Why primes? A shared game across cultures

I believe that building this game around prime numbers is far better than making it depend on a particular language. Primality is universal: a number is prime regardless of the language we speak or the culture we come from. With the same decimal digits and rules, we can all share the same mathematical playing field, without needing to learn a language-specific word list.

Like Scrabble, Primebble gives you a seven-tile rack, a board with bonus squares, and the challenge of making intersecting sequences. Here the tiles are digits, and every new horizontal or vertical number of at least two digits must be prime. The browser checks this mathematically, rather than looking it up in a dictionary.

How large is the “dictionary”?

For this comparison, a Primebble “word” means an ordinary prime written in decimal with no leading zero. The number of n-digit primes is exactly π(10ⁿ) − π(10ⁿ⁻¹), where π(x) counts primes at most x.

The Spanish counts use FILE 2017, an official Spanish Scrabble lexicon. The English counts use TWL06, the 2006 North American tournament word list, as a reproducible historical reference. English Scrabble has different regional lexicons and editions; these figures are specific to TWL06, not to the current NASPA or Collins lists.

Counts are exact; ratios are rounded. On a narrow screen, scroll the table horizontally to see both comparisons.

The four one-digit primes are 2, 3, 5, and 7; this game requires numbers of at least two digits. The current board is 11 × 11, so lengths 12–15 are included for comparison only.

What matters for the seven-tile rack?

The crossover occurs between five and six symbols in both lexicons. For lengths two through five there are fewer primes than Spanish or English words. At six and seven digits the prime counts are already much larger.

• Spanish, lengths 2–7: 88,140 words.
• English, lengths 2–7: 53,901 words.
• Primebble, lengths 2–7: 664,575 primes — about 7.54× the Spanish count and 12.33× the English count.

At length seven alone there are 586,081 primes, compared with 48,609 Spanish words and 24,029 English words: about 12.1× and 24.4× as many, respectively.

This raises an interesting game-design question: does accepting every prime make long plays easier or more common? Dictionary size alone cannot answer it. Digits and letters have different tile distributions, many sequences require the same collection of tiles, and every crossing must also be valid. In fact, although the number of primes grows rapidly with length, the proportion of all numbers that are prime decreases.

I would like to study the probability that a random seven-tile rack admits a legal prime play, using the actual tile bag and board constraints, and compare that with Scrabble. That is likely to tell us much more about how the games feel than the size of their dictionaries alone.

Sources and counting conventions

• Spanish: the FILE 2017 word-length counts supplied for this comparison, computed from the published lexicon file.
• English: Peter Norvig’s published TWL06 text file. Counts were computed from distinct alphabetic entries, grouped by letter count: 178,691 entries of lengths 2–15 in total. For a newer North American edition, see NASPA’s NWL2023 description.
• Primes: differences of the exact PrimePages values of π(10ⁿ), with π(1) = 0. No approximation from the prime number theorem is used in the counts.
• Word length here means letters in the lexicon entries, not necessarily physical tiles: Spanish Scrabble also has CH, LL, and RR tiles. The 2–7 totals compare sequence lengths, rather than the probability of making a play from a rack.