Small children, armed with pencil and paper, often execute what adults call "scrawling", but the topologist calls "a tangle". Consider, please, a pencil scrawl, round and around, crossing over penciled curvings, to achieve a complicated tangle that connects back to its starting point. Now indicate by penciled dot some point "within" the tangle, labeling it "D". And ask the question, "Is dot D topologically inside the tangle or outside?"

Answer: Choose a dot some distance from the tangle on the paper, labeling it "E". Clearly, dot E is outside the tangle. (This resembles Whitehead's No-No Principle: The negative perception is the height of judgment!. Meaning that, we can often easily tell "what something is not", but it's often difficult to "tell what is".)

Now you connect dot E to dot D by a pencilled path and count the boundaries (pencilled scrawls) crossed (passed over) between E and D. If the number is even, then D and E are in topologically equivalent regions. Since E is outside, so is D. But, if the number of boundaries crossed is odd, E and D are in topologically different regions. Since E is outside, D is inside the tangle.

The topologist says that outside and inside are the two different values of a measure called PARITY. Dots D and E are either of the same parity or of different parity.

Why does an even number of crossings mean same parity, and why does an an odd number of crossings mean different parity?

Start from outside-E and draw a path down to the tangle. When you cross one pencil-curve of the tangle, you have crossed a boundary, changing the connectedness, changing the parity. Since you were outside before the crossing, you are now inside the tangle. When you cross another pencil-curve of the tangle, you have crossed another boundary, changing the connectedness, changing the parity. Since you were inside the tangle before the crossing, you are now outside the tangle. But notice(!) that TWO crossings (an EVEN number) returns you to the starting parity, whereas ONE crossing (an ODD number) changed the parity from the starting value. In general, AN EVEN NUMBER OF CROSSINGS OF BOUNDARIES RETAINS THE STARTING-PARITY; AN ODD NUMBER OF CROSSINGS CHANGES THE PARITY. By fixing the starting-parity as outside, you can easily, by "evens-and-odds", tell "where you're at".

There's an old "Mississippi River Boat Gambler's Belt Trick" which resembles the tangle problem, involving a matchstick placed "inside" a looped belt. "When I unloop this belt, is the match inside or outside the belt?" No matter what the gullible bettor says, the gambler could unloop it from the top or bottom so as to win the bet.