English, French, and German (Posted on 2010-09-20)

	Submitted by broll
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Surprisingly, even though only 5 out of 9 different letters are repeated (with only A and R occurring in all 3 words) and C,S,D, and T are ‘wild’, there is no consistent alphametic that can be used to replace the remaining letters. The problem is soluble without a brute-force search. There are only 16 possible candidates for CARRE between 10000 and 99999; if these rules are then applied to those candidates: a. RULE 1: If the penultimate known digit is odd, the last digit is 6; b. RULE 2: If the last digit of SQUARE is 5, then A is 0,2, or 6; c. RULE 3: If the last digit of SQUARE is 1, then the last 3 digits are e01,e41, e81 if the number of hundreds is even, or o21,o61 if the number of hundreds is odd - corresponding rules apply if the last digit is 9; then 6 possibilities (excluding repeated digits, etc.) are left for CARRE/SQUARE. Possible values for T can then be looked up from a table of squares, mod 1000, and the formula x^2-ARE = 1000k, k<1000 used to narrow values for Q and U in QUADRAT, giving 16 'hold-outs' that can easily be checked manually.

	Subject	Author	Date
	exhaustive search	ed bottemiller	2010-09-20 17:39:04