Cubic and Square Numbers and Multiples of 9 — Four in a Row Game and Sum of Cubes = Triangular Squares

After studying square and cubic numbers, to bring it all together, I created a game that combines these two sequences of numbers and introduces a third sequence of numbers: multiples of 9. These are numbers into which 9 is divisible such as 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108… The rule for this sequence of numbers is to add the digits of the number and if that sum is a multiple of 9, then the number is a multiple of 9. For example, each of the numbers listed above has a digital sum of 9 except for 99 which has a digital sum of 18, still a multiple of 9. 111,111,111 is a multiple of 9 since the digits add up to 9. It is also helpful to recognize square numbers that are also cubic numbers such as 1, 64, 729, 4096, 15,625 and so on. 1 is 1 squared and 1 cubed; 64 is 8 squared and 4 cubed; 729 is 27 squared and 9 cubed; and 4096 is 64 squared and 16 cubed. 729 is the first number that is a square, a cube and a multiple of 9. Knowing which number are both squares and multiples of 9 is important to this game such as 9, 36, 81, 144, 225 and so on; also cubic and multiples of 9 such as 27, 216, 729, 1728 and so on. 

The game is played with a spinner using a paper clip and a pencil. The object of the game is to get four in a row so the children will need two different counters such as coins, buttons, candy, etc. The directions and game boards (3 different ones provided) are in the attached pdf. To help the children recognize square, cube and multiples of 9, I created these three lists of numbers in a building with three apartments: apartment 1D (multiples of 9), apartment 2D (squares), and apartment 3D (cubes). 1D is one dimensional with just a door, 2D is two dimensional with a flat square apartment, and 3D is three dimensional with a cubic structure. The children also received a blank game board to create their own. 

Finally, the 5th-7th graders received an additional challenge to add consecutive cubic numbers starting with 1 cubed which will always yield the square of the sum of those base numbers. For example, 1 cubed plus 2 cubed plus 3 cubed (1 + 8 + 27) equals the square of 1 + 2 + 3 = 6 squared or 36. The worksheet provided challenges them to prove that this works for any nth number.

The older children were also tasked to find an algorithm to find numbers that are both square and cubic. Since square numbers are to the second power, and cubic numbers to the third power, they were able to discover that these numbers are to the sixth power (2 x 3 = 6). So 1, 64, 729, 4096, 15,625 are 1 to the sixth power, 2 to the sixth power, 3 to the sixth power, 4 to the sixth power, 5 to the sixth power, and 1,000,000 is 10 to the sixth power. 

The Square, Cubic, Multiples of 9 Game introduces probability since the spinner lands on 1 of 3 possibilities so they have to determine if the number they desire has a 33.3% chance of occurring, 66.6% chance of occurring, 100% % chance of occurring, or 0% chance of occurring.