Problem 360

Solution 1

The integers 1 to 32 are spaced evenly and in order around the outside of a circle.

Consider drawing a first straight line that passes through the centre of the circle and joins any one pair of these 32 numbers.

This leaves $32-2=30$ numbers still to be paired.

Since this first line passes through the centre of the circle, it divides the circle in half.

In terms of the remaining 30 unpaired numbers, this means that 15 of these numbers lie on each side of the line drawn between the first pair.

Let the number that is paired with 12 be $n$.

If we draw the line through the centre joining 12 and $n$, then there are 15 numbers that lie between 12 and $n$ (moving in either direction, clockwise or counter-clockwise).

Beginning at 12 and moving in the direction of the 13, the 15 numbers that lie between 12 and $n$ are the numbers $13,14,15,\dots,26,27$.

Therefore, the next number after 27 is the number $n$ that is paired with 12.

The number paired with 12 is 28.

Solution 2

We begin by placing the integers 1 to 32, spaced evenly and in order, clockwise around the outside of a circle.

As in Solution 1, we recognize that there are 15 numbers on each side of the line which joins 1 with its partner.

Moving in a clockwise direction from 1, these 15 numbers are $2,3,4,\dots,15,16$, and so 1 is paired with 17, as shown.

[[IMAGE0]]

Since 2 is one number clockwise from 1, then the partner for 2 must be one number clockwise from 17, which is 18.

Similarly, 12 is 11 numbers clockwise from 1, so the partner for 12 must be 11 numbers clockwise from 17.

Therefore, the number paired with 12 is $17+11=28$.