### Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

### Doodles

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

### Russian Cubes

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

# Knight Defeated

##### Stage: 4 Challenge Level:

Christina sent us her solution:

A knight can't make a tour on a $2\times n$ board, for any $n$, because it must go into and out of a corner square, and it can't do this without going back on itself.

On the $3\times 4$ grid, we must use a path from the loop JAGIBHJ and a path from the loop KDFLCEK. But they only link up between J and C, and between B and K. So the path must start at a neighbour of J, B, K or C, follow round that loop, switch to the other loop and follow round that. Obviously the path can go round the loop in either direction. So there are $16$ possible tours:

HJAGIBKDFLCE

HJAGIBKECLFD

HBIGAJCEKDFL

HBIGAJCLFDKE

AGIBHJCEKDFL

AGIBHJCLFDKE

IGAJHBKECLFD

IGAJHBKDFLCE

ECLFDKBIGAJH

ECLFDKBHJAGI

EKDFLCJHBIGA

EKDFLCJAGIBH

DFLCEKBIGAJH

DFLCEKBHJAGI

LFDKECJAGIBH

LFDKECJHBIGA

Since it's not possible to get from the finish directly back to the start in any of these tours, there is no circuit.