This is a silly proof that my thirteen-year old cousin showed me the other day. I'm ashamed to admit that it took me a full ten minutes to figure it out.

Let a = 1

Let b = 1

therefore:

a = b

multiply both sides by b:

ab = b^2

subtract a^2:

ab - a^2 = b^2 - a^2

factorise:

a(b - a) = (b - a)(b + a)

divide by (b - a):

[a(b - a)]/(b -a) = [(b - a)(b + a)]/(b -a)

cancel:

a = b + a

returning to our initial property:

a = 1

b = 1

therefore:

a = b + a

1 = 1 + 1

1 = 2

What is wrong with the above proof? (It's not difficult, but it's a good one to pull out to confuse people)