How to Deal with Venn Diagrams

This blog post is primarily for my niece Katya, who was asked this nice problem in her homework:

In a camp, there are 79 kids; 27 of them are younger than twelve, 33 are girls, and 30 are boys that are twelve or older. Fill in this chart:

Girls Boys All
Younger than twelve
Twelve or older
All

The real question was to demonstrate that this chart can only be filled in only one possible way. So great, let's first enter in what we know:

Girls Boys All
Younger than twelve 27
Twelve or older 30
All 33 79

Now, since the "All" must be equal to the sum of the parts, you have enough information to find out two of the cells, namely

Girls Boys All
Younger than twelve 27
Twelve or older 30 I must be 79-27=52
All 33 I must be 79-33=46 79

Now we're here:

Girls Boys All
Younger than twelve 27
Twelve or older 30 52
All 33 46 79

Then there are two more cells we know:

Girls Boys All
Younger than twelve I must be 46-30=16 27
Twelve or older I must be 52-30=22 30 52
All 33 46 79

Now we're here and finally you know the last cell:

Girls Boys All
Younger than twelve 16 27
Twelve or older 22 30 52
All 33 46 79

There are two ways in which we can fill in the last cell - either 33-22 or 27-16 - but either way it's 11! That's no coincidence. Now we have the full table:

Girls Boys All
Younger than twelve 11 16 27
Twelve or older 22 30 52
All 33 46 79

Now let's go back to the original question: why was there only one way to generate this table? Remember at each step above there was no choice in what we were able to do. From the outset certain cells were already forced to be a certain value. As we calculated what those values were, new cells had the same property and we really had no choice at all while filling in the table.

So, from a fifth grader's perspective that's sufficient. But realize that this is, at the core, a question about degrees of freedom. We were given a matrix and the constraint that the last entry in each row must be the sum of the other elements, and the same situation holds for columns. And we showed that once you know 4 entries in the table, you know them all.

But not quite - you must be given 4 independent entries from the table. For example, you could start with this information and still generate the full table:

Girls Boys All
Younger than twelve 11 16
Twelve or older 22 30
All

However, you can't start here and fill out the whole table:

Girls Boys All
Younger than twelve 11 16 27
Twelve or older 22
All

What's the difference? Think about putting the entries in one at a time. Once you have 11 and 16 in the table above, adding in 27 to the table doesn't count as adding in new information. It doesn't count because it's already established.