Polynomials

Where is the monomino?

• Imagine we tile a $4 \times 4$ board with five $3 \times 1$ pieces and a single $1 \times 1$ piece, a monomino. What are the possible monomino positions?
• What are the possible monomino positions if we tile a $5 \times 5$ board with eight $3 \times 1$ pieces and a monomino?

Two dice

The possible sums of two standard dice (with numbers $1, \dots, 6$ on their faces) are $2, 3, \dots, 12$ which occur with frequencies $$1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1.$$ Do there exist two non-standard dice with positive integers on their faces such that the sum of the two has the same frequencies as above?

(Hint: If this sounds complicated, try starting with tetrahedral dice first.)