## The brainbuster

*Find the solution to this brainbuster and win £200 of Amazon vouchers to spend on high-tech!*

It's time for the ACME Ltd 'Spring into Summer' staff event. Amongst the refreshments, there's a 10 litre punchbowl filled to the brim with ACME's legendary punch. Now, a jug (capacity J litres) is filled from the bowl, and the punchbowl topped up with water. Assume there's no spillage, and the punchbowl's contents mix completely.

Then the process is repeated once more - the jug is filled with liquid from the punchbowl, and the bowl topped up with water.

Unfortunately for latecomers to the event, the proportion of punch to water in the punchbowl is now 1:1.

What is the capacity of the jug?

*Show your workings. Only complete entry forms will be considered. The judge's decision is final.*

**The competition closed on Sunday 22nd July 2018.**

## The entries

There was again a good number of entries for this competition. Nearly all entrants worked out the right answer, although one somehow managed to come up with 2.5.

## The solution

A solution as provided by our winner was:

The amount of punch left in the bowl after the nth dilution step is

p_n = p_{n-1} - p_{n-1}*J/B

where J and B are the capacities of the jug and bowl respectively. From this it follows that

p_n = p_0 (1 - J/B)^n

where p_0 is the original amount of punch.

Now with the provided data p_2 = 5L and p_0 = B = 10L we have

J = B*(1-1/sqrt(2)) = **2.93 litres**

## And the winner is...

**Jonas Verschueren, studying for a PhD at Imperial College London, is the winner of ecm's Brainbuster no. 43 competition.**

Jonas wins £200 of Amazon vouchers to spend on high-tech. Congratulations again to him, and watch out for **ecm**'s next brainbuster!