Toilet Paper Orientation

I would graciously accept your refreshments, however given the time and travel costs involved, I am afraid I will have to defer as to such time as we have developed cost-effective teleportation technology.I concur with your analysis of initial setup time.However, the distance to sheet is based highly upon the current state of the roll. You consider problems such as knuckle scraping, however this is a rare problem, entirely avoidable with proper technique, and seems to indicate a worst-case analysis (1) of the system. However, given that we wish to consider total expected expenditure over the lifetime of the system, we would do better to consider average-case analysis (2), and to do so we need to individually analyze each of the possible states of the roll, and furthermore what the probability of the roll being in those states is given either under or over configuration. I will discuss three common states: 9 formation (slightly unrolled), spaghetti formation (greatly unrolled), and 0 formation (entirely rolled).'9' formationThe most efficient state for the roll to be in, and hence the one the user should aim to leave the roll in after retrieving paper, is partially unrolled, with a small tail of paper hanging off. I shall henceforth refer to this as the '9' formation (which can either be 'over' or 'under' depending on what side you're looking at the '9' from). From 9 formation, the user need only pull a set amount of paper, then separate it. Separation can be done either by the two-handed tear method, by the one-handed angled rip across method, by the one-handed shift-and-rip method, or by the two-handed stop-and-tear method. Regardless of the method used, the difference in time spent retrieving paper given an over-configuration, versus an under-configuration, is insignificant. What can differ is the probability that the roll will be left in different formations.Two-handed tear methodThe two-handed tear method is characterized by grabbing two adjacent squares with separate hands and separating them meticulously along the perforation. This is the most time-consuming method, however has the advantage of providing a near-guarantee that the roll will return to 9 formation. If this method is used, it does not matter if the user has configured the roll to over or under.One-handed rip methodThe one-handed rip method is characterized by leaving the single grasping hand in the pulling location and using a quick and violent ripping motion to attempt to separate the squares. This is the fastest method, however has two disadvantages: firstly the number of squares removed is unreliable, and secondly the likelihood that the roll will be left in spaghetti formation is high. This is largely based on the ease with which the squares separate. Now, if the roll is in over configuration, the angle of the paper rip is less extreme since the point of separation from the roll is on the same side as the user's hand. Thus, over configuration is disadvantaged in that it is more likely to leave the roll in an undesired spaghetti formation.One-handed shift-and-rip methodLike the one-handed rip method, this is characterized by a violent one-handed ripping of the paper off the roll. However, this is preceded by first sliding the gripping hand up to the desired perforation of separation. This is more time consuming, but less likely to result in spaghetti formation. Regardless of if the roll is in an under or over configuration, the angle of rip is extreme enough to make differences in likelihood of producing this undesirable formation insignificant.Two-handed stop-and-tear methodThis method is characterized by pulling the paper out with one hand, then stopping the spinning of the roll with the other, and tearing off the paper with the pulling hand (possibly preceded by a shift-up). This method provides a good balance between speed and guaranteeing the roll does not go into spaghetti formation. It does, however, leave the roll in 0 formation with high frequency, regardless of if the roll is in over or under configuration. It is, nonetheless, the method that users should be using most of the time.Full analysisAlthough the differences between under and over configuration are slight when in 9 formation, it is never disadvantageous to be in under, and depending on what ripping method is used, can be bad to be in over configuration.Spaghetti formationThe least desirable state for a roll to be in, this is characterized by having an excessive amount of paper hanging off the roll, often puddling into a spaghetti-like pile on the floor. This is both wasteful of paper (as for sanitary reasons the excess often needs to be discarded), and for time purposes as it now requires two rips to retrieve a usable section of toilet paper from the roll. Generally, the user proceeds by resetting the roll to 9 formation with either a two-handed tear, or a two-handed stop-and-tear. The difference between over and under configuration in this state is negligible, it is only interesting to note that this state occurs more frequently from other states if the roll is in over configuration.'0' formationA frequent and undesirable state of the roll, this is characterized by the paper having no dangling material from the roll. A roll will often reset itself to 0 formation between bathroom visits by the user, even if it was last left in 9 formation. As discussed previously, it is a frequent result of using the two-handed stop-and-tear method from 9 formation. The user must now precede paper extraction by locating the tail-end of the paper, and can then proceed using the methods described in the 9 formation section. Thus, the likelihood that the roll will be left in various states mirrors the discussion in 9 formation. What differs is the time spent locating the end, based on the method used.One-handed spider crawlThis method is characterized by placing the fingers of one hand on the roll, then running them along the roll to make it spin until the end appears and grasping it. It is most efficient to do this such that when the end appears, the fingers rather than the thumb can slip under the edge and pull it off. If this is done under-handed palm facing up, that will occur in over configuration. If this is done over-handed with palm facing down, then that will occur in under configuration. Since the over-handed method allows greater visibility of the fingers (and thus the point of grasping relative to the edge of the roll), it is superior to have the roll in under formation and do an over-handed roll.Two-handed palm rollThis method is characterized by flattening one hand out and rolling it along the roll from fingertips to palm to spin it, while using the other hand to grasp the end when it appears. It is really only practical to do this with the rolling hand sliding down under the roll while the other hand grasps the end on top. When doing this, it is easier to grasp the end when the roll is in under configuration, because then the opening you wish to grasp is further away meaning you can grip it with your fingertips rather than your thumb as you would have to do in over configuration.Full analysisAs we have seen, regardless of method used, under configuration for the roll always results in faster retrieval time. This is largely due to the fact that it makes it easier to exploit the greater dexterity that fingertips have over thumbs.ConclusionWhile not every state and method results in improvements using under configuration for the roll, under configuration is never a bad thing, and often shows efficiency advantages. There is no reason to use over configuration. Each of these method/state combinations could merit further discussion, however in the interest of brevity I shall hold off on that pending which points you wish to contend.(1) http://en.wikipedia.org...(2) http://en.wikipedia.org...