## Tuesday, December 8, 2009

### Dissecting a tetrahedron

Let $T_r$ denote a regular tetrahedron whose side length is $r$.

1. Prove that it is impossible to assemble eight $T_1$ into a $T_2$.

2. Prove that a tetrahedron can be cut into four optically congruent pieces. Two solids $A$ and $B$ are considered optically congruent if either $A$ or its mirror image can be rotated into $B$.

3. Prove that a $T_2$ can be cut into four $T_1$ and a regular octahedron whose side length is 1.