What are some classic fallacious proofs?

Question

If you know it, also try to include the precise reason why the proof is fallacious. To start this off, let me post the one that most people know already:

1. Let $a = b$.
2. Then $a^2 = ab$
3. $a^2 - b^2 = ab - b^2$
4. Factor to $(a-b)(a+b) = b(a-b)$
5. Then divide out $(a-b)$ to get $a+b = b$
6. Since $a = b$, then $b+b = b$
7. Therefore $2b = b$
8. Reduce to $2 = 1$

As @jan-gorzny pointed out, in this case, line 5 is wrong since $a = b$ implies $a-b = 0$, and so you can't divide out $(a-b)$.

Answer 1

Wikipedia has a long list of these:

http://en.wikipedia.org/wiki/Mathematical_fallacy

Answer 2

The two envelopes problem is a good one.

See also:

Card doubling paradox

and:

https://mathoverflow.net/questions/9037

Answer 3

Well, one I've read quite some time ago (sorry, can't give a reference) is a proof that any bag that contains peas either contains only green or only yellow peas.

The proof goes by induction:

Assume the bag holds only one pea. That pea is of course either green or yellow. Therefore for only one pea, the statement is true.

Now assume we've proven it for $n$ peas, and we are given a bag of $n+1$ peas. In that case, we first take one pea out, so we now have a bag of $n$ peas, so by assumption, they all have the same colour. To find out that colour, we take another pea out, and put our first pea back in. Then by looking at the other pea, we can determine the colour of the peas in the sack, and since there are again $n$ peas in the sack, the one we had removed first also has the same colour as the others.