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# An Amazing Coincidence

As we've seen, there are 56 possible visibly distinct combinations when three ordinary dice are thrown. Also, there are 21 possible visibly distinct combinations when two ordinary dice are thrown.

As it happens, the 78-card Tarot deck traditionally consists of a Fool, which is unnumbered, 21 trumps numbered from I to XXI, and 56 cards similar to an ordinary deck of playing cards, but with an additional rank of court card, the Knight.

This, in itself, is not a particularly amazing coincidence; it could mean that the Tarot was based on an ancient method of divination by dice, even if current serious historical research on the history of playing cards makes that appear unlikely.

However, there is a well-attested form of divination using astragali (the ankle bones of sheep, sometimes misnamed knuckle-bones) which were used as four-sided dice. Pillars on which the interpretations of the various throws were written had been unearthed at several locations within Asia Minor (Turkey).

A survey of the various discoveries of this nature is contained in the doctoral dissertation Würfel- und Buchstabenorakel in Griechenland und Kleinasien by Franz Heinevetter.

Five of them were thrown at once. As it happens, the number of visibly distinct combinations of five four-sided dice is also equal to fifty-six.

Three six-sided dice can come up in 216 possible ways that are equal in probability, and when they are visibly identical, these combinations are merged into 56 distinguishable combinations as follows:

``` 6 triples,    * 1 =   6
30 pairs,      * 3 =  90
20 singletons, * 6 = 120
--                   ---
56                   216
```

Five four-sided dice can come up in 1,024 possible ways that are equal in probability, and when they are visibly identical, these combinations are merged into 56 distinguishable combinations as follows:

``` 4 quintuples, *  1 =   4
12 AAAAB,      *  5 =  60
12 AAABB,      * 10 = 120
12 AAABC,      * 20 = 240
12 AABBC,      * 30 = 360
4 AABCD,      * 60 = 240
--                   ----
56                   1024
```

Also, apparently, at least according to some sources, the faces on the astragali were considered to have the numerical values 1, 3, 4, and 6, which suggests they were used (at least for some purposes, if not necessarily when they were used for divination) as substitutes for six-sided dice, which would have been more difficult to make and hence more expensive.

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