Fultiples
Fultiples is a two player Tic-Tac-Toe-like game with a numeric twist. It is played on 10 x 10 grids and each cell of these grids contains a positive integer (in case you don’t know, positive integers are the numbers you count with: 1, 2, 3, etc.).
As in Tic-Tac-Toe, players select cells during their turn and the goal is to connect cells horizontally, vertically, or diagonally. Unlike Tic-Tac-Toe, to win, a player must connect five cells instead of only three (after all, the grids are larger). More importantly, the selection of a cell during a player’s turn depends on the numbers in the cells: to legally select a cell, its number must be a factor or multiple of the number in the cell chosen by the other player on the immediately prior turn.
I’ve had fun playing Fultiples with several of my students and they enjoy it too.
Getting Started
Beginning Grids
You need only two things to play:
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A 10 x 10 number grid. You can download a grid to the left: note that there are beginning, intermediate, and advanced grids. Beginning grids contain only numbers less than or equal to 50; intermediate, less than or equal to 100; and advanced, less than or equal to 200.
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Something to mark selected cells with so that the numbers are still visible. I have a collection of translucent, glass markers so that the same grid can be used over and over, but you can also mark on the grids directly using colored pencils or pens or just graphite pencils with different kinds of marks for the two players.
Every number in a grid is a multiple of one or more of a small set of prime numbers and this set varies among the grids.
Recall: a multiple of a number is the result of multiplying that number by any other number(s) – so 6 is a multiple of 2 since 6 is the result of multiplying 2 times 3 and 70 is a multiple of 5, since 70 is the result of multiplying 5 by 2 and then 7. Similarly, a factor of a number is one that evenly divides into it (divides into it without remainder). So 2 and 4 are factors of 16 but 5 and 9 are not. A prime number is one that has only two factors: one and itself. Two, 7, and 11 are prime numbers, but 6 and 8 are not since 8, for example, is evenly divisible by 2 and 4 as well as by one and itself.
If a grid is based on the prime numbers 2 and 3, every number in the grid is a multiple of either 2 or 3 or both but no other prime number. This grid can contain the numbers 6 ( = 2 x 3) and 12 ( = 2 x 2 x 3) but not 15, since fifteen, besides being a multiple of 3, is also a multiple of 5 (a prime number other than 2 or 3). But 15 is allowed on a grid based on the set of prime numbers 2, 3, and 5.
The set of prime numbers for a grid is written at the top of each one.
Once you have found a grid to play with and decided who will go first, the first player starts the game by marking or shading the cell of any number on the grid. Players continue a game by taking turns marking or shading cells, but each marked cell must contain a number that is a factor or multiple of the number in the cell that was marked on the immediately previous turn. Once a player gets 5 adjacent cells in a straight row, either horizontally, vertically, or diagonally, that player wins (note that the rows must be straight as in Tic-Tac-Toe: each winning row must be completely horizontal, completely vertical, or completely diagonal).
Here is an example of how a game might start. The first player selects and marks a cell on the grid containing the number 48. The second player can now mark any cell containing a factor or multiple of 48. She could look for cells containing the numbers 1, 2, 3, 4, 6, 8, 12, 16, 24, or 48, since each of these numbers is a factor of 48. On an intermediate or advanced grid, she could also look for cells containing multiples of 48 such as 96, 144, or 192. So the second player looks around and selects a cell containing the number 144. The first player continues the game by choosing a cell containing a factor or multiple of 144. And so on, until one of the players gets five in a row.
The initial Inspiration for Fultiples came from the following website:
University of Cambridge Nrich Team's [Factors and Multiples Game]