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Self-referential

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Consider the following table:

 

The numbers in the first column are fixed and $a$, $b$, $c$ and $d$ should be chosen so that in the entire table the total number of $1$s is $a$, the total number of $2$s is $b$, the total number of $3$s is $c$ and the total number of $4$s is $d$.

Here is an example:








In the entire table there are two $1$s, three $2$s, two $3$s and one $4$.

How many other ways can you find to fill in the right-hand column? Can you find them and explain why there are no others?



If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.

This problem is taken from the UKMT Mathematical Challenges.
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