Fultiples
If you know how to play Tic-Tac-Toe or Connect 4, then you nearly know how to play Fultiples.
Fultiples is played on 10 x 10 grids and each cell of these grids contains a positive integer (reminder: positive integers are the numbers you count with: 1, 2, 3, etc.).
I’ve had fun playing Fultiples with several of my students and they enjoy it too.
Description
As in Tic-Tac-Toe or Connect 4, players select cells during their turn with the goal of connecting five (rather than three or four) cells horizontally, vertically, or diagonally. Unlike Tic-Tac-Toe or Connect 4, the selection of a cell during a player’s turn depends on the numbers in the cells: to be legally selected, a cell’s number must be a factor or multiple of the number in the cell chosen by the opposing player on the immediately prior turn.
Getting Started
Free Grids
You need only two things to play:
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A 10 x 10 customized number grid. 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. 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. The prime numbers that a grid is based on appear at the top of the grid.
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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 maybe graphite pencils with different kinds of marks for the two players.
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 14.
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 shows up 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.
Instructions (though you can make up your own or modify these)
Once you have found a grid to play with and decided who will go first (there is not necessarily any advantage to going 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.
Of course, there is nothing sacred about the rules I state here, though I find that they work pretty well. But you might want to experiment with different rules.
Inspiration
The initial Inspiration for Fultiples came from the following website:
University of Cambridge Nrich Team's [Factors and Multiples Game]