Eric Angelini <bk263401@skynet.be> wrote:
this game (hope not old hat) is played on a Scrabble board where all squares are white, and no square has any text on it. The starting "star" square, in the center of the grid, has a 7 on it (the number, not the word, as this game is played with digits only that will form numbers ? instead of letters forming words).
The single player must now form prime numbers at every turn on the board ? at the first turn placing the integer 1, at the second turn placing 2, then 3, then 4, then 5, etc.
Interesting. But why not keep as many rules of Scrabble as possible? Two alternating players, non-blank board, random selection of seven tiles, allowed to play any number of tiles in a straight line, no checking the "dictionary" during a game unless to challenge the opponent's alleged word or prime, etc.? The main problem with this game, as with yours, is that very few people have memorized vast numbers of primes. Your proposed game inspired me to look into building up primes one digit at a time, with every partial result being a prime. I learned that numbers built only leftwards are called left-truncatable primes (A024785), and numbers built only rightwards are called right-truncatable primes (A024770). There are exactly 4260 left-truncatable primes, the largest of which is 35768631264621656762913 (25 digits). (That's in base 10. In base 24, there are about a billion of them.) There are exactly 83 right-truncatable primes, of which the largest is 73939133. Fifteen primes are both left- and right-truncatable; the largest is 739397. The largest prime of which every substring is a prime is 373. But what I'm really looking for is a prime from which any number of digits can be removed, one at a time, from either the left or the right, in any order, without ever giving a non-prime. A137812, with 149677 terms, ending with 8939662423123592347173339993799, seems to be claiming to be such a sequence, but it includes terms that end with 1 or with 9, which are of course not prime. Nor is 99, which that largest term ends with, nor is 8, which it begins with, so I must be misundertanding what that sequence is. Is my proposed sequence not yet in OEIS? Since it could end with any of its digits, every digit must be 2, 3, 5, or 7, and only the leftmost digit could be 2 or 5. On second thought, that's not what I'm looking for, since I'm building primes, not taking them apart, and I can build them in any (gap-free) order. So maybe A137812 is what I'm looking for. But if so, it's phrased rather unclearly.