Binary lot

Unlike most binary lots — which are typically cast multiply affording a variety of possible outcomes — coins are most often cast (flipped or spun) singly, resulting in a simple yes/no, win/lose outcome. Both the lot and its outcome are binary. Further, a coin's two sides are very nearly symmetrical, so that they can each be expected to appear reasonably close to 50% of the time, unlike cowries, half-round staves, and some other forms of binary lots.

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,^[2] and classical Rome, where it was known as Caput aut Navis^[3] ('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').^[4][2]

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'.^[4]

• 
  Roman coin, displaying caput, 'head' (of Nero) and navis, 'ship'
• 
  English half groat featuring Edward III on the pile
• 
  U. S. Washington quarter which, as it happens, includes a tail on its tail

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.^[5]

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.^[6] 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 Ling Ch'i Ching is consulted using 12 wooden disks, strictly, Chinese Chess pieces made from a lightning-struck tree;^[7] unsurprisingly, other congruent objects such as home-made disks, wooden checkers, and coins are normally substituted.^[8][9] 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).

Staves,^[a] lengths of wood (also cane, bone, or other materials) typically semicircular in section, are found in many regions and time periods, being used as randomizers (for example) in many Native American board games (of which the Kiowa game Zohn Ahl is often used as an exemplar),^[17] in the ancient Egyptian Senet^[18] as well as the modern Egyptian Tâb,^[19] in the ancient Chinese Liubo,^[20] and the ancient — and still current — Korean Yunnori.^[21] They are easy to make, usually being formed simply by splitting a stick in half lengthwise, though additional finishing or decoration is often applied.

The majority of games documented use 3 or 4 staves, though H. J. R. Murray notes games requiring as many as 8.^[22] Liubo in fact means '6 rods', which is the number of staves employed in the game (though 18-faced dice were sometimes substituted).^[23]

The shells of cowries, sea snails of the family Cypraeidae, often function as lots. Their durable shell is rounded on one side. The other (flat) side features a long narrow aperture (commonly called the mouth) running from end to end, which the animal may emerge from and withdraw into.

Various species of cowrie are used as dice for a variety of board games in India,^[24] 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.^[25]

In owo mȩrindinlogun, a form of Yoruba 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".^[26]

Any object that may be cast to land distinctly on 1 of 2 sides may function as a binary lot.

In Games of the North American Indians, Stewart Culin provides descriptions and engravings of over 200 sets of binary lots. The majority are half-round staves, but other lots are fashioned from bone, stone, nut shell, fruit stone, corn kernel, mollusk shell, woodchuck and beaver tooth, claw, brass, and china, as well as wooden lots worked to shapes other than the typical half-round stave.^[27]

• A selection of Native American binary lots documented in Games of the North American Indians
• 
  Bone
• 
  Bone
• 
  Stick
• 
  Mollusk shell
• 
  Corn kernel
• 
  Walnut shell
• 
  Plum stone
• 
  Beaver tooth

The Biblical Urim and Thummim might have been binary lots, but their form and function remain unclear.^[28]

Divinatory use of binary lots in the form "four small rectangular or triangular tablets made out of wood, bone or ivory" is widespread in Southern Africa,^[29] likely originating with the Shona people some time before 1561.^[30] These are flat, or slightly lenticular in section. They are cast multiply, but unlike many sets of binary lots, they are each individually marked; thus these 4 tablets yield 16 possible outcomes, not 5 (as would, for example, 4 undifferentiated cowries).^[31]

Several West African divinatory traditions use divining chains featuring multiples of 4 ordered binary lots (often 8 or 16), in the form of half seed pods or half mango seeds, but also pieces of calabash, metal, or other objects.^[32] The most prominent is the Ifa divination of the Yoruba people, using an Opele (divining chain) featuring 8 lots, most commonly the pear-shaped half seed pods of Schrebera trichoclada (locally called Opele, whence the chain gets its name).^[33] Although the lots are visually similar, they are differentiated by position, being fixed to the chain and the chain being marked with a right and left side; therefore 1 cast of the chain yields 1 of 256 (=2⁸) possible outcomes, each of which is associated with memorized verses.^[34]

Any lot with more than two faces can function as a binary lot if all its faces are grouped into two sets. For example, a cubic die can deliver odds 1:1 like a fair coin if three faces are marked yes and the other three are marked no, or, with a die with normal pip markings, if one only observes whether there is a pip at the center (as on the faces 1, 3 and 5) or not. The dice game Bell and Hammer requires 8 cubic dice, each blank on five faces and featuring only a single marked face,^[35] each die thus delivering odds 1:5.

Some sets of the Royal Game of Ur, dating from the mid–3rd millennium BCE, include roughly regular tetrahedral (4-faced) dice with 2 vertices marked, and 2 vertices unmarked.^[36]