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Numbered Discs

Problem

Two discs have four different numbers written on them.


As the discs are spun their totals are added together, e.g. 5 + 8 = 13. By spinning the discs the totals 10, 11, 13, and 14 are obtained.

Use this information to work out which numbers are on the other side of the discs.


Solution

Let the numbers on the first disc be 5 and a (on the reverse), and the second disc be 8 and b. As we already know that 5 + 8 = 13, there are three algebraic totals: a + b, a + 8, and 5 + b, that need to be matched, in some order, with: 10, 11, and 14.

The remaining three possible totals add to 10 + 11 + 14 = 35.

So (a + b) + (a + 8) + (5 + b) = 2a + 2b + 13 = 35,
therefore 2a + 2b = 22, hence we know that a + b = 11.

This leaves two possible totals for 5 + b: 10 and 14. As all of the numbers are different, 5 + b = 14, leading to b = 9 and a = 2.

That is, the numbers 2 and 5 are on the first disc and 8 and 9 are on the second disc.

What if the totals were 11, 12, 13, and 14?

Problem ID: 157 (Mar 2004)     Difficulty: 2 Star

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