such that as viewed from the putting location in question,  303   a  through  303   f,  all of the surface topologies discussed thus far are, in fact, properly addressed. (Of note, in  FIG. 4G  the magnitude of this incline with respect to gravity is illustrated in a rather aggressive manner for the sake of illustration and clarity of drawing only; it is unlikely that such an aggressive incline would ever be met on a real golf course—or at least not very often.) To create a putting scenario with uphill slope but without right-to-left or left-to-right pitch, the putting location  303   a  must be situated directly below the cup  101  on the plane  401   g.  Conversely, for a downhill putting scenario without pitch, the putting location  303   d  must be situated directly above the cup  101  on the plane  401   g.    
     Introducing pitch into these scenarios simply requires offsetting another putting location either to the left or to the right (depending upon the type of pitch desired) of these two putting locations  303   a  and  303   d.  An uphill putting scenario with left-to-right pitch can therefore be created by situating the putting location  303   b  below the cup  101  on the plane  401   g  but also offset to the left (as viewed while facing the cup  101 ). A downhill putting scenario with left-to-right pitch can be created by situating the putting location  303   e  above the cup  101  on the plane  401   g  but offset to the right (as viewed while facing the cup  101 ). Similarly, an uphill putting scenario with right-to-left pitch can be created by situating the putting location  303   c  below the cup  101  on the plane  401   g  but also offset to the right (as viewed while facing the cup  101 ). And, finally, a downhill putting scenario with right-to-left pitch can be created by situating the putting location  303   f  above the cup  101  on the plane  401   g  but also offset to the left (as viewed while facing the cup  101 ). Trajectories  302   a  through  302   f  of a golf ball as it approaches the cup  101  are also drawn within  FIG. 4G  and correspond to those discussed in connection with  FIGS. 4A through 4F . 
     One could, conceivably, take each of the surfaces  401   a  through  401   f  from 
       FIGS. 4A through 4F  and line up the cup  101  and target zone  102  on each surface to produce a figure that looks very much like  FIG. 4G . The harmony of this situation is so striking in fact that similar ovular outlines  401   a  through  401   f  to each of these putting scenarios are included within  FIG. 4G  so that the similarity to the putting surfaces  401   a  through  401   f  of  FIGS. 4A through 4F  can be clearly shown. Each oval shape is, in fact, nothing but a surface boundary drawn around a particular putting topology (and, hence, the use of identical reference numerals to identify corresponding elements is proper). To the extent that the ovular outlines to the putting surfaces  401   a  through  401   f  overlap, boundaries within the overlap region are not shown, so as to maintain clarity of  FIG. 4G . 
     Furthermore,  FIG. 4G  illustrates the manner in which several apparently divergent principles of the present invention now harmonize into a single, coherent conceptual system. Notably, it is now seen that the high-percentage region  106  of the target zone  102  is placed on the side of the cup  101  with higher elevation, and that this principle holds true irrespective of the putting topology involved. Likewise low-percentage region  107  of the target zone  102  is placed on the side of the cup  101  with lower elevation, irrespective of putting topology. It is further illustrated that trajectories  302   d,    302   e,  and  302   f  for the three downhill putting locations  303   d,    303   e,  and  303   f , respectively, converge approximately into a single trajectory as they near the cup  101 . This convergence would coincide with an aggregated line of ideal approach for these three putting locations, if one were included in  FIG. 4G , which it is not. (This convergence, moreover, lends further credence to the concept of an aggregated line of ideal approach, as discussed in connection with  FIG. 3E .) 
     Additionally, the pitchless uphill putting location  303   a  also has a trajectory  302   a  that coincides with the trajectories from the downhill putting locations  303   d,    303   e,  and  303   f,  but the ball would be traveling from the opposite direction. Target zone  102  is therefore oriented such that dividing line  105  is situated perpendicular to these convergent trajectories  303   a,    303   d,    303   e,  and  303   f,  and the corresponding aggregated line of ideal approach (not shown). Furthermore, for the pitched uphill putting locations  303   b  and  303   c,  it can be seen how their trajectories  302   b  and  303   c  enter the target zone  102  from the sides and thereby necessitate the placement of the dividing line  105  coincident with their lines of ideal approach (not shown), instead of perpendicular to them, as is the case for the remaining putting locations  303   a,    303   d,    303   e , and  303   f.  In each of these cases, a single orientation and placement of the target zone  102  will suffice. 
     Describing placement of the cup  101  within the target zone  102  by use of a watch or clock face analogy can also be illustrated clearly in reference to  FIG. 4G . Positional keys  402   a  through  402   f  are shown in  FIG. 4G  as being associated with each of the putting locations  303   a  through  303   f.  The keys consist both of a circle representing a watch or clock face and of the numerals “3,” “6,” “9,” and “12” representing the four cardinal time positions on the watch or clock face. For each key  402   a  through  402   f  the twelve-o&#39;clock position (denoted as “12” inside the key) is illustrated as oriented on the page in the direction in which the hole  101  lies as viewed from the putting location  303   a  through  303   f.  The circular region inside the key is then analogized both to a watch or clock face and to the target zone  101  simultaneously. A black dot is then placed inside of each key  402   a  through  402   f  at the approximate location where a hypothetical golfer standing at the corresponding putting location  303   a  through  303   f  would view the golf cup  101  within the target zone  102 . In this manner it can be seen that the dot or “cup”  101  is located at approximately the six-o&#39;clock position for putting location  303   a,  at approximately the four-o&#39;clock position for putting location  303   b,  at approximately the eight-o&#39;clock position for putting location  303   c,  at approximately the twelve-o&#39;clock position for putting location  303   d,  at approximately the two-o&#39;clock position for putting location  303   e,  and at approximately the ten-o&#39;clock position for putting location  303   f.  A convenient, accurate, and perfectly consistent shorthand system can therefore be used to describe the orientation of the target zone around a specific golf cup when viewed from a particular putting location. This shorthand system will be used in the following appended claims in connection with the orientation of the target zone  102  in light of different putting topologies. 
     While the uses of the presently described apparatus are many and varied, experience has taught that particular methods of use within a golf-putting game with special rules results in optimal teaching performance. Therefore two methods of use in particular are worth noting in connection with the presently disclosed apparatus. 
       FIG. 5A  illustrates the procedure for playing one type of game, using a feature referred to as a “super putt” with a target zone specified in accordance with the present invention. The game begins with a player putting one or more balls from one or more putting locations marked out on a putting surface, step  501 . The putting locations are typically situated toward one side of a golf cup that is surrounded by a target zone in accordance with the present invention. In a preferred embodiment, the locations are different distances to the cup and do not necessarily lie in a straight line, but are clustered in sufficiently close alignment that a line of ideal approach can still be reasonably approximated by the averaging methods described herein in connection with  FIG. 3E . In a preferred embodiment, the number of balls putt is nine (9). When putting is completed (or simultaneously while putting is taking place, since the steps easily overlap), each ball is scored according to where the putt lands. Only one putt is taken on each ball. It should also be noted that in some particular embodiments the putting surface is bounded, and there is a clearly defined region which is considered out of bounds. 
     In accordance with step  502 , if a ball lands in the cup, it is given a score of A points, which, in a preferred embodiment that simulates actual golf play, equals minus one point (corresponding to a birdie putt). If a ball lands in inside the target region but not in the cup, it is scored B points, which, in a preferred embodiment that simulates actual golf play, equals zero points (corresponding to a par putt). If the ball lands outside the target zone but remains on the putting surface, it is scored C points, which, in a preferred embodiment that simulates actual golf play, equals plus one point (corresponding to a bogie putt). And, lastly, if the ball lands out of bounds in an embodiment that has a clearly bounded putting surface, it is scored D points, which, in a preferred embodiment that simulates actual golf play, equals plus two points (corresponding to a double-bogie putt). Of note, although the specific point scores offered for a preferred embodiment are listed here as being in ascending order of value so as to simulate regulation golf play, there is no requirement that the point values be so ordered. In an embodiment with alternative scoring arrangements, the point values decrease in value as the corresponding putts fall farther and farther away from the hole. All that matters for another preferred embodiment of the present invention is that, whether a high score is considered “better” than a low score or a low score is considered “better” than a high score, A is “better” than B, which in turn is “better” than C, which in turn is “better” than D. Conceivably, however, even this rule could be broken, and alternating “good” and “bad” point values could hypothetically be assigned to the different regions out of any strict ascending or descending order, although one is hard pressed to see the instructional value of such a putting game. 
     In accordance with step  503 , one or more putts (although in a preferred embodiment it is only one putt) are taken from each of one or more locations significantly farther away from the cup than the locations used for putting in step  501 . If more than one further-away putting location is used, in a preferred embodiment they too are different distances to the cup and do not necessarily lie in a straight line, but are clustered in sufficiently close alignment that a line of ideal approach can still be reasonably approximated by the averaging methods described herein in connection with  FIG. 3E . 
     Then, in accordance with step  504 , the one or more additional further-away putts are scored according to a point system that is similar to the one used in step  502 , but reflecting the fact that the putts were made from longer distances. As such, putts into the cup are scored at E points (or minus two points in a preferred embodiment that simulates real golf play, corresponding to an eagle putt), putts into the target region are scored at F points (or minus one point in a preferred embodiment that simulates real golf play, corresponding to a birdie putt), putts that remain on the surface are scored at G points (or zero points in a preferred embodiment that simulates real golf play, corresponding to a par putt), and putts that leave the surface are scored at H points (or plus one point in a preferred embodiment that simulates real golf play, corresponding to a bogie putt). As with the arrangement of point values in step  502 , point values E, F, G, and H need not necessarily be in ascending or descending numerical order, but only in an order that reflects the relative quality of the putt according to the scoring system used. That is, E should be “better” than F, which in turn should be “better” than G, which in turn should be “better” than H. But what is more, these values should also be selected such that they reflect the relative difficulty of the further-away putts made in step  502  than the putts made in Step  502  from a closer distance. As such, E should also be “better” than A, F should also be “better” than B, G should also be “better” than C, and H should also be “better” than D. 
     In Step  505 , a final score is tabulated that includes the sum of all point values for all putts taken. In a preferred embodiment, lower scores are better than higher scores, as they are in actual golf play. Furthermore, if the preferred point values are followed as outlined above (i.e., if A=−1, B=0, C=1, D=2, E=−2, F=−1, G=0, and H=1), then the final score of a game played according to this embodiment of the present invention will correlate nicely to the putting performance the same player can expect during an actual round of golf. This is so because the present method of use is designed with the, assumption that the player&#39;s driving game was sufficient to land the ball from the tee into the assigned putting positions within the standard par values for a hole. As such, requiring two putts to sink the ball from any one starting location will preserve par, whereas a single putt to sink the ball will result in one stroke under par, and more than two putts will leave the player an equal number of strokes over par. From this assumption it can be seen that assigning values of A=−1, B=0, C=1, and D=2 will achieve this goal. It is assumed, of course, that if a player makes a putt into the target zone, the next putt will be virtually guaranteed to be successful. (This is because the size of the target zone was selected with the two-putt distance in mind, as discussed in connection with  FIG. 1A .) It is further assumed, quite conservatively, that any putt that lands outside the target zone but remains in bounds will require two additional putts to complete. These assumptions are made so as to provide a realistic but conservative estimate of the player&#39;s putting score and so as to form an approximate upper bound on such score. Performance during actual play might be better than the performance estimated by this method, inasmuch as the player may be able to sink a putt that lands outside of the equivalent target zone but remains on the putting surface in one putt instead of two (e.g., if the putt lands almost, but not quite, inside the target zone), thereby achieving a lower score than expected. Of course, on occasion a player will miss a putt that lands inside of the equivalent target zone, and will require two additional strokes instead of just one. Nevertheless, the score achieved in accordance with this preferred method will provide a realistic approximation to regulation play. 
       FIG. 5B  illustrates a variation on the aforementioned method using a feature that is referred to as a “bonus putt.” It begins with steps  501  and  502 , which are identical to those discussed in connection with  FIG. 5A . One or more balls are putt and scored from a set of specified putting locations in accordance with the aforementioned discussion. However, instead of immediately proceeding to putting from the more distant putting locations of step  503 , an interim score is tabulated in step  506  and compared to some predefined threshold score in step  507 , which in a preferred embodiment is zero points or lower (corresponding to a score of par or better), and in another preferred embodiment it is negative one points or lower (corresponding to one putt under par, or better, for the holes putted thus far). If the threshold score is not reached, the interim score is made final in steps  508  and  505 , and no other putts are taken. The resulting score can then be compared to actual golf play if a preferred embodiment for scoring is used (i.e., if A=−1, B=0, C=1, D=2, E=−2, F=−1, G=0, and H=1). If, on the other hand, the interim score does reach the predetermined threshold, then the player continues the game in step  503  by taking the additional long putts as described in step  503  in connection with  FIG. 5A , above, which are then scored in step  504  according to the same arrangement as described in connection with  FIG. 5A , above, as well. Hence, it is only if the player&#39;s interim score meets a minimum requirement that the player is rewarded with the chance to lower his or her score with taking long putts that are scored more preferentially. 
     Furthermore, in constructing games out of the component parts of the present invention, it is often desirable to practice putting from multiple putting surfaces having different slopes and pitches within the same game. The following embodiments permit precisely such combinations. 
       FIG. 6A  illustrates one preferred embodiment in which a plurality of putting locations  303   a  through  303   h  is situated on a downhill putting surface  401  with no pitch. Golf cup  101  and the surrounding circular target zone  102  are located downhill on putting surface  401  from putting locations  303   a  through  303   h.  As such the larger subregion  106  of circular target zone  102  is placed on the close side of the circular target zone  102  to the plurality of putting locations  303 , which is, in fact, the high side of the cup  101 . Smaller subregion  107  is placed on the far side of the target zone  102 , thereby placing cup  101  at the 12 o&#39;clock position inside target zone  102 , in accordance with the foregoing discussion and  FIG. 4D . A hypothetical and aggregated line of ideal approach  301  (not shown) can be estimated for the plurality of putting locations  303 . The dividing line  105  between the high-percentage side  106  and low-percentage side  107  of the cup situated perpendicularly to this line of ideal approach  301 . Notable too is the fact a separate putting location  601  is illustrated such that the distance from  601  to the cup  101  is considerably longer (25 feet in a preferred embodiment) than any of the putting locations within the plurality  303  (which are located at 3, 5, 7, 10, 12, 14, 16, and 18 feet, respectively, from the cup  101  in a preferred embodiment). Putting location  601  thereby acts as the longer putting location from which to take superputts or bonus putts according to an embodiment of the present invention as described in connection with  FIGS. 5A and 5B , respectively. 
     Hypothetical line  602  demarcates the boundary of a playable surface for putting surface  401 , such that once any ball putted from the region of the plurality  303  or the longer putting location  601  passes the boundary  602 , it is considered out of play. In a preferred embodiment of the present invention, boundary  602  is located 18 inches behind the furthest edge of the circular target zone  102  and 18 inches in front of the farthest edge (as viewed from the putting locations  303  and  601 ) of putting surface  401 . In a preferred embodiment, boundary  602  is made to be distinctly visible, either through coloration on the putting surface  401  or by a change in turf type from smooth to rough. Alternatively, in another embodiment, hypothetical line  602  can be physically implemented as a sand or water trap or other equivalent golf-course obstacle. Additionally, in a preferred embodiment, putting locations  303  are made to be distinctly visible on the putting surface through the use of a coloration technique, which, in another preferred embodiment consists of yellow or red circles. In another preferred embodiment, longer putting location  601  is also made to be distinctly visible on the putting surface, optionally in another preferred embodiment as a larger yellow or red putting marker. In yet another preferred embodiment, putting distances (i.e., distances to the golf cup  101 ) are indicated on the putting surface  401  next to each of the putting locations  303  and  601 , as is illustrated in  FIG. 6A . 
       FIG. 6B  illustrates another preferred embodiment in which a plurality of putting locations  303   a  through  303   h  are situated on a putting surface  401  that slopes upward toward the golf cup  101  and that has no pitch. Analogous components to those shown in  FIG. 6A  are provided, and all features and embodiment disclosures discussed in connection with  FIG. 6A  also apply to  FIG. 6B . Specifically, there is a plurality of putting locations  303   a  through  303   h  situated uphill from golf cup  101 . Of note, however, is the fact the larger subregion  106  of circular target zone  102  is placed on the uphill side of the cup  101 . Consequently, the cup  101  is located at the 6 o&#39;clock position, in accordance with the foregoing discussion and  FIG. 4A . A hypothetical and aggregated line of ideal approach  301  (not shown) can be estimated for the plurality of putting locations  303 . The dividing line  105  between the high-percentage side  106  and low-percentage side  107  of the cup  101  is situated perpendicular to this line of ideal approach  301 . An analogous separate putting location  601  is illustrated such that the distance from  601  to the cup  101  is considerably longer (30 feet in a preferred embodiment) than any of the putting locations within the plurality  303  (which are located at 4, 6, 8, 12, 14, 16, 18, and 20 feet, respectively, from the cup  101  in a preferred embodiment). Putting location  601  thereby acts as the longer putting location from which to take super putts or bonus putts according to an embodiment of the present invention as described in connection with  FIGS. 5A and 5B , respectively. 
     Hypothetical line  602  similarly demarcates the boundary of a playable surface from putting surface  401 , such that once any ball putted from the region of the plurality  303  or the longer putting location  601  passes boundary  602 , it is considered out of play. In a preferred embodiment of the present invention, boundary  602  is located 18 inches behind the furthest edge of the circular target zone  102  and 18 inches in front of the farthest edge (as viewed from the putting locations  303  and  601 ) of putting surface  401 . In a preferred embodiment, boundary  602  is made to be distinctly visible, either through coloration on the putting surface  401  or by a change in turf type from smooth to rough. Additionally, in a preferred embodiment, putting locations  303  are made to be distinctly visible on the putting surface through the use of a coloration technique, which, in another preferred embodiment, consists of yellow or red circles. In another preferred embodiment, longer putting location  601  is also made to be distinctly visible on the putting surface  401 , optionally in the form of a larger yellow or red putting marker. In yet another preferred embodiment, putting distances (i.e., distances to the golf cup  101 ) are indicated on the putting surface  401  next to each of the putting locations  303  and  601 . 
       FIG. 6C  illustrates another preferred embodiment in which a plurality of putting locations  303   a  through  303   h  is situated on a flat putting surface  401  with right-to-left pitch. Analogous components to those shown in  FIGS. 6A and 6B  are provided, and all features and embodiment disclosures discussed in connection with  FIGS. 6A and 6B  also apply to  FIG. 6C . Specifically, there is a plurality of putting locations  303   a  through  303   h  situated at varying distances from golf cup  101 . Of note, however, is the fact the larger subregion  106  of circular target zone  102  is placed on the uphill side of the cup  101 . Consequently, cup  101  is located at the 8 o&#39;clock position, in accordance with the foregoing discussion and  FIG. 4C . A hypothetical and aggregated line of ideal approach  301  (not shown) can be estimated for the plurality of putting locations  303 . The dividing line  105  between the high-percentage side  106  and low-percentage side  107  of the target region  102  is situated along this line of ideal approach  301 . An analogous separate putting location  601  is illustrated such that the distance from  601  to the cup  101  is considerably longer (35 feet in a preferred embodiment) than any of the putting locations within the plurality  303  (which are located at 3, 6, 9, 12, 15, 18, 21, and 24 feet, respectively, from the cup  101  in a preferred embodiment). Putting location  601  thereby acts as the longer putting location from which to take super putts or bonus putts according to an embodiment of the present invention as described in connection with  FIGS. 5A and 5B , respectively. 
     Hypothetical line  602  similarly demarcates the boundary of a playable surface from putting surface  401 , such that once any ball putted from the region of the plurality  303  or the longer putting location  601  passes boundary  602 , it is considered out of play. In a preferred embodiment of the present invention, boundary  602  is located 18 inches behind the furthest edge of the circular target zone  102  and 18 inches in front of the farthest edge (as viewed from the putting locations  303  and  701 ) of putting surface  401 . In a preferred embodiment, boundary  602  is made to be distinctly visible, either through coloration on the putting surface  401  or by a change in turf type from smooth to rough. Additionally, in a preferred embodiment, putting locations  303  are made to be distinctly visible on the putting surface through the use of a coloration technique, which, in another preferred embodiment, consists of yellow or red circles. In another preferred embodiment, longer putting location  601  is also made to be distinctly visible on the putting surface  401 , optionally in the form of a larger yellow or red putting marker. In yet another preferred embodiment, putting distances (i.e., distances to the golf cup  101 ) are indicated on the putting surface  401  next to each of the putting locations  303  and  601 . 
       FIG. 6D  illustrates another preferred embodiment in which a plurality of putting locations  303   a  through  303   h  are situated on a putting surface that slopes upward toward the golf cup  101  and that has a left-to-right pitch. Analogous components to those shown in  FIGS. 6A through 6C  are provided, and all features and embodiment disclosures discussed in connection with  FIGS. 6A through 6C  also apply to  FIG. 6D . Specifically, there is a plurality of putting locations  303   a  through  303   h  situated downhill from golf cup  101 . Of note is the fact the larger subregion  106  of circular target zone  102  is placed on the high side of the cup  101 . Consequently, cup  101  is located at the 4 o&#39;clock position, in accordance with the foregoing discussion and  FIG. 4B . A hypothetical and aggregated line of ideal approach  301  (not shown) can be estimated for the plurality of putting locations  303 . The dividing line  105  between the high-percentage side  106  and low-percentage side  107  of target region  102  coincides with this line of ideal approach  301 . An analogous separate putting location  601  is illustrated such that the distance from  601  to the cup  101  is considerably longer (40 feet in a preferred embodiment) than any of the putting locations within the plurality  303  (which are located at 5, 7, 10, 14, 18, 22, 26, and 30 feet, respectively, from the cup  101  in a preferred embodiment). Putting location  601  thereby acts as the longer putting location from which to take super putts or bonus putts according to an embodiment of the present invention as described in connection with  FIGS. 5A and 5B , respectively. 
     Hypothetical line  602  similarly demarcates the boundary of a playable surface from putting surface  401 , such that once any ball putted from the region of the plurality  303  or the longer putting location  601  passes boundary  602 , it is considered out of play. In a preferred embodiment of the present invention, boundary  602  is located 18 inches behind the furthest edge of the circular target zone  102  and 18 in front of the farthest edge (as viewed from the putting locations  303  and  601 ) of putting surface  401 . In a preferred embodiment, boundary  602  is made to be distinctly visible, either through coloration on the putting surface  401  or by a change in turf type from smooth to rough. Additionally, in a preferred embodiment, putting locations  303  are made to be distinctly visible on the putting surface  401  through the use of a coloration technique, which, in another preferred embodiment, consists of yellow or red circles. In another preferred embodiment, longer putting location  303  is also made to be distinctly visible on the putting surface  401 , optionally in the form of a larger yellow or red putting marker. In yet another preferred embodiment, putting distances (i.e., distances to the golf cup  101 ) are indicated on the putting surface  401  next to each of the putting locations  303  and  601 . 
     The embodiments illustrated in  FIGS. 6A through 6D  can be used individually or in combination with one another. In fact, a preferred embodiment of the present invention combines all four embodiments illustrated in the aforementioned figures into a single, coherent golf-putting practice game by utilizing the methods described in connection with  FIGS. 5A and 5B . In such an embodiment, a round using either the feature “superputt” or the feature “bonus putt” is played on each of the four surfaces, in any order (although in a preferred embodiment, it is the order presented here) and the scores combined. When used in combination with one anther, the game played on surfaces illustrated in  FIGS. 6A through 6E  is known commercially as “BirdZone® Tour 4.” 
       FIG. 7  illustrates another preferred embodiment of the present invention in which a plurality of putting locations  303   a  through  303   h  are situated on a pitchless putting surface  401  that either slopes upward to the hole or downward to the hole depending upon the putting location in question. It too shares several features in common with the embodiments illustrated in  FIGS. 6A through 6D , but is referred to commercially as “BirdZone® 9.” Analogous components are provided, and many of the features and embodiment disclosures discussed in connection with  FIGS. 6A through 6D  also apply to  FIG. 6E . Specifically, there is a plurality of putting locations  303   c  through  303   e  situated uphill from golf cup  101 . There is also a plurality of putting locations  303   f  through  303   h  located downhill from golf cup  101 . As such, larger subregion  106  is therefore placed on the uphill side of the putting surface  401 . Therefore, depending upon the putting location in question, placement of the cup  101  inside target zone  102  is either in the 6 o&#39;clock position (for uphill putts) or the 12 o&#39;clock position (for downhill putts), in accordance with the foregoing discussion and  FIGS. 4A and 4D , respectively. 
     Furthermore, a hypothetical and aggregated line of ideal approach  301  (not shown) can be estimated for the plurality of putting locations  303 . The dividing line  105  between the high-percentage side  106  and the low-percentage side  107  of the cup  101  situated perpendicularly to this line of ideal approach  301 . An analogous separate putting location  601  is illustrated such that the distance from  601  to the cup  101  is considerably longer (20 feet in a preferred embodiment) from the cup  101  than any of the putting locations within the plurality  303  (which are located at 4, 6, 9, 12, and 15 feet, respectively, from the cup  101  in a preferred embodiment). Putting location  601  thereby acts as the longer putting location from which to take super putts or bonus putts according to an embodiment of the present invention as described in connection with  FIGS. 5A and 5B , respectively. In a preferred embodiment, putting locations  303  are made to be distinctly visible on the putting surface  401  through the use of a coloration technique, which, in another preferred embodiment, consists of yellow or red circles. In another preferred embodiment, longer putting location  601  is also made to be distinctly visible on the putting surface  401 , optionally in the form of a larger yellow or red putting marker. In yet another preferred embodiment, putting distances (i.e., distances to the golf cup  101 ) are indicated on the putting surface  401  next to each of the putting locations  303  and  601 .