24 points and six-sided dice: the mathematics of a backgammon board
Some Roman gaming boards had 36 playing spaces, arranged as six words of six letters. By the fifth century, the Byzantine emperor Zeno was playing on a board with the familiar 24 points, though he used three dice.
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A Roman marble gaming board described in the British Museum’s 1920 guide carries an inscription about a chariot-racing crowd: the venue is full and the noise is enormous. The words are arranged in three rows, with two words in each row. Each word has six letters, giving the board 36 playing spaces.
The lettering makes the arrangement easy to see. Before the familiar two rows of twelve points, related Roman games used three rows of twelve, already divided into groups of six.
Six words, six letters each
CIRCVSPLENVSCLAMORINGENSIANVAETENSAE
Schematic transcription of the British Museum board described in its 1920 guide. Each letter occupies one of the 36 spaces. The final word, shown with a dashed outline, follows the guide’s proposed restoration; this is not a photograph or a facsimile. British Museum, 1920.
From 36 spaces to 24 points
In his account of the Roman game duodecim scripta, the archaeologist Ulrich Schädler interprets the use of three dice and the later omission of the middle row as ways of speeding up play. His discussion draws on surviving boards, including one from Ostia, Rome’s ancient port. The playable rules he proposes include reconstruction where the evidence is incomplete.
The game described in Agathias’s poem about Zeno, a fifth-century Byzantine emperor, has a 24-point track and fifteen checkers per player, although there are three dice. The board’s familiar shape therefore belonged to games whose rules differed from ours. We can trace these layouts more confidently than we can recover an original decision to choose exactly twenty-four.
Six faces and six home points
On a modern board, the four groups of six have a clear relationship to the dice. A checker on the bar enters on one of the opponent’s six home points, according to a die’s number. At the other end of its journey, each of your own six home points has a matching face for bearing a checker off once all your remaining checkers are home.
Those rules put the home board within the range of a single die. If there were a seventh home point, it would have no matching die face for entry or exact bearing off. You would need another rule to say what happens there.
Why a six-point prime cannot be jumped
Two checkers together close a point to the opponent. Make several consecutive closed points and you have a prime, which can trap an opposing checker behind it. The crucial distance is from the checker to the first open point beyond the block.
A checker immediately behind five closed points can get across with a 6. Add a sixth closed point and the same escape requires a 7. You cannot add a 6 and a 1 into a single jump: each die is a separate move and each landing point must be open. Even double six supplies four separate sixes, each of which needs a legal landing.
The difference between five blocked points and six
A complete six-point prime takes at least twelve checkers, leaving three of your fifteen elsewhere. Those spare checkers can give you moves to spend while the prime stays intact. Once you have to move a checker out of one of its pairs, an escape may open. This helps explain the importance of timing: a block that stops everything today may be difficult to preserve through the next few rolls.
Why the last closed point matters so much
When the six closed points are all in your home board, an opposing checker on the bar has nowhere to enter. Before that last point closes, the chances change by much more than a simple one-sixth per point.
Suppose four home points are closed and the opponent has one checker on the bar. A roll fails to enter only when both dice name closed points, giving 4 × 4 = 16 failed rolls out of the 36 equally likely outcomes. The other 20 rolls allow entry. Close a fifth point and there are 5 × 5 = 25 failures, leaving just 11 entering rolls.
Close the home points and count the entering rolls
Ivory has one checker on the bar. Choose how many of Red’s home points are closed, then inspect all 36 rolls. A tick means at least one die lets that checker enter.
The highlighted points are closed. Numbers are from Ivory’s perspective: a 1 enters on 24, a 6 on 19. These are separate example positions, not successive turns. We count entry of the single bar checker, not the chance of winning the game.
Closing the fifth point removes nine of the rolls that could enter; closing the sixth removes the remaining eleven. The final point has the largest effect because either die could previously have supplied the one number that got the checker back in.
For this narrow question, the number of closed points determines the answer. Which particular points are closed will matter once the checker enters, especially if one roll hits a blot or leaves a useful anchor. With more than one checker on the bar, you also need to distinguish entering at least one from entering them all.
Does double six explain the 24 points?
There is an attractive numerical fit: double six provides four moves of six, totalling 24 pips if you can use them all. Under the right conditions, one checker can travel from 24 to 18, then 12, then 6, and finally off the board. The intermediate points must be open, and your other remaining checkers must be home before that last move.
The equality is real, while its role in choosing the board’s size remains unproven. The early 24-point game described in Schädler’s Zeno discussion used three dice, and the older 36-space boards were also organised in sixes. These are important limits on a story that derives the whole board from a modern 6–6.
What about 24’s many divisors?
Twenty-four has eight positive divisors: 1, 2, 3, 4, 6, 8, 12 and 24. You can divide it into equal halves, thirds or quarters, which makes it convenient for regular layouts. That observation alone does not establish a special balance between board size and dice; five, for example, does not divide twenty-four.
A different board length would change how far checkers travel and how players meet, so comparing its strategy with backgammon would require a specified set of rules. The surviving sources describe layouts and changes in play, but none of those cited here gives a calculation establishing twenty-four as the uniquely best board length.
