### Cushion Ball

The shortest path between any two points on a snooker table is the straight line between them but what if the ball must bounce off one wall, or 2 walls, or 3 walls?

### Retracircles

Four circles all touch each other and a circumscribing circle. Find the ratios of the radii and prove that joining 3 centres gives a 3-4-5 triangle.

### 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?

# Matrix Meaning

##### Age 16 to 18 Challenge Level:

This problem involves the algebra of matrices and various geometric concepts associated with vectors and matrices. As you consider each point, make use of geometric or algebraic arguments as appropriate. If there is no definitive answer to a given part, try to give examples of when the question posed is or is not true.

In the five questions below: $R, S$ are rotation matrices; $P, Q$ are reflection matrices; $M,N$ are neither rotations nor reflections. Consider each part in 2D and 3D.

1. Is it always the case that $M+N = N + M$?
2. It it always the case that $RS= SR$?
3. It it always the case that $RP= PR$?
4. It it always the case that $PQ= QP$?
5. Is it ever the case that $MN = NM$?

How do the values of the determinants of the various matrices affect the results of these questions?