Solve this alphametic, where each of the capital letters in bold represents a different decimal digit from 0 to 9. Each of S and T is nonzero.

√(SEE) = √(THE) + √( TOP)

SQRT(SEE) = SQRT(THE) + SQRT(TOP)

As SEE, THE, and TOP are all integers, there must be an even number of each distinct prime factor for the composite number of (THE)(TOP).
It is already given that neither S nor T can be 0, and as the value of (300 + 2*[SQRT(300)]*[SQRT(300)] + 300) is greater than a 3-digit number, T must be < 3.
One can try the various permutations of the pairs of distinct prime factors that will result in the correct composite. As it is, the only solution that works for this problem is where (THE)(TOP) = 2⁸*3⁴. With further trial, it is found that
THE = 2²*3³ = 108
SQRT(THE) = 6*SQRT(3)
TOP = 2⁶*3¹ = 192
SQRT(TOP) = 8*SQRT(3)

SQRT(SEE) = SQRT(108) + SQRT(192)
SQRT(SEE)² = [SQRT(108) + SQRT(192)]²
SEE = 108 + 2*[6*SQRT(3)]*[8*SQRT(3)] + 192
SEE = 108 + 288 + 192 = 588

E H O P S T
8 0 9 2 5 1

Posted by Dej Mar on 2009-05-21 22:02:07