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Sunday, 20 July 2014

Sunday Afternoon Maths XXI

Here's this week's collection. Answers & extensions here. Why not discuss the problems on Twitter using #SundayAfternoonMaths or on Reddit. This Tuesday is Maths Jam, make sure you're there!

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 $$a+b=a\times b$$?

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?