On this page we house Strange But True mathematical factoids.
A goofy graphic - author/source unknown. Go here to view a most peculiar animated .gif, of an old visual puzzle. It takes about 10 seconds to view before it does anything. While you wait: count the number of people you see. Poof!
4-4/3+4/5-4/7+4/9-... is exactly pi. That is: take the reciprocals of the odd integers. Alternately add and subtract them. Multiply by 4. You get precisely pi. The mystery is: what do the odd integers have to do with pi?? Why should there be any connection whatsoever between reciprocals of odd integers, and pi - where's the circle?? (As a pure mathematical fact, it's not too hard to show, if you've had calculus II/III. The derivation may not really shed light on the mystery, though.) A related oddity: the probability that 2 positive integers are relatively prime, i.e. that they have no common factors, is 6/pi2, exactly. (Again, this fact is not too hard to show, using elementary number theory.)
There are 49 ways to make change for a half-dollar coin. (I think that's a lot.) Even worse: there are 242 ways to make change for a dollar, using just pennies, nickels, dimes and quarters - 292 ways if you allow use of half-dollar coins as well). (Not too hard to show, but not that easy.)
The Banach Tarski paradox: It's possible to take a solid ball, say of radius 1; chop it into finitely many 'pieces' (meaning disjoint subsets); rigidly translate those, and/or rotate those, pieces; then recombine them; and get two solid balls, of the same radius. (This should seem to violate your basic instinct about "conservation of mass").
This is fairly hard to show - here's one key step: Imagine a circle of radius 1/(2 pi) centered at the origin in the x-y plane, so that the circumference is 1, with one point missing, namely the point where the circle would have struck the x-axis with positive x-coordinate (i.e., with x0=(1/(2pi),0) removed). Call the circle-minus-one-point CMP. Let S be the set of points in CMP, as follows: start at x0 and 'walk' counterclockwise along the circumference in steps of size 1/sqrt(2). That creates a countable infinity of points. Next, imagine removing S from the circle-minus-one-point by pulling straight upwards; rotating the whole set S rigidly by angle (2pi)/sqrt(2) radians clockwise, and so swinging x3 to now lie over the hole where x2 was, and similarly having x2 lie over the hole where x1 was, etc; and placing the set S back down on the circle. Note that x1 has been rotated to lie over the x-axis, i.e. right over where the 'original hole' was - and thus we have an example of taking a set, CMP, chopping it into two pieces, namely S and CMP-S, rigidly rotating one of them, namely S, and getting a 'bigger' set - namely CMP, with one extra point.
Godel's (first) incompleteness theorem ("On Formally Undecidable Propositions"): Crudely, in any consistent axiomatic system that contains basic set theory and the Peano axioms (and hence the integers, and arithmetic), there is a statement that is true, and yet cannot be derived to be true from the axioms and derivation rules. Here's a rough example: Let P be the proposition that says "There is no proof of P" - yep, it's a self-referential statement. P is true: P can't be proved, because if P could be proved, P is false, a contradiction of the consistency assumption. So either the axiomatic system is inconsistent, OR there are statements that one can pose within the system that are true, and yet cannot be deduced within that system.
If you wonder "How on earth can you prove something like this Godel incompletenss??", the answer is that Godel came up with a way to show that if the axiomatic system contains the integers, and ordinary arithmetic, he could then list every proposition as an integer, which is now called the Godel number; this isn't too strange, think of taking any statement, saving it as a computer file, then noting that it is, after all, just a long string of zeroes and ones - an integer. Then one proposition can refer to itself (as in the comment above) because it's a proposition about a statement with a certain Godel number.
It's easy to tell if a random 200-digit integer is prime or not; but it's hopelessly impossible to factor a random 200-digit integer completely into primes. (No one knows how intrinsically hard a problem factoring is; there are no known algorithms that are fast. Primality testing is not particularly hard.)
