Three secret numbers, two questions

Question

I am thinking of three positive whole numbers x, y and z. You may, twice, choose three whole numbers a, b, c (any size, positive or negative), and I will tell you the value of ax + by + cz. Choose the two questions so that you can always determine x, y and z. Explain why the second question depends on the answer to the first.

Accepted Answer

First ask with a = b = c = 1 to learn S = x + y + z. Then ask with a = 1, b = S + 1, c = (S + 1) squared. The answer, written in base S + 1, has digits z, y, x. Why. From the first answer, each of x, y and z is at most S (they are positive, so each is less than S in fact). The second answer is x + y(S + 1) + z(S + 1) squared. Since x, y, z are each between 0 and S, this is exactly the number whose base-(S + 1) digits are z, y, x. Reading the digits off gives all three unknowns. Example. Suppose the first answer is 12. Ask for x + 13y + 169z and receive 1,238. In base 13: 1,238 = 7 × 169 + 4 ×…

Three secret numbers, two questions

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