For a box of width w, height h, and depth d, what is the best sized sheet of wrapping paper to cut out to minimise wasted paper?

Work out the sum of all the positive whole number from 1 to 100 as shown.

There's a cool trick for doing this quickly, see if you can figure it out!

How would you do this up to some arbitrary number 'n'?

This problem has some relevance to an important concept in the quantum mechanics of many particle systems. Where particles being ‘identical’ carries a much stronger meaning than in our everyday usage of the word. This can go a way to helping us explain the electron shell model of the atom. (But enough physics for now)

Suppose I have two pairs of dice. Pair number 1 consists of two completely identical blue dice that we will call dice 1A and dice 1B, with pair number 2 being one blue die called 2A and one red die called 2B.

When I roll either of the pairs of dice, I do so with my eyes closed, opening my eyes only once the dice have settled.

Find the probability that the sum of the number landed on by dice 1A and 1B in pair 1 is 6. Do the same for dice 2A and 2B in pair 2.

Now consider the case where we roll each pair separately, and they land such that the sum over the two dice is 6. Find the probability that the dice landed in the configuration 1A = 2, and 1B = 4. Repeat this, but for pair 2, with 2A = 2 and 2B = 4.

(Hint: Pay close attention to that fact that the dice in pair 1 are identical. Would we be able to distinguish 1A = 2 with 1B = 4 and 1A = 4 with 1B = 2?)

Imagine you have 100 light bulbs in a row, all initially off. Each is labelled in order with a number 1 to 100.

Now suppose you flick the switch on all the bulbs whose number is a multiple of 1 (i.e. all the bulbs are now on), then you flick all the multiples of 2 (so now 2,4,6,8,... are off again), then 3, and so on until you get up to 100.

Which bulbs will be left on after this process is finished? (Don’t try and do this by going through and working it out with brute force! Try and think of a way of explaining this for any number, potentially much bigger than 100!)