Answer: O=2, N=3, E=6, T=4, W=7, S=5, V=9, I=1
... so that ONE + ONE = TWO is 236 + 236 = 472,
the prime number SEVEN is 56963,
and the perfect square NINE is 3136=56^2.
Solution: By scanning the squares of the number 32 to 99, we find that there
are only 4 four-digit squares which match the pattern NINE -- that is, only
4 of these squares have the same first and third digits. These are 2025,
3136, 6561, and 8281. This means that N is one of 2, 3, 6, or 8, but since
SEVEN is to be prime, N must be odd. Thus, N=3, I=1, and E=6.
Then immediately we have O=2, since 2*E must end in the digit O ("oh"), so
that W=7 and T=4.
Finally, the number SEVEN is _6_63, with the first blank chosen from 5, 8,
or 9 (the only remaining nonzero digits), and the second blank chosen from
0, 5, 8, or 9. Searching a table of primes can reveal which of these 9
possible numbers is prime; alternatively, use a TI-92 and the factor command
(or corresponding methods on a similar tool):
factor(56063+{0,800,900})
reveals that 56963 is prime; the other two are composite;
factor(86063+{0,500,900})
reveals that all three are composite;
factor(96063+{0,500,800})
reveals that all three are composite.
(If leading zeros were allowed, we could have S=0 and V=5 or 8, since both
06563 and 06583 are prime.)
There are no solutions base 8, 9, 11, 12, 14, 15, or 20. Here are the
solutions in bases 10, 13, and 16 through 19 (a=10, b=11, etc.):
one two nine seven base 10: 236 472 3136 56963 base 13: 589 b45 8089 29398 base 13: 589 b45 8089 29798 base 13: 589 b45 8089 39c98 base 13: 589 b45 8089 79298 base 16: 231 462 3931 51d13 base 16: 231 462 3931 71e13 base 16: 231 462 3931 81013 base 16: 231 462 3931 81d13 base 16: 231 462 3931 a1713 base 16: 231 462 3931 b1f13 base 16: 231 462 3931 c1013 base 16: 231 462 3931 c1513 base 16: 231 462 3931 d1b13 base 16: 231 462 3931 d1e13 base 16: 231 462 3931 e1713 base 16: 231 462 3931 f1b13 base 17: 139 271 3a39 49g93 base 17: 139 271 3a39 d9f93 base 17: 139 271 3a39 f9b93 base 17: 1c9 381 cfc9 2959c base 17: 1c9 381 cfc9 29b9c base 17: 1c9 381 cfc9 59e9c base 17: 1c9 381 cfc9 69b9c base 17: 1c9 381 cfc9 7909c base 17: 1c9 381 cfc9 79g9c base 17: 1c9 381 cfc9 b929c base 17: 1c9 381 cfc9 e959c base 17: 462 8c4 6b62 12g26 base 17: 462 8c4 6b62 72a26 base 17: 462 8c4 6b62 e2326 base 17: 462 8c4 6b62 e2726 base 17: 462 8c4 6b62 e2d26 base 17: 462 8c4 6b62 g2126 base 17: 4e2 9b4 e6e2 1282e base 17: 4e2 9b4 e6e2 5282e base 17: 4e2 9b4 e6e2 7202e base 17: 4e2 9b4 e6e2 8252e base 17: 4e2 9b4 e6e2 a232e base 17: 4e2 9b4 e6e2 a252e base 17: 4e2 9b4 e6e2 a2f2e base 17: 4e2 9b4 e6e2 c232e base 17: 4e2 9b4 e6e2 c272e base 17: 4e2 9b4 e6e2 f2c2e base 17: 4e2 9b4 e6e2 g232e base 17: 864 gc8 6064 24746 base 17: 864 gc8 6064 24b46 base 17: 864 gc8 6064 54246 base 17: 864 gc8 6064 54a46 base 17: 864 gc8 6064 a4546 base 17: 864 gc8 6064 e4b46 base 18: 25a 4b2 515a 3aha5 base 18: 25a 4b2 515a 6a7a5 base 18: 25a 4b2 515a 6a8a5 base 18: 25a 4b2 515a 6aca5 base 18: 25a 4b2 515a 7a0a5 base 18: 25a 4b2 515a 7ada5 base 18: 25a 4b2 515a 7afa5 base 18: 25a 4b2 515a 7aga5 base 18: 25a 4b2 515a 8aca5 base 18: 25a 4b2 515a 8aea5 base 18: 25a 4b2 515a ca6a5 base 18: 25a 4b2 515a da0a5 base 18: 25a 4b2 515a da9a5 base 18: 25a 4b2 515a eafa5 base 18: 25a 4b2 515a fa3a5 base 18: 25a 4b2 515a fada5 base 18: 25a 4b2 515a faha5 base 18: 25a 4b2 515a ga9a5 base 18: 25a 4b2 515a gaha5 base 18: 25a 4b2 515a hada5 base 18: 25a 4b2 515a haga5 base 19: 2b1 532 bhb1 4161b base 19: 2b1 532 bhb1 41e1b base 19: 2b1 532 bhb1 41i1b base 19: 2b1 532 bhb1 71f1b base 19: 2b1 532 bhb1 8141b base 19: 2b1 532 bhb1 81a1b base 19: 2b1 532 bhb1 81g1b base 19: 2b1 532 bhb1 91d1b base 19: 2b1 532 bhb1 c1g1b base 19: 2b1 532 bhb1 e1a1b base 19: 2b1 532 bhb1 e1g1b base 19: 2b1 532 bhb1 f1d1b base 19: 2b1 532 bhb1 g1c1b base 19: 2b1 532 bhb1 i141b base 19: 2d1 572 dad1 6141d base 19: 2d1 572 dad1 61g1d base 19: 2d1 572 dad1 8101d base 19: 2d1 572 dad1 b1h1d base 19: 2d1 572 dad1 c141d base 19: 2d1 572 dad1 e161d base 19: 2d1 572 dad1 h1b1d base 19: 2d1 572 dad1 h1f1d base 19: 2d1 572 dad1 i181d base 19: 3gb 7e3 g6gb 4bdbg base 19: 3gb 7e3 g6gb 5b4bg base 19: 3gb 7e3 g6gb 5babg base 19: 3gb 7e3 g6gb 5bcbg base 19: 3gb 7e3 g6gb 9b0bg base 19: 3gb 7e3 g6gb fb0bg base 19: 3gb 7e3 g6gb fb8bg base 19: 3gb 7e3 g6gb fbibg base 19: 3gb 7e3 g6gb hb0bg base 19: 3gb 7e3 g6gb ibhbg base 19: 814 g28 1714 64a41 base 19: 814 g28 1714 64e41 base 19: 814 g28 1714 94h41 base 19: 814 g28 1714 b4341 base 19: 814 g28 1714 b4941 base 19: 814 g28 1714 b4f41 base 19: 814 g28 1714 d4f41 base 19: 814 g28 1714 e4c41 base 19: 814 g28 1714 f4541 base 19: 814 g28 1714 h4941 base 19: 864 gc8 6964 24346 base 19: 864 gc8 6964 54046 base 19: 864 gc8 6964 54a46 base 19: 864 gc8 6964 54i46 base 19: 864 gc8 6964 d4a46 base 19: 864 gc8 6964 e4146 base 19: 864 gc8 6964 i4b46 base 19: 864 gc8 6964 i4f46