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A binary lot is an object that, when cast, comes to rest with 1 of 2 distinct faces uppermost. These can range from precisely-machined objects like modern which produce balanced results (each side coming up half the time over many casts), to naturally-occurring objects like cowrie shells which may produce a range of unbalanced results depending upon the species, individual, and even circumstances of the cast.
Binary lots may be used for divination, impartial decision-making, gambling, and game playing, the boundaries of which (as David Parlett suggests) can be quite blurred. They may be cast singly, yielding a single binary outcome (yes/no, win/lose, etc.), but often they are cast multiply, several in a single cast, yielding a range of possible outcomes.
The coin flipping game now known as Heads or Tails is ancient, going back at least to classical Greece, where Aristophanes knew it as Artiasmos, and classical Rome, where it was known as Caput aut Navis ('Head or Ship'), the two images on either side of some Roman coins. In the medieval period, various nations stamped various images on their coins, so that Italians played Fiori o Santi ('Flower or Saint'), Spaniards played Castile or Leon, Germans played Wappen oder Schrift ('Weapon or Writing'), and the French played Croix ou Pile ('Cross or Reverse').
Whereas most of these terms describe the images stamped on both sides, both the earlier English Cross and Pile (equivalent to the French, above) and the current English Heads or Tails describe only one side. Pile does not describe what is pictured: it merely indicates 'the reverse side'; likewise Tail indicates 'the side opposite the head'.
For centuries, coin tosses have served both as complete games, and as preliminaries to actions in other games: as early as the 1660s Francis Willughby notes Cross & Pile being played by children as an independent game, but also cases in which Cross & Pile is used to determine who takes a turn first in other games.
Coins are commonly used in I Ching divination (although the tallying of Achillea alpina (yarrow) stalks is the older method). The usual method involves casting three coins to generate each of the six lines of a hexagram. Historically, Chinese coins had only one marked side (stamped with writing), and in this procedure it is regarded as yin and given a numerical value of 2, while the unmarked reverse is yang and given a value of 3. The sum of the values of the three cast coins will be between 6 and 9; an even sum means one of the six lines of the hexagram is yin, while odd means yang, with equal probabilities. The cast simultaneously gives a second binary result with unequal probabilities: The sums 7 and 8 mean the line is "young", where as the less likely sums 6 and 9 mean the line is "old" and about to change to its opposite.
The oracular text Lingqijing is consulted using 12 wooden disks, strictly, Xiangqi pieces made from a lightning-struck tree; unsurprisingly, other congruent objects such as home-made disks, wooden checkers, and coins are normally substituted. The 12 disks comprise 4 each of 3 types (say, 4 quarters, 4 nickels, and 4 pennies), so that a single cast is equivalent to 3 differentiated casts of 4 undifferentiated lots, yielding 1 of 125 possible outcomes (=(4+1)3).
The majority of games documented use 3 or 4 staves, though H. J. R. Murray notes games requiring as many as 8. Liubo in fact means '6 rods', which is the number of staves employed in the game (though 18-faced dice were sometimes substituted).
Various species of cowrie are used as dice for a variety of board games in India, perhaps most prominently in the traditional Indian game of Pachisi. Here, either 6 or 7 cowries are cast simultaneously, resulting in a single move value, depending upon the number landing mouth up.
In owo mȩrindinlogun, a form of Yoruba people divination, 16 cowries are cast, yielding 1 of 17 possible outcomes, each of which is "associated with memorized verses which contain myths and folktales that aid in their interpretation".
Some sets of the Royal Game of Ur, dating from the mid–3rd millennium BCE, include roughly regular Tetrahedron (4-faced) dice with 2 vertices marked, and 2 vertices unmarked.
When the lots are undifferentiated, then n lots produce n+1 possible outcomes: thus, casting 4 staves yields 1 out of 5 (=4+1) possible outcomes. These outcomes are defined by the number of marked faces uppermost, but the value of these outcomes may differ from the simple count of marked faces. For example, in the modern Egyptian board game Tâb, the following schedule is used:
This schedule is typical of most board games using multiple-binary casts in that: 1) the move values are based on, but modified from, the simple count of marked faces, and 2) it is the more extreme counts (which are statistically rarer, see below) that are bumped up in value.
When the lots are differentiated, then n lots produce 2 possible outcomes: thus casting 4 distinct Hakata divination tablets yields 1 out of 16 (=2) possible outcomes.
These methods are not always strictly exclusive. Several Native American board games make use of 3 staves, only 1 of which is differentiated, resulting in 6 possible outcomes — midway between the 4 if undifferentiated, and the 8 if fully differentiated. The most common coin-based method of I Ching divination begins with a cast of 3 undifferentiated coins (4 possible results), but utilizes 6 casts (differentiated by order) to produce a complete hexagram, representing 1 of 4096 (=(3+1)) possible outcomes.
| + All possible casts |
| 2 |
| 1 |
| 1 |
| 0 |
Here, there are 4 possible casts, but these yield only 3 outcomes, which have unequal odds, higher for the central outcome(s) and lower for the extreme outcomes: 2 = 25% and 1 = 50% and 0 = 25%. This pattern holds for all casts of undifferentiated binary lots, as shown below:
The graphical flattening can be deceptive: using 2 lots, the central (most common) outcome is 2 times as likely as the extreme outcome. But using 8 lots, the central (most common) outcome is 70 times more likely than the extreme outcome. Using cubic dice (or any dice with more than 2 faces) flattens this curve somewhat, making the odds more even, as shown below:
David Parlett notes: "Cubes have always tended to oust binaries where both are known, probably because they are more convenient, but perhaps also because they bring the rarer numbers more frequently into play."
The odds of "mouth up" for each cowrie may vary by species, individual, and even casting method. During tumbling, a mouth-up cowrie will have an unstable base and high center of gravity, increasing the likelihood of more tumbling; conversely, a mouth-down cowrie will have a stable base and a low center of gravity, increasing the likelihood of coming to rest. The likelihoods of the 7 possible outcomes can be compared between hypothetical cases in which the mouth-up probabilities are 1/3 versus 2/5 versus 1/2:
At first glance, it appears that uneven odds will make for an extremely slow game. However, Pachisi, like most games calling for binary lots, rewards the extreme throws more than the central throws, for example in this schedule, which Murray asserts to be the most common:
If the 4 extreme outcomes are collectively considered the "good" ones, then uneven odds actually increase the chances of a "good" cast (where the bulk of the gain is from 1 mouth up):
| 22% |
| 27% |
| 37% |
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