UNO + UNO + TRES = CINCO Solution: As UNO < 1000 and TRES < 10000 we have CINCO < 12000. Therefore C=1 and I=0. Now T is in {8, 9}. At position (1) we get O + S = 10. So {S,O} is {2,8}, {3,7}, or {4,6}. At position (2) we get 2N + E = 10 or 2N + E = 20. At position (3,4) we get 10T + 2U + R + (2N + E)/10 = 100 + N. That is 2U + R = 100 + N - 10T - (2N + E)/10. Now make the seach in order N, E, {S,O}, T, R, U. N E 2N+E {S,O} T 2U+R R U ---------------------------------------- 2 6 10 {3,7} 8 21 5 8- 9 6- 9 11 5 3- 9 1- 3 5 10 {2,8} 9 12 4 4- 6 3- 4 2 10 {3,7} 8 23 5 9 9 7- 9 13 5 4- 6 8 20 {3,7} 9 14 2 6- 4 5 7 6 20 {2,8} 9 15 3 6- 5 5- 8 4 20 {3,7} 9 16 2 7- 6 5 9 2 20 {3,7} 8 27 9- 9- 9 2 20 {4,6} 8 27 9- 9- The minus sign indicates that this value is in use already. So we get the following three pairs of solutions 943 + 943 + 8527 = 10413 23*41 + 23*41 + 8527 = 3^2*13*89 947 + 947 + 8523 = 10417 947 + 947 + 3^2*947 = 11*947 563 + 563 + 9487 = 10613 563 + 563 + 53*179 = 10613 567 + 567 + 9483 = 10617 3^4*7 + 3^4*7 + 3*29*109= 3*3539 583 + 583 + 9647 = 10813 11*53 + 11*53 + 11*877 = 11*983 587 + 587 + 9643 = 10817 587 + 587 + 9643 = 29*373 References: - Puzzle 179: Prime Alphametics http://www.primepuzzle.net/puzzles/puzz_179.htm - double true and magic square alphametics using prime numbers