### Which Twin Is Older?

A simplified account of special relativity and the twins paradox.

### Snookered

In a snooker game the brown ball was on the lip of the pocket but it could not be hit directly as the black ball was in the way. How could it be potted by playing the white ball off a cushion?

### Spokes

Draw three equal line segments in a unit circle to divide the circle into four parts of equal area.

# Scale Invariance

### Why do this problem?

This problem offers a fascinating exploration into probability density functions for real world data. Whilst the individual steps are quite simple, the problem draws together many strands from distribution theory. The results can be tested on any set of data from any geography book, giving an interesting relevance to the mathematics.

### Possible approach

The first obstacle to overcome is that of notation: can the students understand what is being asked?

The question involves little computation but requires clear thinking of the ideas. This might be facilitated in a group discussion, but might also require individual work.

### Key questions

• If a function is to be a probability density function, what is the major property it must possess?
• What ranges of values will start with a digit $1$?

### Possible extension

Consider carefully why this problem involves 'scale invariance'. Consider the restriction of scale invariance on real world data. Which sets of real world data do you think will be modelled by this distribution? Why?

### Possible support

Skip the first part and provide students with the scale invariant functions. Also, first use the range 1< x < 10 in the last part of the question.