# Question about the “Cat in the Hat” problem

I'm looking at the Cat in the Hat problem. Can anyone explain to me how a simple math logic is used to derive this equation? $$\left(\frac{N}{N+1}\right)^{M-1} = \frac{B}{A}.$$

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This is terribly "uncontained"! –  The Chaz 2.0 Sep 16 '11 at 19:21
Nobody can tell you how to derive such an equality unless you say something about what $N$, $M$, $A$, and $B$ are. –  Michael Hardy Sep 16 '11 at 19:26
@pcraft: To explain The Chaz' comment: Questions should ideally be self-contained. This question is doubly un-self-contained in that it not only merely links to another page that states the question, but in fact links to another page that links to another page that states the question. Please avoid doing that. It's already a potential problem with just one link, since the content or address of the other page could change any time, and the problem is obviously even worse if it goes through two stages. –  joriki Sep 16 '11 at 19:32
@joriki: exactly! –  The Chaz 2.0 Sep 16 '11 at 20:05
@pcraft: when posting the same question to two groups, please cross-link them. The other one got migrated here and got answered by somebody else. –  Ross Millikan Sep 16 '11 at 23:07

Beware of sentences like "Using some simple math logics we can ...". First, there's no such thing as "math logics", so that person more likely than not doesn't know what he or she is talking about. Second, what's simple for one person may be not so simple for someone else, so someone telling you that something is simple either hasn't understood this yet, or is trying to impress and/or intimidate you by saying that something is simple when it might not be for you.

In the present case, we have a hierarchy of $M$ levels of cats. With each of the $M-1$ steps down the hierarchy, the cats become more numerous by a factor $N$ and smaller by a factor $N+1$. So since the number of cats on the highest level is $1$ and the height of a cat on the lowest level is also $1$, the number $B$ of cats on the lowest level is $N^{M-1}$ and the height $A$ of the cat on the highest level is $(N+1)^{M-1}$. Taking the ratio of these two yields the equation.

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Or the author was being really lazy. –  anon Sep 16 '11 at 19:50
Thanks joriki for helping out the 'simple' way :) –  pcraft Sep 17 '11 at 19:56

It is simple recursion or induction.

If the number of levels is $M=2$ then the one big cat is of height $H_2=N+1$ and has $N$ cats of height one in its hat.

If the number of levels is $M=3$ then the one big cat is of height $H_3=(N+1)H_2=(N+1)^2$ where and has $N$ cats of height $H_2$ in its hat and so $N^2$ cats of height one.

So in general, with $M$ levels then the one big cat is of height $A=H_M=(N+1)H_{M-1}=(N+1)^{M-1}$, and has $N$ cats of height $H_{M-1}$ in its hat and so $B = N^{M-1}$ cats of height one.

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Note that my answer is one minute older than this, so the paragraph on claiming that something is simple wasn't specifically referring to this :-) –  joriki Sep 16 '11 at 19:51