# 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.