Hamse Adam, Mugisha Uwiragiye, and Dion Miller found full solutions, and Taija Williams a partial answer.

Two questions:

**The first question: is 3599 a prime?**

**Second question: a spider is at the centre of a base of a box which is a 1-metre cube. A fly is at one of the top corners. The spider can’t fly, but only walk on the surfaces inside the box. What is the shortest distance by which the spider can reach the fly?**

How do you answer them *quickly*?

**Solution:**

(1) 3599 = 60^{2}−1 = (60−1)×(60+1) = 59×61

therefore 3599 is not prime.

There is another way to do this. There is no quick method to prove that a number is prime (the security system for ATMs and similar depends on this fact). So, if 3599 were prime, there would be no way to prove it quickly. On the other hand, there are many quick ways to prove that a number is *not* prime, by just finding a factor. (For example, if the last digit is even, or 5, or the digits add up to a multiple of 3, the number is not a prime). Just because there is *some* quick method to answer the question, you know 3599 *must be not prime*, even if you don’t know what the quick method is!

(2) Imagine the top taken off the box and the sides dropped flat onto the floor. The shortest route now from the spider to the fly is a straight line, the hypotenuse of a triangle of sides ½ and 1½. That hypotenuse has length ½√10, by Pythagoras. But it’s still the shortest route if the sides were left up. So the answer is ½√10.

The picture below shows you how to work out the shortest distance in a slightly different problem, where the spider is at one corner of the base and the fly is at the opposite corner of the top of the cubical box.

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