# Gender ratio

A country is preparing for a possible future war. The country s tradition is to send only males into battle and so they want to increase the proportion of males to females in the population through regulating births. A law is passed that requires every married couple to have children and they must continue to have children until they have a male.

What effect do you expect this law to have on the makeup of the population?

(Thanks to Wilbur.)

**Solution**

A bit of a trick question, this, and open to plenty of interesting discussion.

The obvious answer is that there is no effect on the gender ratio. However, this would only be true under certain assumptions about the distribution of the sexes of offspring among couples. Consider a population in which each couple can only ever have boys or only ever have girls. Those who have boys could stop after one child, whereas those who have girls can never stop having children, with the end result being more girls than boys. (Of course, this might not matter since the goal is for there to be more males, there is no requirement on the number of females.) And if there is any autocorrelation between births this will also have an impact. If autocorrelation is 1, so that a male child is always followed by a male, and a female by a female, then the ratio of males to females decreases, but with a negative correlation the ratio increases.

# Covering a chessboard with dominoes

You have a traditional chessboard, eight by eight square. From a single diagonal, any diagonal, you remove two squares. The board now has just 62 squares. You also have 31 domino tiles, each of which is conveniently the same size as two of the chessboard squares. Is it possible to cover the board with these dominoes? (Thanks to alphaquantum.)

**Solution**

No, since a domino always occupies a white and a black square! If you remove two from the same diagonal then they will have the same colour, leaving you with 32 of one colour and 30 of the other, so it is impossible to cover two squares.

# Aircraft armour

Where should you reinforce the armour on bombers? You can t put it everywhere because it will make the aircraft too heavy. Suppose you have data for every hit on planes returning from their missions, how should you use this information in deciding where to place the armour reinforcement? (Thanks to Aaron.)

**Solution**

The trick here is that we only have data for aircraft that survived. Since hits on aircraft are going to be fairly uniformly distributed over all places that are accessible by gunfire one should place the reinforcements at precisely those places which appeared to be unharmed in the returning aircraft. They are the places where hits would be 'fatal. This is a true Second World War story about the statistician Abraham Wald who was asked precisely this.

# Hanging a picture

You have a framed picture with a string attached to it in the usual manner. You have two nails on the wall. The problem is to try and hang the picture on the wall such that if you remove either one of the nails then the frame falls down. (Thanks to wannabequantie.)

**Solution**

Here's one solution:

It's quite simple to 'mathematize' this problem as follows. Use *x* to denote wrapping once around the first nail in the clockwise direction, with x2 meaning wrap the string around the first nail twice and, crucially, x-1 means wrapping anticlockwise around the first nail. Similarly *y* etc. for the second nail. To solve this problem you need an expression involving products of *x* s and *y* s and their inverses which is not the identity (for the 'identity means no wrapping and the picture falls!) but such that when either the *x* or the *y* are replaced with the identity (i.e. removed!) the result becomes the identity! (You have that multiplication by the identity leaves *x* and *y* unchanged, that xx-1 = x-1 *x* = 1, that *xy* = *yx* and that 1-1 = 1.)

One such solution is *xy* x-1 y-1. Check the maths and then draw the picture.

The above picture is represented by *xy-1 x-1 y.*