Skip over navigation
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.
Knight Defeated
Age 14 to 16
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.
You may also like
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?
Polycircles
Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?