a) A piece of wire 120 cm long is given. Can a wire frame of a cube with an edge of 10 cm be made without breaking the wire?

b) What is the minimum number of times the wire will have to be broken to make the required frame?
a) If this were possible, the wire would run along the edges of the cube without overlapping, meaning we would have drawn the frame of the cube without lifting the pencil from the paper. However, this is impossible because the cube has eight odd vertices.

b) Since there are eight odd vertices, at least four such pieces are needed.

Four pieces are sufficient: for example, in the cube $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$, the wire can follow the broken line $A B C D A A^{\prime} B^{\prime} C^{\prime} D^{\prime} A^{\prime}$.

The remaining three edges $B B^{\prime}, C C^{\prime}, D D^{\prime}$ can be covered by three separate pieces of wire.

## Answer

a) It is impossible; b) three times.