smc

Question

A cryptographer devises the following method for encoding positive integers. First, the integer is expressed in base 5. Second, a 1-to-1 correspondence is established between the digits that appear in the expressions in base 5 and the elements of the set V, W, X, Y, Z. Using this correspondence, the cryptographer finds that three consecutive integers in increasing order are coded as VYZ, VYX, VVW, respectively. What is the base-10 expression for the integer coded as XYZ?

Accepted Answer

Since VYX + 1 = VVW, i.e. adding 1 causes the "fives" digit to change, we must have X = 4 and W = 0. Now since VYZ + 1 = VYX, we have X = Z + 1 Z = 4 - 1 = 3. Finally, note that in VYX + 1 = VVW, adding 1 will cause the "fives" digit to change by 1 if it changes at all, so V = Y + 1, and thus since 1 and 2 are the only digits left (we already know which letters are assigned to 0, 3, and 4), we must have V = 2 and Y = 1. Thus XYZ = 413_5 = 4 5^2 + 1 5 + 3 = 100 + 5 + 3 = 108, which is answer 108.