Ways to Make 0 Through 12 on the 2x2 Board


 

Ways to make the numbers zero through 12 on the 2 by 2 board above using at most one token per square (A white token signifies a positive number, and a red token signifies a negative number.)

Supplies: Four two-color tokens and a board

A list of the ways to make zero through twelve:

0 = 3-2-1 6-3-2-1 1+2-3 3+2+1-6 empty board 5 ways








1 = 1 2-1 3-2 6-3-2
4 ways
2 = 2 3-1 6-3-1 3+1-2 6+1-3-2 5 ways
3 = 3 1+2 6-3 6-2-1
4
4 = 3+1 3+2-1 6-2 6+1-3 6+2-3-1 5
5 = 3+2 6-1 6+1-2 6+2-3
4
6 = 6 1+2+3 6+1+2-3 6+3-1-2
4
7 = 6+1 6+2-1 6+3-2

3
8 = 6+2 6+3-1 6+3+1-2

3
9 = 6+3 6+2+1


2
10 = 6+3+1 6+2+3-1


2
11 = 6+2+3



1
12 = 6+1+2+3



1__  







38 ways


                                                           

How do we know we have them all?

On each of the four squares, there can be "no token," a white token, or a red token. So for each square there are 3 possibilities. 3*3*3*3 = 81, so there are 81 possible configurations of tokens.

As listed above (and shown below!), 5 of them make zero, so 81-5 = 76 are not zero. 76/2 = 38 are positive, and 38 are negative. The 38 positive configurations are listed above.


Postscript.

Here are the five ways to make zero:


 


Webpage Maintained by Owen Ramsey
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