Fresh from regular slots on Radio 4’s Puzzle for Today, Imperial’s best minds have set you the ultimate prize puzzle challenge...

## 1: HARD

I have a rectangular bar of chocolate consisting of four rows of three small chocolate squares each. Being of a generous disposition I want to divide the bar into the 12 small squares to give to my friends. I can break the bar in many different ways. For example, I might start by breaking it into two 2x3 bars or perhaps three 1x4 bars and then subdivide the resulting pieces. At each stage a cut is made along a single straight line between some of the small squares until all 12 squares of chocolate are separated (no stacking of one piece on top of another is allowed as it makes too much of a mess!).

Is there a method for doing this that involves the least number of such cuts?

From: Dr Lynda White, Principal Teaching Fellow in Experimental Design in the Department of Mathematics

However you break the chocolate you will take eleven cuts to separate the twelve squares of chocolate because at each stage you increase the number of pieces of chocolate (which may consist of more than one square) by one until they are all separated.

## 2: VERY HARD

A general decides to count the number of soldiers in his battalion. When he tries to line the soldiers up in threes, he finds that he has one soldier left over. He then tries to line them up in fives and there are three soldiers left over. Finally, he attempts to line them up in rows of seven and there are two soldiers left over. How many soldiers are in the battalion? Is this the only answer?

From: Melissa Lee, Research Postgraduate, Department of Mathematics

The answer is any number of soldiers of the form 58+105k, where k is an integer greater than or equal to zero (eg, the first few valid answers are 58, 163, 268 and 373).

## 3: FIENDISH

A prime is a whole number only divisible by itself and 1, so the lowest primes are 2, 3, 5, 7, 11, etc. The number of primes thins out quite dramatically when you go higher – there are 25 primes between 0 and 100, 16 primes between 1,000 and 1,100 and just 11 between 10,000 and 10,100. But do they run out altogether or are there infinitely many? And how can you be sure?

From: Dr Daniel Mortlock, Lecturer in Astro-Statistics, Department of Physics

There are infinitely many primes, even though the fraction becomes tiny for larger numbers. The simplest way to see that there is no largest prime number is to assume that this is the case and then show that this leads to a contradiction.

If you multiply all the prime numbers, from 2 up to the hypothetical largest prime, and then add 1 to this product you’ll have found a number that is not divisible by any of this list of primes. But that means that it must be a prime number itself, as it is clearly larger than the hypothetical largest prime that hypothesis is contradicted.

So, there must be infinitely many primes (not that large primes are easy to find: in late 2017 researchers working with the Great Internet Mersenne Prime Search (Gimps) collaboration showed that 2^77,232,917 - 1, a number with 23,249,425 digits, to be a prime.)

### Prize puzzlers

Entries to our puzzle competition closed on Friday 27 July 2018.

Senders of the first ten correct solutions for two or more of the puzzles will receive a copy of Professor Jonathan Haskel and Stian Westlake's new book, Capital Without Capitalism: The Rise of the Intangible Economy. Details of the winners will be printed in Imperial 45 in November and on the magazine homepage from August 2018.