Continuous Play:

In this variation, as soon as one player forms their shape, they must select a coin from their shape and move it to any other position on the board (not necessarily just one space or jumping another coin). Play then continues. Keep track of how many of their shapes each player has made. First to make 3 wins.

Solitaire Puzzles:

After kids are familiar with the game, have them investigate the following:

• How many different triangle positions are there on the game board? (4)
• How many different hexagon positions are there on the game board? (3)
• From the initial coin setup, what is the fewest number of moves needed to make a hexagon? (3)
• From the initial coin setup, what is the fewest number of moves needed to make a triangle (4)?
• Make a triangle, then place the remaining coin anywhere on the board. What is the fewest number of moves needed to make a different triangle? (As few as 2, depending on the setup.)
• Make a hexagon, then place the remaining coin anywhere on the board. What is the fewest number of moves needed to make a different hexagon? (As few as 3, depending on the setup.)
• Make a triangle, then place the remaining coin anywhere on the board. What is the fewest number of moves you can make to form a hexagon? (As few as 1, depending on the setup.)

Triangular Numbers:

The triangle formation above uses 6 coins, so 6 is one of the "triangular" numbers. Let kids try forming larger and smaller triangles with their coins to determine the sequence of triangular numbers: 1, 3, 6, 10, 15, 21, etc. (Note that the triangular numbers are "solid," without a hole in the middle.)

Triangular numbers are found arithmetically by adding consecutive counting numbers: 1 = 1, 3 = 1 + 2, 6 = 1 + 2 + 3, 10 = 1 + 2 + 3 + 4, and so on.

Hexagonal Numbers:

6 is also a "hexagonal" number. Let kids use their coins to find the hexagonal numbers: 1, 6, 15, 28. (Note that hexagonal numbers form only the outline of a hexagon, leaving a hole in the middle, and each hexagonal number is built by adding a larger hexagon to the previous.)

Can they find the list of hexagonal numbers within the list of triangular numbers? The first, third, fifth, and so on, triangular numbers are hexagonal numbers! This means that every hexagon can be formed into a triangle, but there are triangles that can't be formed into hexagons.

Children can explore other figurate numbers. Ask them to guess what the first few "square" numbers are.