Abstract:
A game is provided which requires the player to analyze and solve physical problems confronted while playing the game. The game may be played by all persons, regardless of age. The game features the construction of a track assembly which will enable a sphere to roll on the track assembly from the first end of the assembly to the second end of the assembly without falling off of the assembly. The track assembly is comprised of a pair of circular cylinders in parallel alignment and at least two braces positioned in the assembly to maintain the alignment of the cylinders. The sphere is caused, by gravity, to roll on top of the aligned cylinders from one end of the assembly to the other end without falling off of the assembly. There is also provided a method of designing the operative elements of the game (cylinder diameter, sphere diameter and cylinder separation) to change the level of skill required to play the game.

Description:
BACKGROUND OF THE INVENTION 
       [0001]    1. Technical Field of the Invention 
         [0002]    This invention relates to a game. The invention further relates to apparatus employed in playing the game. The invention still further relates to a method of playing the game. The invention also relates to a mathematical method of designing articles and selecting conditions to be employed to play the game. 
         [0003]    2. Prior Art and Problems Solved 
         [0004]    Playing a game requires the player to be aware of the goal to be achieved by playing the game. Playing the game requires skill, persistence, stamina, personal desire and sufficient interest to achieve that goal. A player must know and understand the rules and standards of the game, and ultimately requires the player to develop the mental and physical abilities to be employed to achieve the goal 
         [0005]    Playing a game can be principally physical in nature, principally mental in nature or a combination of physical and mental ability. A game which is principally mental in nature requires a player to analyze and solve problems encountered while playing the game in order to educate the player to know when and how to change tactics and/or to design, move or manipulate parts and pieces. 
         [0006]    A game principally involving mental analysis and problem solving can be played and enjoyed by all persons young and old, regardless of whether such person possess exceptional physical ability. 
       SUMMARY OF THE INVENTION 
       [0007]    This invention provides a game which can be played by all persons. The game principally involves mental analysis and problem solving to educate, indicate, or at least to suggest, the manipulation, movement and/or design of articles employed in the game. Playing the game of this invention features the first step of forming a track assembly and the second step of causing a sphere to roll on the track assembly. 
         [0008]    The track assembly has a first end and a second end. The goal of the game is to enable a sphere to roll on the track assembly from the first end to the second end without falling off of the track. 
         [0009]    The track assembly is comprised of a first circular cylinder, that is, a cylinder having a circular cross section, a second circular cylinder, also having a circular cross section, at least a first brace and at least a second brace. The first brace, the second brace and all other braces which may be employed in the track assembly are identical. The cylinders are positioned side-by-side in the track assembly so that the longitudinal axes of the cylinders are in substantial parallel alignment. 
         [0010]    The first circular cylinder has an exterior radius which is constant through out the entire length of the track assembly from the first end to the second end and the second circular cylinder has an exterior radius which is constant through out the entire length of the track assembly from the first end to the second end. The exterior radius of the first circular cylinder is substantially equal to the exterior radius of the second cylinder. The length of the first circular cylinder is equal to the length of the second circular cylinder. 
         [0011]    Each of the first and second cylinders has a proximal side, a distal side, a bottom side, a top side, a first end and a second end. In the track assembly, the first end of the first cylinder and the first end of the second cylinder are adjacent to form the first end of the track assembly and the second end of the first cylinder and the second end of the second cylinder are adjacent to form the second end of the track assembly. The proximal sides of each cylinder in the track assembly face each other, wherein the respective proximal sides may be in contact or may be spaced apart. The distance between the proximal sides of the cylinders is defined as the track gap. The track gap has a value in the range of zero to a value which is less than the diameter of the sphere. 
         [0012]    The distance, that is, the track gap, between the proximal side of the first circular cylinder and the proximal side of the second circular is established by the mentioned braces to thereby maintain the longitudinal axes of the cylinders in substantial parallel alignment. There can be a plurality such braces, but there must be one brace at the first end of the track assembly and a second brace at the second end of the track assembly. The number of braces in excess of two to be employed is a decision to be made while playing the game. 
         [0013]    When the track is assembled, the first end of the track assembly is raised relative to the second end. Thereafter, a sphere is positioned on the top side of the track at the first end thereof and permitted to roll on the top side of the track toward the second end of the track. As the sphere rolls along the track it must be understood that the sphere is in tangential contact with successive points on the top side of the first cylinder and is in simultaneous tangential contact with successive points on the top side of the second cylinder. The distance between the simultaneous points of tangential contact is defined as the track width. There is no other contact between the sphere and any other surface. The braces employed in the track assembly not only maintain track gap, but also maintain track width at a substantially constant value through out the entire length of the track assembly from the first end to the second end thereof. 
         [0014]    There is no mechanical connection between the sphere and the track assembly. The only elements causing the sphere to remain in rolling contact with the track is the natural force of gravity between the sphere and the earth and force developed by the combination of the velocity of the rolling sphere and the curvature of the track. 
         [0015]    Energy employed to cause the sphere to roll on the track assembly from the first end to the second end is potential energy supplied by the elevation difference between the first end of the track and the second end of the track. The game is complete when the sphere rolls on the track assembly from the first end to the second end without falling off. 
         [0016]    If the sphere fails to roll along the track assembly from the first end to the second end, and instead ceases to roll or leaves the track, then the need for problem analysis and solution is indicated. The solutions include such adjustments as maintaining track width, changing the velocity of the sphere, changing the weight of the sphere, banking the track assembly and combinations thereof. 
         [0017]    In another aspect, the game provides a mathematical method of selecting conditions to be employed to play the game. Such conditions include brace design, cylinder radius, sphere radius and track gap. 
     
    
     
       DESCRIPTION OF THE DRAWINGS 
         [0018]      FIG. 1  is an end view of the track assembly of this invention showing a first circular cylinder and a second circular cylinder each mounted in a brace to form a track assembly.  FIG. 1  also shows a sphere positioned on the upper portion (top side) of the track wherein a point on the top side of the first cylinder is in contact with a first point on the sphere and a point on the top side of the second cylinder is in contact with a second point on the sphere. 
           [0019]      FIG. 2  is the right side view of  FIG. 1  showing a first brace at one end of the track assembly and a second brace at another end of the track assembly, wherein one end of a cylinder is shown mounted in the first brace and another end of the same cylinder is shown mounted in the second brace. A portion of the sphere extends into the space between each cylinder. That portion of the sphere is shown as a dashed curved line. The portion of the sphere extending into the space is shown in  FIG. 1 . That portion lies below the mentioned tangential contact points between the sphere and the cylinders. 
           [0020]      FIG. 3  is the top view of  FIG. 2  showing the sphere positioned on the upper portion (top side) of the track assembly. The sphere partially covers the track gap between the proximal sides of the first cylinder and the second cylinder. The limits of the track gap are shown as a pair of dashed lines. The space between each cylinder is directly shown in  FIG. 1 . A second pair of dashed lines, outside of the first mentioned pair, represents the path of the mentioned tangential contact points as the sphere rolls along the track. (Line “k,”  FIG. 5 ) 
           [0021]      FIG. 4  is an end view of a second embodiment of the track assembly of this invention showing a first circular cylinder and a second circular cylinder mounted in a brace to form a track assembly. A sphere is shown positioned on the upper portion (top side) of the track assembly wherein a point on the top side of the first cylinder is in contact with a first point on the sphere and a point on the top side of the second cylinder is in contact with a second point on the sphere.  FIG. 4 , when compared to  FIG. 1 , shows the effect on the track assembly and the sphere by increasing the track gap between the first cylinder and the second cylinder. As mentioned, track gap, sometimes referred to as spaced separation, is the distance between the proximal side of the first circular cylinder and the proximal side of the second circular cylinder. 
           [0022]      FIG. 5  shows various spacial relationship between the first circular cylinder, the second circular cylinder and the sphere, and provides visual basis of a method of calculating the values of the distances shown in  FIG. 5 . The calculated values vary upon adjusting track gap, cylinder diameter and sphere diameter. 
       
    
    
     DESCRIPTION OF THE INVENTION 
       [0023]    Referring now to  FIGS. 1 ,  2  and  3 , first circular cylinder  1  is shown positioned in the interior of open ring  2  of brace  3 , and second circular cylinder  4  is shown positioned in the interior of open ring  5  of brace  3 . Cylinder  1  and cylinder  4  are positioned side-by-side and, in combination with brace  3  and brace  6 , as shown in  FIGS. 2 and 3 , form a section of track assembly  7 . It is to be understood that track assembly  7  can consist of a multiplicity of sections as shown in  FIGS. 2 and 3 . Cylinder  1  and cylinder  4  each have a circular cross section and the external radius of cylinder  1  is equal to the external radius of cylinder  4 . 
         [0024]    Brace  3  and brace  6  are identical in size, shape and construction. Track assembly  7  includes at least one of such braces placed at the first end thereof and at least one of such braces placed at the second end thereof. Depending upon the operation of the invention, as disclosed below, there can be a number of such braces between the first end and the second end of track assembly  7 . 
         [0025]    Brace  3  and brace  6  are identical. Any reference to or mention of brace  3  in this disclosure shall mean brace  3  or brace  6 , unless otherwise stated. With this admonition in mind, the following description applies to brace  3  and brace  6 . Accordingly, for simplicity, separate reference numerals shall not be included and a separate description of brace  6  is not provided. 
         [0026]    Brace  3  consists of base  8 , open ring  2 , open ring  5 , leg  9  and leg  10 . Base  8  includes top side  8   a , bottom side  8   b , midpoint  8   c , left hand side  8   d  and right hand side  8   e . Open ring  2  is rigidly attached to top side  8   a  on right hand side  8   e , and open ring  5  is rigidly attached to top side  8   a  on left hand side  8   d . Open ring  2  and open ring  5  are also rigidly attached each to the other at a position above midpoint  8   c  of base  8 . 
         [0027]    Base  8  is rectangular in shape and has a longitudinal axis and a transverse axis.  FIG. 1  provides a view of the longitudinal axis of base  8 .  FIG. 3  provides a view of the transverse axis of base  8 .  FIGS. 1 and 3  indicates the relative proportions of the longitudinal axis and the transverse axis of base  8 . 
         [0028]      FIG. 3  shows first cylinder  1  and second cylinder  4  mounted in open ring  2  and open ring  5 , respectively. The longitudinal axis of each cylinder is substantially perpendicular to the longitudinal of base  8  and substantially parallel to the transverse axis of base  8 . 
         [0029]    Open ring  2  consists of curved distal side  2   a  and curved proximal side  2   b . Distal side  2   a  and proximal side  2   b  join at right side  8   e  of top  8   a  to thereby form open ring  2 . Open ring  5  consists of curved distal side  5   a  and curved proximal side  5   b . Distal side  5   a  and proximal side  5   b  join at left side  8   d  of top  8   a  to thereby form open ring  5 . Proximal side  2   b  contacts and is attached to proximal side  5   b  above midpoint  8   c  of base  8 . The contact lies within the vertical line between the center of sphere  16  and midpoint  8   c.    
         [0030]    The end of curved proximal side  2   b  and the end of curved proximal side  5   b  form plane horizontal surface  11 . Horizontal surface  11  lies along a line joining the centers of open ring  2  and open ring  5 . Accordingly, proximal side  2   b  does not extend above the center of open ring  2 , and proximal side  5   b  does not extend above the center of open ring  5 . In contrast, distal side  2   a  does extend above the center of open ring  2 , and distal side  5   b  does extend above the center of open ring  5 . Accordingly, the distance from edge  14  to edge  15  on the inner surface of open ring  5  is less than the inside diameter of open ring  5 . Similarly, the distance from edge  12  to edge  13  on the inner surface of open ring  2  is less than the inside diameter of open ring  2 . The distance from edge  14  to edge  15  is less than the inside diameter of open ring  5 . The distance from edge  12  to edge  13  is less than the inside diameter of open ring  2   
         [0031]    Horizontal surface  11  is shown in  FIGS. 1 and 3  and the edge of horizontal surface  11  is shown as a dotted line in  FIG. 2 . The bottom of sphere  16  does not contact surface  11 . 
         [0032]    The external radius of cylinder  1  is equal to or slightly greater than the internal radius of open ring  2 . The external radius of cylinder  4  is equal to or slightly greater than the internal radius of open ring  5 . Accordingly, the external diameter cylinder  1  is greater than the distance from edge  12  to edge  13  and the external diameter of cylinder  4  is greater than the distance from edge  14  to edge  15 . 
         [0033]    The center of open ring  2  is substantially coincident with the center of cylinder  1 , and the center of open ring  5  is substantially coincident with the center of cylinder  4 . 
         [0034]    As shown in  FIGS. 1 and 3  the inner edge of distal side  2   a  terminates on the upper side of cylinder  1  at the outer surface of cylinder  1  to form line  12 . The inner edge of proximal side  2   b  terminates on the upper side of cylinder  1  at the outer surface of cylinder  1  to form line  13  at surface  11 . Open ring  2  is described as being open due to the absence of structural material in ring  2  between line  12  and line  13  on the upper side of open ring  2 . The angular distance between line  12  and line  13  is more than 90 and less than about 160 degrees. The lineal distance between line  12  and line  13  is less than the outside diameter of cylinder  1 . 
         [0035]    As shown in  FIGS. 1 and 3  the inner edge of distal side  5   a  terminates on the upper side of cylinder  4  at the outer surface of cylinder  4  to form line  14 . The inner edge of proximal side  5   b  terminates on the upper side of cylinder  4  at the outer surface of cylinder  4  to form line  15  at surface  11 . Open ring  5  is described as being open due to the absence of structural material in ring  5  between line  14  and line  15 . The angular distance between line  14  and line  15  is more than 90 and less than about 140 degrees. The lineal distance between line  14  and line  15  is less than the outside diameter of cylinder  4 . 
         [0036]    Surface  11  is substantially parallel with the longitudinal axis of base  8 . The distance along surface  11  between line  13  and line  15  is defined as the distance between the proximal side of cylinder  1  and the proximal side of cylinder  4 . It is evident, therefor, that the proximal side of cylinder  1  faces the proximal side of cylinder  4 . This distance is referred to as the track gap and is shown in  FIG. 1  and  FIG. 3 . Track gap is graphically illustrated in  FIG. 5  as the distance between points H and L and is equal to distance “2c.” A function of brace  3  is to maintain track width at a substantially constant value which is accomplished by maintaining track gap at a substantially constant value through out the entire length of the track assembly. 
         [0037]    In one embodiment, cylinder  1  and cylinder  4  are not rigid. Each cylinder is sufficiently flexible and compressible to enable it to be manually formed into curves and loops and to enable slight deformation of cross section. An example of such a flexible and compressible cylinder is an ordinary garden hose which may be compressed so that can be manually inserted between lines  12  and  13  and between lines  14  and  15  of open rings  2  and  5 , respectively. 
         [0038]    In operation, track assembly  7  is constructed as previously described employing a garden hose as flexible cylinders  1  and  4 . Thereafter, the first end of track assembly  7  is elevated to a level above the second end of track assembly  7 . In this regard legs  9  and  10  can be placed on any support structure and the flexible cylinders  1  and  4  are permitted to naturally bend or slope to the second end of game structure  7 . As the track assembly traverses the distance from the first end to the second end the assembly may be formed into one or more curves, loops, slopes and hills. 
         [0039]    Sphere  16  is placed on the upper portion (top) of track assembly  7  where it is supported by cylinders  1  and  4 . As shown in  FIG. 1 , sphere  16  tangentially contacts cylinder  1  at point E and tangentially contacts cylinder  4  at point D. Track width, the distance between point D and point E, is graphically illustrated in  FIG. 5 . Distance DE is the chord of a circle whose center, O, is the center of sphere  16 . Angle DOE is the central angle, 2θ, of a vertical section taken through the center of sphere  16 . Central angle DOE subtends chord DE and arc DE each of which is graphically illustrated in  FIG. 5 . Chord DE is equal to distance 2f. Arc DE is the product of central angle DOE, expressed in radians, and radius, a, of sphere  16 . In one embodiment, sphere  16  can be an ordinary golf ball. 
         [0040]    Track width and central angle DOE vary with change in hose radius, ball radius, track gap and combinations of such changes. Central angle DOE can vary from about 10 to about 170 degrees, but the central angle is preferably in the range of from about 30 to about 160, more preferably from 60 to about 140, and still more preferably from about 70 to about 120 degrees. The most preferred central angle is in the range of from about 85 to about 95 degrees. 
         [0041]    Track gap was defined as the distance along surface  11  between line  13  and line  15 . With regard to  FIGS. 1 ,  2  and  3 , due to the description of brace  3 , track gap and radius, b are fixed. However, track width is not constant. If the radius of ball  16  is constant, then track width is constant, but the radius of ball  16 , as employed with brace  3 , can be adjusted, which such adjustment will effect chord DE and arc DE.  FIG. 5 , discussed below, addresses the physical results of changing ball radius. 
         [0042]    Refer to  FIG. 5  and note that for a given track width, ball  16  will roll along hoses  1  and  4  so that successive tangent points D follow path DF and successive tangent points E follow path EG. It is preferred that path DF and path EG remain substantially equal in length and parallel from the first end of the track assembly the second end of the track assembly. Like chord DE, the distances DF and EG are chords of a circle whose center, O, is the center of ball  16 . The chords are graphically illustrated in  FIG. 5  and are equal to distance k which is distance 2r, e.g., k=2r. 
         [0043]    If the ball does not leave the track, but simply stops rolling, then the first end of the track assembly must be further raised relative to the second end of the track assembly. If the ball leaves the track then there are several problems to be confronted and solved. In one instance, stabilization of track width is required by taking steps to adjust track gap to place paths DF and EG in substantial parallel alignment. In this regard it is desired, and, in fact preferred, that track width remain constant through out the length of the track assembly. Track width stabilization can be accomplished by adding braces to portions of the track which reveal a variation in track gap. 
         [0044]    In another instance, the velocity of the ball may be sufficient to overcome the gravitational force which functions to hold the ball on the track assembly. Ball velocity can be reduced by lowering the first end of the track assembly relative to the second end of the track assembly. In an optional adjustment, ball velocity can be accommodated by banking the track assembly (increasing the angle of the track relative to ground) at the point where the sphere leaves the track. 
         [0045]    In another option, with track gap fixed, the distance from the top of the ball to the line tangent to the bottom surfaces of cylinders  1  and  4  ( FIG. 5 , distance h) can be reduced by decreasing track width. This can be accomplished by reducing the radius, a, of ball  16 . 
         [0046]    A combination of the mentioned adjustments, as well as a change in sphere weight or mass, can be adopted. 
         [0047]    Referring now to  FIG. 4 , cylinder  1  is shown positioned in the interior of open ring  17  of brace  18 , and cylinder  4  is shown positioned in the interior of open ring  19  of brace  18 . Cylinder  1  and cylinder  4  are positioned side-by-side and, in combination with brace  18  and a brace which is not shown, form a section of a track assembly which is also not shown. In this regard, the section not shown would be the same as the section shown in  FIG. 2 , but with an increase in track gap. It is to be understood that the track assembly can consist of a multiplicity of sections. Cylinder  1  and cylinder  4  each have a circular cross section and the external radius of cylinder  1  is equal to the external radius of cylinder  4 . 
         [0048]    Brace  18 , and the brace not shown, are identical in size, shape and construction. The track assembly, not shown, includes at least one such brace  18  placed at the first end thereof and at least one such brace  18  placed at the second end thereof. Depending upon the operation of the invention, as disclosed below, there can be a number of such braces between the first end and the second end of the track assembly. Any reference to or mention of brace  18  in this disclosure shall mean brace  18  or the brace not shown, unless otherwise stated. With this admonition in mind the following description of brace  18  applies to brace  18  and the brace which is not shown. 
         [0049]    Brace  18  consists of base  20 , open ring  17 , open ring  19 , leg  21 , leg  22  and spacer element  28 . Base  20  consists of top side  20   a , bottom side  20   b , midpoint  20   c , left hand side  20   d  and right hand side  20   e . Open ring  17  is rigidly attached to top side  20   a  on right hand side  20   e , and open ring  19  is rigidly attached to top side  20   a  on left hand side  20   d . Open ring  17  and open ring  19  are also rigidly attached to spacer element  28 . 
         [0050]    Base  20  is rectangular in shape. Base  20  has a longitudinal axis and a transverse axis.  FIG. 4  provides a view of the longitudinal axis of base  20 . First circular cylinder  1  and second circular cylinder  4  are mounted on base  20 , wherein the longitudinal axis of each cylinder is substantially perpendicular to the longitudinal axis of base  20  and substantially parallel to the transverse axis of base  20 . 
         [0051]    Open ring  17  consists of curved distal side  17   a  and curved proximal side  17   b . Distal side  17   a  and proximal side  17   b  join at right side  20   e  of top  20   a  to thereby form open ring  17 . Open ring  19  consists of curved distal side  19   a  and curved proximal side  19   b . Distal side  19   a  and proximal side  19   b  join at left side  20   d  of top  20   a  to thereby form open ring  19 . 
         [0052]    Spacer element  28  is rigidly attached to top side  20   a  intermediate open ring  17  and open ring  19  at a position above midpoint  20   c  of base  20 . Midpoint  20   c  lies within the vertical line passing through the center, O, of sphere  16 . Spacer element  28  consists of right side arm  28   a  and left side arm  28   b  which are identical in shape and length. Proximal side  17   b  contacts and is attached to right side arm  28   a . Proximal side  19   b  contacts and is attached to left side arm  28   b . The end of curved proximal side  17   b  and the end of right side arm  28   a  form plane horizontal surface  22 . Horizontal surface  22  lies along a line joining the centers of open ring  17  and open ring  19 . The end of curved proximal side  19   b  and the end of left side arm  28   b  form plane horizontal surface  23 . Horizontal surface  23  lies along a line joining the centers of open ring  17  and open ring  19 . In addition, the center of open ring  17  is substantially coincident with the circular center of cylinder  1 , and the center of open ring  19  is substantially coincident with the center of circular cylinder  4 . 
         [0053]    The inner edge of distal side  17   a  terminates on the upper side of cylinder  1  at the outer surface of cylinder  1  to form line  24 . The inner edge of proximal side  17   b  terminates on the upper side of cylinder  1  at the outer surface of cylinder  1  to form line  25  at surface  22 . Open ring  17  is described as being open due to the absence of structural material in ring  17  between line  24  and line  25 . The angular distance between line  24  and line  25  is more than 90 and less than about 160 degrees. The lineal distance between line  24  and line  25  is less than the diameter of cylinder  1 . 
         [0054]    The inner edge of distal side  19   a  terminates on the upper side of cylinder  4  at the outer surface of cylinder  4  to form line  26 . The inner edge of proximal side  19   b  terminates on the upper side of cylinder  4  at the outer surface of cylinder  4  to form line  27  at surface  23 . Open ring  19  is described as being open due to the absence of structural material in ring  19  between line  26  and line  27 . The angular distance between line  26  and line  27  is more than 90 and less than about 140 degrees The lineal distance between line  26  and line  27  is less than the diameter of cylinder  4 . 
         [0055]    The distance between line  25  and line  27  along surfaces  22  and  23  (parallel with the longitudinal axis of base  20 ) is referred to as the track gap, which is the distance between the proximal side of cylinder  1  and the proximal side of cylinder  4 . Track gap is graphically illustrated in  FIG. 5  as the distance between points H and L and is equal to distance 2c. As mentioned earlier, a function of the braces employed in the track assembly is to maintain the track gap and, accordingly, the track width at a substantially constant value through out the entire length of the track assembly. In one embodiment, cylinder  1  and cylinder  4  are not rigid, but each is sufficiently flexible and compressible to enable the formation of curves, hills and loops and to permit slight deformation of the cross section. An example of such a flexible cylinder is an ordinary garden hose. 
         [0056]    In operation, track assembly  18  is constructed as described employing a garden hose as circular cylinders  1  and  4 . Thereafter, the first end of track assembly  18  is elevated to a level above the second end of track assembly  18 . In this regard legs  21  and  22  can be placed on any support structure and cylinders  1  and  4  are permitted to naturally bend or slope to the second end of track assembly  18 . In traversing from the first end to the second end the assembly may be formed into one or more curves, loops, slopes and hills. 
         [0057]    Sphere  16  is placed on the upper portion (top side) of the track assembly so that sphere  16  is supported by the top sides of cylinders  1  and  4 . Sphere  16  tangentially contacts cylinder  1  at point E and tangentially contacts cylinder  4  at point D. The distance between point D and point E is the defined track width. In one embodiment, sphere  16  can be an ordinary golf ball. 
         [0058]    The curved surfaces of the first circular cylinder and the second circular cylinder result in variation of track width upon variation of ball radius. The distance between line  25  and line  27 , previously defined as track gap, is fixed due to the description of brace  18 . However, track width is the length of line DE. If the radius of ball  16  is constant, then track width is constant, but the radius of ball  16 , as employed with brace  18 , can be changed which such change will cause the length of line DE, track width, to change.  FIG. 5 , discussed below, addresses the physical results of changing ball radius. 
         [0059]    For a given track width, ball  16  will roll along circular cylinders  1  and  4  so that successive tangent points D follow path DF and successive tangent points E follow path EG (as suggested in  FIG. 3 ). It is preferred that path DF and path EG remain substantially parallel from the first end of the track assembly to the second end of the track assembly. 
         [0060]    If the ball does not leave the track assembly, but simply stops rolling, then the first end of the assembly must be further raised relative to the second end of the assembly. If the ball does leave the track assembly then there are several problems to be confronted and solved. In one instance, stabilization of track width is required by taking steps to adjust paths DF and EG to a substantially parallel alignment. In this regard it is desired, and, in fact preferred, that track width remain constant through out the length of the track. This stabilization can be accomplished by adding braces to portions of the track which reveal a variation in track gap. 
         [0061]    In another instance, the velocity of the ball may be sufficient to overcome the gravitational force which acts to hold the ball on the track assembly. Ball velocity can be reduced by lowering the first end of the assembly relative to the second end of the assembly. In an optional adjustment, ball velocity can be accommodated by banking the assembly (increasing the angle of the track relative to ground) at the point where the sphere leaves the track. 
         [0062]    In another option, with track gap fixed, the distance from the top of the ball to the line tangent to the bottom surfaces of circular  1  and  4  can be reduced by decreasing track width by reducing the radius of the ball. 
         [0063]    A combination of the mentioned adjustments, as well as a change in sphere weight or mass, can be adopted. 
         [0064]    The differences between the track assemblies illustrated in  FIGS. 1 and 4  are: track width (line DE), track gap ( FIG. 5 , line HL), width of track path ( FIG. 5 , distance e), vertical distance, h, ( FIG. 5 ) from the top of ball  16  to the line tangent to the bottoms of hoses  1  and  4 , and the lengths of braces  3  and  18 . All of the differences are the result of the insertion of spacer element  28  between open ring  17  and open ring  19  which insertion caused the increase in track gap of  FIG. 4  relative to  FIG. 1 . These terms and the values of the differences are addressed in connection with the description of  FIG. 5 . The discussion of  FIG. 5  will clarify why it is believed that the game requires greater skill when played on an assembly as described in  FIG. 1  as compared when played on an assembly as described in  FIG. 4 . 
         [0065]    The spacial relationships between ball  16  and the track assembly is illustrated in  FIG. 5 . Triangle POR is formed by the lines which connect the centers of ball  16  and hoses  1  and  4 . Accordingly, the lengths of sides OP and OR are equal, wherein each is equal to the sum of radius, a, of ball  16  and radius, b, of cylinder  1  (or  4 ). Base PR of triangle POR is the sum of track gap, 2c, radius b of cylinder  1  and radius b of cylinder  4 . (PR=2b+2c) 
         [0066]    Triangle DOE is similar to triangle POR. Side OP of triangle POR passes through tangent point D and line OR of triangle POR passes through tangent point E. Triangle DOE includes sides OD and OE each of which is equal to radius a of ball  16 . Base DE is formed by joining tangent points D and E wherein side DE is track width 2f. 
         [0067]    Line OP is extended along radius b of cylinder  4  to its intersection with the surface of cylinder  4  at point B to form side OB. Line OR is extended along radius b of cylinder  1  to its intersection with the surface of cylinder  1  at point C to form side OC. Side BC is formed by joining points B and point C to form triangle BOC which is similar to triangle POR. 
         [0068]    Angles BOC, POR and DOE are central angle 2θ which subtends line DE, and arc DE. 
         [0069]    The ability of rolling ball  16  to remain on the track assembly is defined as ball stability. It is believed that ball stability is related to the lengths of line DE, arc DE, and the distance, h, from the top of ball  16  to the bottoms of hoses  1  and  4  ( FIG. 5 ). As previously mentioned, line DE is equal to track width 2f. The length of arc DE is equal to the product of ball radius, a, and the number of degrees in the central angle expressed in radians. Ball stability decreases as line DE and arc DE decrease and as distance h increases. The lengths of line DE and arc DE approach an equal value as central angle BOG decreases. The difference between the lengths of arc DE and line DE is graphically demonstrated in  FIG. 5  by distance t. Stated differently, ball stability decreases as the ratio of arc DE to line DE approaches one. For purposes of this invention, the ratio is defined as the track width ratio, wherein track width ratio is an amount in the range of from about 1.013 to about 1.434, preferably from about 1.047 to about 1.3 and still more preferably from about 1.065 to about 1.211. The most preferred track width ratio is about 1.111. 
         [0070]    A player desiring to design a brace different from the braces shown in  FIGS. 1 and 4  can do so by independent or simultaneous variations in ball radius, hose radius and track gap. As previously disclosed, track gap is the fixed distance between lines  13  and  15  in  FIG. 1 , lines  25  and  27  in  FIG. 4  and the length of line HL (distance 2c) in  FIG. 5 . With regard to  FIGS. 1 and 4  the opportunities for changes in dimensions are limited to changes in radius, a, of ball  16 , because radius, b, of hoses  1  and  4  and track gap are fixed. 
         [0071]    Track gap 2c is a fraction, x, of ball diameter 2a. Accordingly, one-half of track gap is a fraction, x, of one-half of ball diameter. In mathematical terms, 2c=2ax, therefor, c=ax. This relationship is an important consideration in the design of a brace. The minimum value of c is zero, wherein x, defined herein as the hose separation factor, is zero. If hose separation factor is zero, then proximal point H on hose  4  contacts proximal point L on hose  1  ( FIG. 5 ). For any given combination of ball radius and hose radius, if track gap is zero then central angle BOC is a minimum and track width ratio (ration of arc DE to line DE) is a minimum. The maximum value of hose separation factor, x, is less than one. When x is equal to one, then track gap, 2c, is equal to ball diameter, 2a, and track width, 2f. Furthermore, when x is equal to one, then central angle BOC is equal to 180°, and points H and L would be in alignment with ball diameter, 2a. Under this condition ball  16  would have no track on which to roll and would probably slip or fall through points H and L. 
         [0072]    In another aspect, comparison of  FIGS. 1 and 4  illustrates that increase in track gap permits ball  16  to roll on the track in a lower position between hoses  1  and  4 . That is, distance t ( FIG. 5 ) increases. At some combination of ball radius, hose radius and track gap ball  16  will contact surface  11  shown in  FIG. 1 . At that combination, distance, s, of  FIG. 5  will be zero or less and the contact of ball  16  with a surface will not be limited to tangent points D and E. This condition is illustrated in Table 1. Furthermore, at some combination of ball radius, hose radius and track gap the bottom of ball  16  will extend below the line tangent to the bottoms of hoses  1  and  4  and strike base  8  ( FIG. 1 ) or base  20  ( FIG. 4 ). This condition is illustrated in Tables 2 and 3. 
         [0073]    In view of the above comments, central angle BOC has a value in the range of from about 30 to about 160, preferably from about 60 to about 140, still more preferably from about 70 to about 120 and most preferably from about 85 to about 95 degrees. 
         [0074]    The various dimensions shown in  FIG. 5  are defined in Table A. A mathematical program which can be employed in designing a track assembly useful to play the game of this invention is provided in Table B. The terms of the equations in Table B are defined in Table A. 
         [0000]    
       
         
               
             
               
               
             
           
               
                 TABLE A 
               
               
                   
               
               
                 DEFINITIONS of TERMS EMPLOYED in CONNECTION with FIG. 5 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 a = 
                 Radius of sphere/ball. 
               
               
                 b = 
                 Radius of cylinder/hose. 
               
               
                 y = 
                 Radius of ball divided by radius of hose. y = a/b 
               
               
                 x = 
                 Hose Separation Factor. Fraction of ball radius. 
               
               
                 c = 
                 Hose Separation. Perpendicular distance from proximal surface of hose along 
               
               
                   
                 horizontal line through centers of hoses to vertical line through center of ball. 
               
               
                   
                 c = xa 
               
               
                 d = 
                 Track separation. Perpendicular distance from distal surface of hose along horizontal 
               
               
                   
                 line through centers of hoses to vertical line through center of ball. 
               
               
                 e = 
                 Width of track path. e = 2d 
               
               
                 sin θ = 
                 Length of horizontal line through center of hose to vertical line through center of ball 
               
               
                   
                 divided by length of slant line (hypotenuse) from center of ball to center of hose. 
               
               
                 θ = 
                 Ball angle (inverse of sin θ). One-half of central angle of triangle BOC formed by 
               
               
                   
                 connecting centers of ball and hoses. 
               
               
                 2θ = 
                 Central angle of the above defined triangle BOC. 
               
               
                 cos θ = 
                 Length of vertical line through center of ball to horizontal line through center of hose 
               
               
                   
                 divided by length of slant line (hypotenuse) from center of ball to center of hose. 
               
               
                 f = 
                 Horizontal distance from tangent point of ball and hose to vertical line through center 
               
               
                   
                 of ball. 
               
               
                 2f = 
                 Width of ball track. 
               
               
                 r = 
                 Distance along vertical line through center of ball to ball track. 
               
               
                 k = 
                 Diameter of vertical section through ball at tangent point of ball and hose. 
               
               
                   
                 k = 2r 
               
               
                 g = 
                 One-half length of base of triangle BOC whose sides are ball radius and hose 
               
               
                   
                 diameter. 
               
               
                 2g = 
                 Length of base of triangle BOC. 
               
               
                 L = 
                 Length of ball movement per revolution. (Circumference of ball section at distance f) 
               
               
                 t = 
                 Distance along vertical line through center of ball from ball track to bottom of ball. 
               
               
                 ON = 
                 Perpendicular distance from center of the ball to line connecting centers of hoses. 
               
               
                 s = 
                 Difference between distance ON and ball radius. If “s” is a negative value, then the 
               
               
                   
                 bottom of the ball extends distance “s” below the line between the centers of the 
               
               
                   
                 hoses, and ON is less than ball radius. If “s” is a positive value, then the bottom of the 
               
               
                   
                 ball extends distance “s” above the line between the centers of the hoses, and ON is 
               
               
                   
                 greater than ball radius. 
               
               
                 j = 
                 Distance along vertical line through center of ball from line connecting center of hoses 
               
               
                   
                 to line connecting intersection of slant (hypotenuse) lines and outer surfaces of hoses. 
               
               
                 h = 
                 Vertical distance from top of ball to bottom of hoses. 
               
               
                 b − j = 
                 Distance along vertical line through center of ball between line tangent to the bottoms 
               
               
                   
                 of the hoses and the line connecting intersection of slant (hypotenuse) lines and outer 
               
               
                   
                 surfaces of hoses. (Extension of ball below base of triangle BOC.) 
               
               
                 b + s = 
                 Distance along vertical line through center of ball between line connecting center of 
               
               
                   
                 hoses and line tangent to bottoms of hoses. A negative answer means that the ball 
               
               
                   
                 extends below the line tangent to the bottoms of hoses. Hose separation factor, “x,” is 
               
               
                   
                 too large. 
               
               
                 h/e = 
                 Vertical distance from top of ball to the bottom of the hoses divided by width of track 
               
               
                   
                 path. 
               
               
                 rad = 
                 Central angle, 2θ, expressed in radians. 
               
               
                 arc DF = 
                 Length of arc between points D and E. (Length of circumference of ball which lies 
               
               
                   
                 below line DE. Arc DF is subtended by central angle 2θ. Length of Arc DF 
               
               
                   
                 approaches length of chord 2f and “t”diminishes as, “y,” increases. 
               
               
                 seg = 
                 Number of chords and arcs subtended by central angle 2θ. 
               
               
                 R = 
                 Diameter of ball divided by width of track 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
             
               
               
               
             
           
               
                 TABLE B 
               
               
                   
               
               
                 PROGRAM for CALCULATING VALUES 
               
               
                 of TERMS DEFINED in TABLE A 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 a 
                 radius of ball 
               
               
                 b 
                 radius of hose 
               
               
                 y = 
                 a/b 
               
               
                 b = 
                 a/y 
               
               
                 a = 
                 by 
               
               
                 x 
                 Value in the range of from about 0 to about 1 
               
               
                 c = 
                 xa 
               
             
          
           
               
                 c/b = 
                 xa/(a/y) = 
                 xy 
               
               
                 d = 
                 2b + c = 
                 (a/y)*(2 + yx) 
               
               
                 e = 
                 2d = 
                 (2a/y)*(2 + yx) 
               
               
                 sin θ = 
                 (b + c)/(a + b) = f/a = 
                 (1 + yx)/(1 + y) 
               
               
                   
                 g/(2b + a) = 
               
               
                 θ = 
                 the angle whose sin is 
                 sin −1 θ 
               
               
                   
                 (1 + yx)/(1 + y) = 
               
               
                 2θ = 
                   
                 2(sin −1 θ) 
               
               
                 cos θ = 
                 r/a = ON/(a + b) = j/b = 
                 cos(sin −1 θ) 
               
               
                 f = 
                 a(sin θ) = 
                 a(1 + yx)/(1 + y) 
               
               
                 2f = 
                 2a(sin θ) = 
                 2a(1 + yx)/(1 + y) 
               
               
                 r = 
                   
                 a (cos 8) 
               
               
                 k = 
                 2r = 
                 2a (cos θ) 
               
               
                 g = 
                 (2b + a)(sin θ) = a[(2 + 
                 (a/y)*(2 + y)*[(1 + 
               
               
                   
                 y)/y]*(sin θ) = 
                 yx)/(1 + y)] 
               
               
                 2g = 
                   
                 (2a/y)*(2 + y)*[(1 + 
               
               
                   
                   
                 yx)/(1 + y)] 
               
               
                 L = 
                 2πr = 
                 2π a (cos θ) 
               
               
                 t = 
                 a − r = 
                 a(1 − cos θ) 
               
               
                 ON = 
                 (a + b)(cosθ) = r + t + 
                 (a/y)*(y + 1)*(cos θ) 
               
               
                   
                 s = a + s = 
               
               
                 s = 
                 ON − a = ON − (r + t) = 
                 (a/y)*(y + 1)*(cos θ) − a 
               
               
                 j = 
                 r/y = b(cos θ) = 
                 (a/y)*(cos θ) 
               
               
                 h = 
                 a + b + ON = 
                 (a/y)*(y + 1)*(1 + cos θ) 
               
               
                 b − j = 
                   
                 (a/y)*(1 − cos θ) 
               
               
                 b + s = 
                   
                 (a/y)*(1 + (y + 
               
               
                   
                   
                 1)*(cos θ) − y) 
               
               
                 h/e = 
                   
                 (1 + y)*(1 + cos 
               
               
                   
                   
                 θ)/(2)/(2 + xy) 
               
               
                 Rad = 
                 (2π)(2θ)/360 = 
                 (π)*(θ)/90 
               
               
                 arc DF = 
                 (radians)*(radius) = 
                 [(π)*(θ)/90]*a 
               
               
                 arc DF/2f = 
                 [(π)*(θ)/90]*a/(2a)(sin θ) = 
                 (π)*(θ)/(180)*sinθ 
               
               
                 seg = 
                 (2)*(π)*(a)/[(π)*(θ)/90]*a = 
                 180/θ 
               
               
                 R = 
                 2a/((2b + c) + (2b + c)) = 
                 y/(2 + xy) 
               
               
                 y = 
                 [(1 + xy)(sin θ)] − 1 
               
               
                   
               
             
          
         
       
     
       Example 1 
       [0075]    A track assembly comprised of two flexible garden hoses and at least two brace members was constructed. Each garden hose was about 25 feet long and had an outside diameter of about 0.85 inches. Accordingly, referring to  FIG. 5 , the radius, b, of each hose was about 0.425 inches. Each brace member employed in the structure is described in connection with reference numeral  3  of  FIG. 1 . The track gap, 2c, in each brace was fixed at about 0.176 inches. Accordingly, as shown in  FIG. 5 , c, was about 0.088 inches. 
         [0076]    The garden hoses were positioned side-by-side to form a precursor track assembly which extended 25 feet from the first end of the assembly to the second end of the assembly. A brace member was placed at the first end of the assembly and at the second end of the assembly to form a track assembly. 
         [0077]    The first end of the track assembly was raised so that it was about 18 inches higher than the second end of the assembly. Thereafter, a golf ball was positioned on the upper portion (top) of the structure at the first end thereof so that a point on the ball contacted a point on the top side one of the two hoses and another point on the ball contacted a point on the top side of the second of the two hoses. The ball was then permitted to roll on the hoses toward the second end. To maintain track width, additional braces were connected to the hoses in the track assembly at points where the ball failed to remain on the hoses. Eventually the ball rolled from the first end of the track assembly to the second end of the rack assembly without falling off. 
         [0078]    The golf ball had an outside diameter of about 1 and 11/16 (1.6875) inches. Accordingly, referring to  FIG. 5 , the radius, a, of the ball was about 0.84375 inches. The hose separation factor (x) was, therefor, about 0.104 inches (Table B, x=c/a) 
         [0079]    Calculations employing the mathematical relationships disclosed in Table B were made for ten hose separation factors (x) ranging from 0.104 to 1.0. The results of the calculations are provided in Table 1. The brace employed in the track assembly was constant ( FIG. 1 ), accordingly, track gap hose, 2c, was constant at 0.176 and hose radius, b, was constant at 0.425. Ball radius was calculated by the formula, a=c/x. The results for hose separation factors 0.2 to 1.0 are simulated. 
         [0000]    
       
         
               
               
             
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Term 
                 Hose Separation Factor 
               
             
          
           
               
                 X 
                 0.104 
                 0.2 
                 0.3 
                 0.4 
                 0.5 
                 0.6 
                 0.7 
                 0.8 
                 0.9 
                 1.0 
               
               
                   
               
             
          
           
               
                 a 
                 0.844 
                 0.440 
                 0.293 
                 0.220 
                 0.176 
                 0.147 
                 0.126 
                 0.110 
                 0.098 
                 0.088 
               
               
                 b 
                 0.425 
                 0.425 
                 0.425 
                 0.425 
                 0.425 
                 0.425 
                 0.425 
                 0.425 
                 0.425 
                 0.425 
               
               
                 y 
                 1.986 
                 1.035 
                 0.689 
                 0.518 
                 0.414 
                 0.345 
                 0.296 
                 0.259 
                 0.230 
                 0.207 
               
               
                 c 
                 0.088 
                 0.088 
                 0.088 
                 0.088 
                 0.088 
                 0.088 
                 0.088 
                 0.088 
                 0.088 
                 0.088 
               
               
                 e 
                 1.876 
                 1.876 
                 1.876 
                 1.876 
                 1.876 
                 1.876 
                 1.876 
                 1.876 
                 1.876 
                 1.876 
               
               
                 sin θ 
                 0.404 
                 0.593 
                 0.714 
                 0.795 
                 0.854 
                 0.897 
                 0.932 
                 0.959 
                 0.981 
                 1.000 
               
               
                 θ 
                 23.828 
                 36.37 
                 45.561 
                 52.655 
                 58.649 
                 63.766 
                 68.749 
                 73.536 
                 78.813 
                 90.000 
               
               
                 2θ 
                 47.656 
                 72.740 
                 91.122 
                 105.310 
                 117.298 
                 127.532 
                 137.497 
                 147.072 
                 157.626 
                 180.000 
               
               
                 cos θ 
                 0.915 
                 0.805 
                 0.700 
                 0.607 
                 0.520 
                 0.442 
                 0.362 
                 0.283 
                 0.194 
                 0.000 
               
               
                 2f 
                 0.682 
                 0.522 
                 0.419 
                 0.350 
                 0.300 
                 0.263 
                 0.234 
                 0.211 
                 0.192 
                 0.176 
               
               
                 r 
                 0.772 
                 0.354 
                 0.205 
                 0.133 
                 0.092 
                 0.065 
                 0.046 
                 0.031 
                 0.019 
                 0.000 
               
               
                 k 
                 1.544 
                 0.708 
                 0.410 
                 0.267 
                 0.183 
                 0.130 
                 0.091 
                 0.062 
                 0.038 
                 0.000 
               
               
                 2g 
                 1.369 
                 1.530 
                 1.633 
                 1.702 
                 1.752 
                 1.789 
                 1.818 
                 1.841 
                 1.860 
                 1.876 
               
               
                 t 
                 0.072 
                 0.086 
                 0.088 
                 0.087 
                 0.084 
                 0.082 
                 0.080 
                 0.079 
                 0.079 
                 0.088 
               
               
                 s 
                 0.317 
                 0.256 
                 0.210 
                 0.171 
                 0.137 
                 0.106 
                 0.074 
                 0.041 
                 0.004 
                 −0.088 
               
               
                 h 
                 2.430 
                 1.561 
                 1.221 
                 1.036 
                 0.914 
                 0.824 
                 0.750 
                 0.686 
                 0.624 
                 0.513 
               
               
                 b + s 
                 0.742 
                 0.681 
                 0.635 
                 0.596 
                 0.562 
                 0.531 
                 0.499 
                 0.466 
                 0.429 
                 0.337 
               
               
                 h/e 
                 1.296 
                 0.832 
                 0.651 
                 0.552 
                 0.487 
                 0.439 
                 0.400 
                 0.366 
                 0.333 
                 0.273 
               
               
                 arc DF 
                 0.702 
                 0.559 
                 0.466 
                 0.404 
                 0.360 
                 0.326 
                 0.302 
                 0.282 
                 0.269 
                 0.276 
               
               
                 seg 
                 7.554 
                 4.949 
                 3.951 
                 3.418 
                 3.069 
                 2.823 
                 2.618 
                 2.448 
                 2.284 
                 2.000 
               
               
                 arc DF/ 
                 1.029 
                 1.070 
                 1.113 
                 1.155 
                 1.199 
                 1.240 
                 1.288 
                 1.338 
                 1.402 
                 1.571 
               
               
                 2f 
               
               
                 R 
                 0.900 
                 0.469 
                 0.312 
                 0.235 
                 0.188 
                 0.156 
                 0.134 
                 0.117 
                 0.104 
                 0.094 
               
               
                   
               
             
          
         
       
     
         [0080]    Referring to Table 1, results obtained for the experimental track assembly are listed under hose separation factor, x=0.104. The track width 2f was 0.682 inches, the central angle (2θ) was 47.656 degrees, track height (h) was 2.43 inches, track width ratio (arc DF/2f) was 1.029 and the ratio of the radius of the ball to the radius of the hose was 1.986. 
         [0081]    In subsequent simulations, ball radius, a, and ratio, y, of ball radius to hose radius decreased from 0.844 to 0.088 and 1.986 to 0.207, respectively, with increase in hose separation factor, x, from 0.104 to 1.0 while holding hose radius and track gap constant. 
         [0082]    The calculated results predict that a hose separation factor in the range of from 0.104 to 0.9 will produce an operative result. That is, a ball of the indicated sizes will roll on the defined track from the first end to the second end without falling off the track. 
         [0083]    At hose separation factor x=1.0, a ball on the defined track will strike surface  11  as shown on  FIG. 1 . This conclusion is based on the negative value of distance “s” ( FIG. 5 , Table A). Also, at hose separation factor of x=1.0, points H and L on hoses  1  and  4  lie in the same straight line produced by alignment of the hose diameters and the ball diameter. The ball could not roll (k=0; e=2g); it is thus believed that the ball would fall through the gap between the hoses ( FIG. 5 ). 
       Example 2 
       [0084]    Example 1 was repeated with the following exceptions. Each brace member employed in the structure is described in connection with reference numeral  18  of  FIG. 4 . The track gap in each brace was fixed at about 1.000 inch. Accordingly, as shown in  FIG. 5 , hose separation, c, was about 0.500 inches. 
         [0085]    The golf ball had an outside diameter of about 1 and 11/16 (1.6875) inches. Accordingly, referring to  FIG. 5 , radius, a, of the ball was about 0.84375 inches. The hose separation factor (x) was, therefor, about 0.593 (Table B, x=c/a) 
         [0086]    Calculations employing the mathematical relationships disclosed in Table B were made to simulate results for five hose separation factors (x) ranging from 0.593 to 1.0. The brace employed in the track structure was constant ( FIG. 4 ), accordingly, hose separation, c, was constant at 0.500 and hose radius, b, was constant at 0.425. Ball radius for each hose separation factor, x, was calculated by the formula, a=c/x. The results of the calculations are provided in Table 2. 
         [0000]    
       
         
               
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 2 
               
               
                   
               
               
                 Term 
                   
                   
                   
                   
                   
               
               
                 X 
                 0.593 
                 0.7 
                 0.8 
                 0.9 
                 1.0 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 a 
                 0.844 
                 0.714 
                 0.625 
                 0.556 
                 0.500 
               
               
                 b 
                 0.425 
                 0.425 
                 0.425 
                 0.425 
                 0.425 
               
               
                 y 
                 1.986 
                 1.680 
                 1.471 
                 1.307 
                 1.176 
               
               
                 c 
                 0.500 
                 0.500 
                 0.500 
                 0.500 
                 0.500 
               
               
                 e 
                 2.701 
                 2.700 
                 2.700 
                 2.700 
                 2.700 
               
               
                 sin θ 
                 0.729 
                 0.812 
                 0.881 
                 0.943 
                 1.000 
               
               
                 θ 
                 46.803 
                 54.292 
                 61.763 
                 70.562 
                 90.000 
               
               
                 2θ 
                 93.605 
                 108.583 
                 123.526 
                 141.123 
                 180.000 
               
               
                 cos θ 
                 0.685 
                 0.584 
                 0.473 
                 0.333 
                 0.000 
               
               
                 2f 
                 1.231 
                 1.159 
                 1.101 
                 1.048 
                 1.000 
               
               
                 r 
                 0.578 
                 0.417 
                 0.296 
                 0.185 
                 0.000 
               
               
                 k 
                 1.155 
                 0.833 
                 0.591 
                 0.370 
                 0.000 
               
               
                 2g 
                 2.471 
                 2.540 
                 2.599 
                 2.652 
                 2.700 
               
               
                 t 
                 0.266 
                 0.297 
                 0.329 
                 0.371 
                 0.500 
               
               
                 s 
                 0.025 
                 −0.049 
                 −0.128 
                 −0.229 
                 −0.500 
               
               
                 h 
                 2.138 
                 1.804 
                 1.547 
                 1.307 
                 0.925 
               
               
                 b + s 
                 0.450 
                 0.376 
                 0.297 
                 0.196 
                 −0.075 
               
               
                 h/e 
                 0.791 
                 0.668 
                 0.573 
                 0.484 
                 0.343 
               
               
                 arc DF 
                 1.379 
                 1.353 
                 1.347 
                 1.368 
                 1.571 
               
               
                 seg 
                 3.846 
                 3.315 
                 2.914 
                 2.551 
                 2.000 
               
               
                 arc DF/ 
                 1.120 
                 1.167 
                 1.224 
                 1.305 
                 1.571 
               
               
                 2f 
               
               
                 R 
                 0.625 
                 0.529 
                 0.463 
                 0.412 
                 0.370 
               
               
                   
               
             
          
         
       
     
         [0087]    Referring to Table 2, results obtained for the experimental track assembly are listed in the column headed x=0.593. Track width (2f) was 1.231 inches, central angle (2θ) was 93.605 degrees, track height (h) was 2.138 inches and track width ratio (arc DF/2f) was 1.120. 
         [0088]    Ball radius, a, and ratio, y, of ball radius to hose radius decreased from 0.844 to 0.500 and 1.986 to 1.176, respectively, with increase in hose separation factor, x, from 0.593 to 1.0 while holding hose radius and track gap constant. 
         [0089]    The calculated results indicate that a hose separation factor in the range of from 0.593 to 0.9 will produce an operative result. That is, a ball of the indicated sizes will roll on the defined track from the first end to the second end without falling off the track. 
         [0090]    Brace  18  ( FIG. 4 ) does not include a surface similar to surface  11  as shown on  FIG. 1 . Accordingly, the ball will not strike such a surface  11  even if the value of distance “s” ( FIG. 5 , Table A) is negative. At hose separation factor x=1.0 the value of the quantity “b+s” (Table A) is negative. Accordingly, ball  16  will strike the bottom of spacer element  28 . Furthermore, at hose separation factor x=1.0, points H and L ( FIG. 5 ) on hoses  1  and  4  lie in the same straight line produced by alignment of the hose diameters and the ball diameter. The ball could not roll (k=0; e=2g); it is thus believed that the ball would fall through the gap between the hoses ( FIG. 5 ). 
       Example 3 
       [0091]    This example simulates a track assembly comprised of two flexible garden hoses. Example 2 was repeated with the following exceptions. Each brace member employed in the structure is described in connection with reference to numeral  18  of  FIG. 4 . The track gap in each brace is fixed at about 1.000 inch. Accordingly, as shown in  FIG. 5 , hose separation, c, was about 0.500 inches. 
         [0092]    The balls employed had simulated outside diameters ranging from 10 inches, at hose separation factor 0.05, to 0.500 inches at hose separation factor 1.0. (a=c/x, Table B) 
         [0093]    Calculations employing the mathematical relationships disclosed in Table B were made for eleven hose separation factors (x) ranging from 0.05 to 1.0. The results of the calculations are provided in Table 3. The brace employed in the track assembly was constant ( FIG. 4 ), accordingly, hose separation, c, was constant at 0.500 and hose radius, b, was constant at 0.425. Ball radius was calculated by the formula, a=c/x. 
         [0000]    
       
         
               
               
             
               
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 3 
               
             
             
               
                   
               
               
                 Term 
                 Hose Separation Factor 
               
             
          
           
               
                 X 
                 0.05 
                 0.1 
                 0.2 
                 0.3 
                 0.4 
                 0.5 
                 0.6 
                 0.7 
                 0.8 
                 0.9 
                 1.0 
               
               
                   
               
             
          
           
               
                 a 
                 10.000 
                 5.000 
                 2.500 
                 1.667 
                 1.250 
                 1.000 
                 0.833 
                 0.714 
                 0.625 
                 0.556 
                 0.500 
               
               
                 b 
                 0.425 
                 0.425 
                 0.425 
                 0.425 
                 0.425 
                 0.425 
                 0.425 
                 0.425 
                 0.425 
                 0.425 
                 0.425 
               
               
                 y 
                 23.529 
                 11.764 
                 5.882 
                 3.921 
                 2.941 
                 2.353 
                 1.960 
                 1.680 
                 1.471 
                 1.307 
                 1.176 
               
               
                 c 
                 0.500 
                 0.500 
                 0.500 
                 0.500 
                 0.500 
                 0.500 
                 0.500 
                 0.500 
                 0.500 
                 0.500 
                 0.500 
               
               
                 e 
                 2.700 
                 2.700 
                 2.700 
                 2.700 
                 2.700 
                 2.700 
                 2.700 
                 2.700 
                 2.700 
                 2.700 
                 2.700 
               
               
                 sin θ 
                 0.089 
                 0.171 
                 0.316 
                 0.442 
                 0.552 
                 0.649 
                 0.735 
                 0.812 
                 0.881 
                 0.943 
                 1.000 
               
               
                 θ 
                 5.106 
                 9.846 
                 18.421 
                 26.232 
                 33.504 
                 40.466 
                 47.307 
                 54.292 
                 61.763 
                 70.562 
                 90.000 
               
               
                 2θ 
                 10.212 
                 19.692 
                 36.842 
                 52.463 
                 67.008 
                 80.932 
                 94.614 
                 108.583 
                 123.526 
                 141.123 
                 180.000 
               
               
                 cos θ 
                 0.996 
                 0.985 
                 0.949 
                 0.897 
                 0.834 
                 0.761 
                 0.679 
                 0.584 
                 0.473 
                 0.333 
                 0.000 
               
               
                 2f 
                 1.775 
                 1.705 
                 1.581 
                 1.474 
                 1.381 
                 1.298 
                 1.225 
                 1.159 
                 1.101 
                 1.048 
                 1.000 
               
               
                 r 
                 9.960 
                 4.925 
                 2.372 
                 1.495 
                 1.042 
                 0.761 
                 0.565 
                 0.417 
                 0.296 
                 0.185 
                 0.000 
               
               
                 k 
                 19.920 
                 9.850 
                 4.744 
                 2.990 
                 2.085 
                 1.521 
                 1.131 
                 0.833 
                 0.591 
                 0.370 
                 0.000 
               
               
                 2g 
                 1.925 
                 1.995 
                 2.119 
                 2.226 
                 2.319 
                 2.402 
                 2.474 
                 2.540 
                 2.599 
                 2.652 
                 2.700 
               
               
                 t 
                 0.040 
                 0.075 
                 0.128 
                 0.172 
                 0.208 
                 0.239 
                 0.268 
                 0.297 
                 0.329 
                 0.371 
                 0.500 
               
               
                 s 
                 0.383 
                 0.344 
                 0.275 
                 0.210 
                 0.147 
                 0.084 
                 0.021 
                 −0.049 
                 −0.128 
                 −0.229 
                 −0.500 
               
               
                 h 
                 20.808 
                 10.768 
                 5.700 
                 3.968 
                 3.072 
                 2.509 
                 2.112 
                 1.804 
                 1.547 
                 1.307 
                 0.925 
               
               
                 b + s 
                 0.808 
                 0.769 
                 0.700 
                 0.635 
                 0.572 
                 0.509 
                 0.446 
                 0.376 
                 0.297 
                 0.196 
                 −0.075 
               
               
                 h/e 
                 7.707 
                 3.988 
                 2.111 
                 1.470 
                 1.138 
                 0.929 
                 0.782 
                 0.668 
                 0.573 
                 0.484 
                 0.343 
               
               
                 arc DF 
                 1.782 
                 1.718 
                 1.608 
                 1.526 
                 1.462 
                 1.413 
                 1.376 
                 1.353 
                 1.347 
                 1.368 
                 1.571 
               
               
                 seg 
                 35.253 
                 18.282 
                 9.771 
                 6.862 
                 5.372 
                 4.448 
                 3.805 
                 3.315 
                 2.914 
                 2.551 
                 2.000 
               
               
                 arc DF/ 
                 1.004 
                 1.008 
                 1.017 
                 1.035 
                 1.059 
                 1.088 
                 1.123 
                 1.167 
                 1.224 
                 1.305 
                 1.571 
               
               
                 2f 
               
               
                 R 
                 7.407 
                 3.704 
                 1.852 
                 1.235 
                 0.926 
                 0.741 
                 0.617 
                 0.529 
                 0.463 
                 0.412 
                 0.370 
               
               
                   
               
             
          
         
       
     
         [0094]    Referring to Table 3, results obtained for the simulated track assembly are listed in the hose separation factors x=0.05 to x=1.0. The track width (2f) varied from 1.775 to 1.000 inches, central angle (2θ) varied from 10.212 to 180 degrees, track height (h) varied from 20.808 to 0.925 inches and track width ratio (arc DF/2f) varied from 1.004 to 1.571 inches. 
         [0095]    The results for hose separation factors 0.7, 0.8, 0.9 and 1.0 are identical to the results shown in Table 2. 
         [0096]    Ball radius, a, and ratio, y, of ball radius to hose radius decreased from 10.000 to 0.500 and 23.529 to 1.176, respectively, with increase in hose separation factor, x, from 0.05 to 1.0 while holding hose radius and track gap constant. 
         [0097]    The calculated results predict that a hose separation factor in the range of from 0.05 to 0.9 will produce an operative result. That is, a ball of the indicated sizes will roll on the defined track assembly from the first end to the second end without falling off the track. The calculated values of track height “h” and track width ratio suggest the requirement of a great deal of skill by a player to achieve a successful result for hose separation factors 0.05, 0.1 and 0.2. 
         [0098]    Brace  18  ( FIG. 4 ) does not include a surface similar to surface  11  as shown on  FIG. 1 . Accordingly, the ball will not strike such a surface  11  even if the value of distance s ( FIG. 5 , Table A) is negative. At hose separation factor 1.0 the value of the quantity “b+s” (Table A) is negative. Accordingly, ball  16  will strike the bottom of spacer element  28 . Furthermore, at hose separation factor of 1.0, points H and L ( FIG. 5 ) on hoses  1  and  4  lie in the same straight line produced by alignment of the hose diameters and the ball diameter. The ball could not roll (k=0; e=2g); it is thus believed that the ball would fall through the gap between the hoses ( FIG. 5 ). 
       Example 4 
       [0099]    This example simulates a track assembly comprised of two flexible garden hoses. Each garden hose has an outside diameter of about 0.85 inches. Accordingly, referring to  FIG. 5 , the radius, b, of each hose is about 0.425 inches. The track gap varied from about 0.176 inches to about 1.688 inches. Accordingly, as shown in  FIG. 5 , hose separation, c, varied from about 0.088 to about 0.844 inches. 
         [0100]    The garden hoses are positioned side-by-side to form a track assembly. A golf ball is positioned on the upper portion (top) of the track assembly on one of the two hoses and another point on the ball contacted a point on the second of the two hoses. 
         [0101]    The golf ball had an outside diameter of about 1 and 11/16 (1.6875) inches. Accordingly, referring to  FIG. 5 , the radius, a, of the ball was about 0.84375 inches. 
         [0102]    Calculations employing the mathematical relationships disclosed in Table B were made for ten different hose separation factors (x) as follows: 0.104, 0.2, 0.3, 0.4, 0.5. 0.593, 0.7, 0.8, 0.9 and 1.0. The results of the calculations are provided in Table 4. The ball radius, hose radius and, therefore, ratio, y, of ball radius to hose radius are held constant for each set of calculation. The hose separation for each set of calculations was calculated by the formula, c=ax. 
         [0000]    
       
         
               
               
             
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 4 
               
             
             
               
                   
               
               
                 Term 
                 Hose Separation Factor 
               
             
          
           
               
                 X 
                 0.104 
                 0.2 
                 0.3 
                 0.4 
                 0.5 
                 0.593 
                 0.7 
                 0.8 
                 0.9 
                 1.0 
               
               
                   
               
             
          
           
               
                 a 
                 0.844 
                 0.844 
                 0.844 
                 0.844 
                 0.844 
                 0.844 
                 0.844 
                 0.844 
                 0.844 
                 0.844 
               
               
                 b 
                 0.425 
                 0.425 
                 0.425 
                 0.425 
                 0.425 
                 0.425 
                 0.425 
                 0.425 
                 0.425 
                 0.425 
               
               
                 y 
                 1.985 
                 1.985 
                 1.985 
                 1.985 
                 1.985 
                 1.985 
                 1.985 
                 1.985 
                 1.985 
                 1.985 
               
               
                 c 
                 0.088 
                 0.169 
                 0.253 
                 0.338 
                 0.422 
                 0.500 
                 0.591 
                 0.675 
                 0.759 
                 0.844 
               
               
                 e 
                 1.876 
                 2.038 
                 2.206 
                 2.375 
                 2.544 
                 2.701 
                 2.881 
                 3.050 
                 3.219 
                 3.388 
               
               
                 sin θ 
                 0.404 
                 0.468 
                 0.534 
                 0.601 
                 0.667 
                 0.729 
                 0.800 
                 0.867 
                 0.934 
                 1.000 
               
               
                 θ 
                 23.828 
                 27.905 
                 32.276 
                 36.942 
                 41.836 
                 46.803 
                 53.130 
                 60.112 
                 69.067 
                 90.000 
               
               
                 2θ 
                 47.656 
                 55.809 
                 64.552 
                 73.883 
                 83.672 
                 93.605 
                 106.260 
                 120.224 
                 138.134 
                 180.000 
               
               
                 cos θ 
                 0.915 
                 0.884 
                 0.845 
                 0.799 
                 0.745 
                 0.685 
                 0.600 
                 0.498 
                 0.357 
                 0.000 
               
               
                 2f 
                 0.682 
                 0.790 
                 0.902 
                 1.014 
                 1.126 
                 1.231 
                 1.351 
                 1.463 
                 1.575 
                 1.688 
               
               
                 r 
                 0.772 
                 0.746 
                 0.713 
                 0.674 
                 0.629 
                 0.578 
                 0.506 
                 0.420 
                 0.301 
                 0.000 
               
               
                 k 
                 1.544 
                 1.491 
                 1.426 
                 1.348 
                 1.257 
                 1.155 
                 1.013 
                 0.840 
                 0.602 
                 0.000 
               
               
                 2g 
                 1.369 
                 1.585 
                 1.811 
                 2.036 
                 2.261 
                 2.471 
                 2.712 
                 2.937 
                 3.162 
                 3.388 
               
               
                 t 
                 0.072 
                 0.098 
                 0.131 
                 0.170 
                 0.215 
                 0.266 
                 0.338 
                 0.424 
                 0.543 
                 0.844 
               
               
                 s 
                 0.317 
                 0.277 
                 0.228 
                 0.170 
                 0.102 
                 0.025 
                 −0.083 
                 −0.212 
                 −0.391 
                 −0.844 
               
               
                 h 
                 2.429 
                 2.390 
                 2.341 
                 2.282 
                 2.214 
                 2.137 
                 2.030 
                 1.901 
                 1.722 
                 1.269 
               
               
                 b + s 
                 0.742 
                 0.702 
                 0.653 
                 0.595 
                 0.527 
                 0.450 
                 0.343 
                 0.213 
                 0.034 
                 −0.419 
               
               
                 h/e 
                 1.295 
                 1.173 
                 1.061 
                 0.961 
                 0.870 
                 0.791 
                 0.705 
                 0.623 
                 0.535 
                 0.375 
               
               
                 arc DF 
                 0.702 
                 0.822 
                 0.951 
                 1.088 
                 1.232 
                 1.378 
                 1.565 
                 1.770 
                 2.034 
                 2.651 
               
               
                 seg 
                 7.554 
                 6.451 
                 5.577 
                 4.873 
                 4.303 
                 3.846 
                 3.388 
                 2.994 
                 2.606 
                 2.000 
               
               
                 arc DF/ 
                 1.029 
                 1.041 
                 1.054 
                 1.073 
                 1.094 
                 1.120 
                 1.158 
                 1.210 
                 1.291 
                 1.571 
               
               
                 2f 
               
               
                 R 
                 0.900 
                 0.828 
                 0.765 
                 0.711 
                 0.663 
                 0.625 
                 0.586 
                 0.553 
                 0.524 
                 0.498 
               
               
                   
               
             
          
         
       
     
         [0103]    Referring to Table 4, results obtained for the simulated track assembly are listed in the hose separation factors x=0.104 to x=1.0. The track width 2f varied from 0.682 to 1.688 inches, the central angle 2θ varied from 47.656 to 180 degrees, track height h varied from 2.429 to 1.269 inches and the track width ratio (arc DF/2f) varied from 1.029 to 1.571 inches. 
         [0104]    Ball radius, a, hose radius, b, and, therefore, ratio, y, of ball radius to hose radius were held constant. Track gap 2c increased from 0.176 to 1.688 with increase in hose separation factor. 
         [0105]    The calculated results predict that a hose separation factor in the range of from 0.104 to 0.9 will produce an operative result. That is, a ball of the indicated size will roll on the defined track assembly from a first end to a second end without falling off the track. 
         [0106]    This simulation did not include a specific brace. Accordingly, to achieve the results of this example, a brace which maintains hoses having a specific radius (such as disclosed in  FIGS. 1 and 4 ) while avoiding a center element (such as disclosed in  FIG. 1 ) and having movable open rings should be provided. This simulation does not include a surface similar to surface  11  as shown on  FIG. 1 . Accordingly, the ball will not strike an element such as surface  11  even if the value of distance s ( FIG. 5 , Table A) is negative. At hose separation factor 1.0 the value of the quantity “b+s” (Table A) is negative. Accordingly, a ball could strike the bottom of a spacer, such as spacer element  28 . Furthermore, at hose separation factor of 1.0, points H and L ( FIG. 5 ) on hoses  1  and  4  lie in the same straight line produced by alignment of the hose diameters and the ball diameter. The ball could not roll (k=0; e=2g); it is thus believed that the ball would fall through the gap between the hoses ( FIG. 5 ).