 ## 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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