# Sunday Afternoon Maths XXI

**Posted on 2014-07-20**

## Wool circles

\(n\) people stand in a circle. The first person takes a ball of
wool, holds the end and passes the ball to his right, missing a
people. Each person who receives the wool holds it and passes the
ball on to their right, missing \(a\) people. Once the ball returns to
the first person, a different coloured ball of wool is given to
someone who isn't holding anything and the process is repeated. This is
done until everyone is holding wool.
For example, if \(n=10\) and \(a=3\):

In this example, two different coloured balls of wool are needed.

In terms of \(n\) and \(a\), how many different coloured balls of
wool
are
needed?

## Sum equals product

\(3\) and \(1.5\) are a special pair of numbers, as \(3+1.5=4.5\)
and
\(3\times 1.5=4.5\) so \(3+1.5=3\times 1.5\).

Given a number \(a\), can you find a number \(b\) such that
\(a+b=a\times b\)?

If you enjoyed these puzzles, check out Advent calendar 2019,

puzzles about people maths, or a random puzzle.

puzzles about people maths, or a random puzzle.