# Witches and Cats

This being Halloween, we will consider the problem of witches and their cats. The whole crowd is out tonight, naturally, and an interested observer, seeing them fly by (yes, witches' cats fly), noticed that the number of legs all told was 5 times the number of heads and the total heads and legs was 72. How many cats and witches are out this Halloween?

This riddle is taken from the Mensa genius riddle book.

• This would seem to be impossible. The leg-to-head ratio is 4 for cats and 2 for witches; no combination of them can result in a leg-to-head ratio of 5, unless a) we can have negative numbers of witches, or b) some of them are headless. – DevOfZot Sep 26 '14 at 19:01
• So, the book gave the answer 12 witches and four cats. I strongly believe that this answer is incorrect as others have signaled to. – user170141 Sep 27 '14 at 6:55
• Or they expect mensans to immediately compute heads+legs=72, legs=5*heads => 6*heads = 72 => heads=12. Or they expect mensans to think outside of the box: If witch cats can fly, why wouldn't they have 6 legs? – Florian F Sep 27 '14 at 12:14
• I was thinking that maybe the riddle was a play on some common fairytale which has flying cars/witches, like the Wizard of Oz. – user170141 Sep 27 '14 at 19:51
• If the witches had been celebrating Halloween with warlocks and their spirits, what if some of the witches were legless? – Ken Y-N Sep 30 '14 at 2:38

Taking into account everything that was already said (that witches and cats with 2 and 4 legs can not lead to the 5 ratio), and that I have never heard about flying cats, and do not know how many legs they have, I would answer simply "There are 12 cats and witches in total".

"The number of legs all told was 5 times the number of heads and the total heads and legs was 72"

From this we can find easily that number of heads are 72/(5+1) = 12. Therefore, independently of how many legs flying cats can have, the total number of creatures must be 12.

P.S. You cannot exact the precise number of cats and witches. For example, it can be 9 6-legged cats and 3 2-legged witches or 6 2-legged witches each with 8-legged cat.

• That was my thought. – user170141 Sep 27 '14 at 6:57

I feel like the question assumes a witch and her cat flying together somehow mask the presence of a single head.

If this line of reasoning is accurate 8 Witch/Cat combinations (8 heads 48 legs), 2 solo cats (2 heads 8 legs) and 2 solo witches (2 heads 4 legs) would suffice (12 heads and 60 legs altogether).

However, this would also suggest that the observer is not an accurate one, and all their other details would also be thrown into question (which isn't surprising since witches aren't supposed to exist at all). This line of reasoning suggests 0 witches.

You decide which witch list fits.

The answer I'm going to go with is: The observer made a typo, he meant to report that there are 3 times as many legs as heads, and hence the number of witches and cats is nine each.