## 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`) = 2`a` + 2`b` + 13 = 35,

therefore 2`a` + 2`b` = 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?