Assume that SQUARE, CARRE, and QUADRAT are alphametic squares, that is, that their component letters (accents being ignored) can be replaced by a mutually consistent, non-duplicating set of base-10 digits, with no leading zeroes. Prove that there is in fact no set of numbers by which these letters can be so replaced.

Alternatively, provide a solution to the alphametic.

A complete search shows no solution for all three terms (under 3 seconds execution time).  Even for the first two alone (CARRE and SQUARE) there are only 12 pairs:

20449, 573049

46225, 180625

46225, 390625

52441, 687241

64009, 815409

70225, 164025

70225, 198025

70225, 416025

70225, 483025

70225, 648025

70225, 819025

70225, 893025

Posted by ed bottemiller on 2010-09-20 17:39:04