Here are two classic questions about dominoes, with variations for each.

1. The classic: Imagine a standard 8×8 chess board. Remove the opposite corners of the board; now there are 62 spaces. Can you cover the board with 31 dominoes, if each domino covers exactly two spaces?

Variations:

- What is the removed corners are adjacent?
- What if you remove 4 spaces at each corner of the board?
- Go back to a full chess board; it can be easily covered with 32 dominoes. When I did this, I found that 16 were placed vertically, and 16 horizontally. Can you place them in any other combination?

2. The classic: Imagine a strip 8 squares long, covered with dominoes and monominoes (monominos are also called ‘squares’).

In this figure, the strip is covered by two brown dominoes and four green monominoes. You could, of course, just use four dominoes, or 1 domino and 6 monominoes, etc. In how many ways can this you cover the strip using only these figures?

Variations:

- What if the strip were 100 squares long?
- What if you had brown and blue dominoes, and yellow and green monominoes?
- What if you had monominoes, dominoes, and triominoes (formed by connecting three squares in row)?
- What if these figures were made by linking together equilateral triangles rather than squares?

The variations are endless!

PS. Online math circles start next week. Learn more here!

And in case you missed it:

Math Circles at Home Take 1: 1d Tic-Tac-Toe

Math Circles at Home Take 2: Knight Moves