# Maths problems for mock university interviews

Thursday 9th December 2021

Mike Baxter, an Area Coordinator for the AMSP North West regional team and experienced Oxbridge maths interview coach, shares two nice maths problems which can be used for mock university interviews.

## The Bean-counter

An urn contains 21 white beans and 38 black beans.

A clever monkey takes 2 beans.

If they're the same colour, he puts a black bean back, and if they're different, he puts a white bean back.

There is a spare supply of white beans, if needed, and the monkey repeats this process until there is one bean is left.

What colour is it?

If the monkey takes two blacks, the exchange is $$-$$ $$BB + B = - B$$

If the monkey takes one of each, the exchange is $$–$$ $$BW + W = - B$$

If the monkey takes two whites, the exchange is $$-$$ $$WW + B =$$ $$(B$$ $$–$$ $$2W)$$

This implies that any selection leads to an even drop in number of whites.

We started with an odd number of whites, so the last pair must be BW and the final pick results in the return of a white.

## How many ways?

How many ways are there to pick a subset of numbers from the set {$$1, 2, 3, 4, 5, 6, 7, 8, 9, 10$$} that does not include any consecutive numbers?

Remember the empty set {-}

Calling this $$S(10)$$, start with $$S(1) =$$ no. subsets with $$1$$ $$+$$ no. subsets without $$1$$

$$S(1) = 1 + 1 = 2$$

$$S(2) = 1 + 2 = 3$$

$$S(3) = 2 + 3 = 5$$

$$S(4) = 3 + 5 = 8$$

$$S(5) = 5 + 8 = 13$$

$$S(6) = 8 + 13 = 21$$

$$S(7) = 13 + 21 = 44$$

$$S(8) = 21 + 34 = 55$$

$$S(9) = 34 + 55 = 89$$

$$S(10) = 55 + 89 = 144$$