Combinatorial Puzzles

These puzzles have a combinatorial or probabilistic basis. They involve finding the number of ways something can be done, or how likely something is to happen. Basically, you just need to be able to count things carefully.

1. Coloured Cube

How many distinct ways are there to colour the six faces of a cube using six different colours, and applying a different colour to each face. Cubes which could be rotated to look the same as each other are not counted as distinct.

Multicoloured Cube

2. Snap!

I have two identical packs of cards. I take each pack and shuffle it separately, placing them both face down in the table. I then proceed to play a game of snap with myself. (Being a mathematician I have no friends.) I compare the top card from each pile, and then compare the second card from each pile, ... and so on. What is the probability that I continue in this way all the way down the piles, and never find an exact match in the 52 pairs of cards?

3. A Birthday Shared...

Given 23 randomly selected people, what is the probability that at least two of them have their birthday on the same day of the year?

[You may assume that none of them was born on 29th February, and that any other day of the year is equally likely to occur as a birthday. ]

4. Take your seats

A group of n passengers are waiting to board an aeroplane. The flight is full so there are exactly n seats on board, and each passenger has a specific numbered seat on their ticket. The passengers board and take their seats one at a time in a random order. The first passenger to board doesn't realise that the seats are allocated and just chooses one of the n seats at random. The other passengers all attempt to sit in their allocated seat when their turn comes. If they find their seat is empty, they sit in it. If it is already occupied, then rather than making a fuss, they just choose another empty seat at random and sit there instead. When the final passenger comes to sit down, there will be exactly one seat left. What is the probability that it is the seat he or she was allocated on their ticket?