This week's puzzle concerned the Quiz Master Shop sock drawer.
We only have black or white socks, and we put them, unpaired, in the communal sock drawer. There is something strange about the number of each colour of socks in the drawer - if we take out two socks at random, there is exactly a 50% chance that they are both black. How many socks of each colour do we have?
Once you've worked out a solution (quite quickly we hope) you will realise that we can't have that small a number of socks. There would not be enough to share between us. In fact the number of socks we have is in double figures. So now work out how many socks of each colour we have.
As a further puzzle, work out how many black socks and how many white socks we would need such that there is a 50% chance of three randomly selected socks all being black.
Is there another solution for this with higher numbers, in the same way as for two random socks?
Can you work out a general solution for four, five, six socks etc.?
The simple solution to the two socks problem is three black socks in a total of four socks. The probability of the first sock being black is 3 / 4 and the second is 2 / 3. Multiplying these gives 1 / 2 or 50%.
But we have more than four socks, and the next solution is 15 black socks out of a total of 21 socks. The probability of the first sock being black is 15 / 21 and the second is 14 / 20. Multiplying these gives 1 / 2 or 50%.
We can find no higher solutions, but perhaps you can - or prove that there are none.
For three socks you need five black socks in a total of six socks. The probability of the first sock being black is 5 / 6, the second is 4 / 5 and the third is 3 / 4. Multiplying these gives 1 / 2 or 50%.
Again we can find no higher solutions - are there any, or is there a proof that there are none?
Finally the general solution for taking N random socks from the drawer is 2N - 1 black socks in a total of 2N socks. That is always one white sock in a total of 2N socks. So to pick five socks, you would have nine black out of ten socks, and have probabilities of 9 / 10, 8 / 9, 7 / 8, 6 / 7 and 5 / 6. Everything cancels leaving 5 / 10 or 50%.