A few years ago, I wrote a puzzle column for parents called Parent's Corner. The idea was for parents to teach their kids some math and logic through puzzles. One of those puzzles, called "Polish Hand Magic," was a method used by Polish schoolchildren to multiply pairs of numbers between 5 and 10 knowing only how to multiply pairs of numbers between 1 to 4 and to add.

Here is how it went: suppose you are multiplying 6 times 8. Hold up one finger in your left hand to represent the 6 (5 + 1 = 6) and then three fingers in your right hand to represent the 8 (5 + 3 = 8). So, we'll represent the 6 by (|....) and the 8 by (|||..).

We'll compute as follows: add up the fingers that are up (there are 1 + 3 = 4 in this case) and multiply that number by 10, yielding 40. Next, multiply the fingers that are down (4 * 2 = 8 in this case) and add the two calculated numbers: 40 + 8 = 48. Thus, the product of 6 times 8.

Try some other pair in which each number is between 5 and 10 inclusive. For example, 6 * 7 yields the finger setup (|....) and (||...). Count 1 + 2 = 3 for the up fingers (yielding 30) and 4 * 3 = 12 for the down fingers (yielding 12) for a total of 42.

Warm-Up:
Prove that this system always works.

Solution to warm-up:
Suppose you want to multiply x times y. You put up x - 5 fingers in your left hand and y - 5 fingers on your right hand. This leaves 10 - x fingers in your left hand that are down and 10 - y fingers in your right hand that are down. The calculation says to multiply the up fingers by 10, yielding 10((x - 5) + (y - 5))I, and to multiply the down fingers together, yielding (10 - x)(10 - y). The total is:

10((x - 5) + (y - 5)) + (10 - x)(10 - y)
= 10(x + y - 10) + (10 - x)(10 - y)
= 10x + 10y - 100 + 100 - 10x - 10y + xy
= xy

Thus, all this manipulation leads to exactly the result we want.

End of solution to warm-up.

Now the question is how to extend this to numbers beyond 10. I have three challenges for you:

1. Still using only two hands (with one hand representing each number), and knowing only how to add and subtract and how to multiply pairs of numbers up to 5, can you devise a method to multiply any pair of numbers between 10 and 15?

Hint: Remember that every such product will equal or exceed 100.

2. Can you extend the method above (under the same constraints) to multiply any pair of numbers between 5 and 15?

3. Using four hands (with each pair of hands representing a number), and knowing only how to add and subtract and to multiply pairs of numbers up to 10, can you devise a method to multiply any pair of numbers between 0 and 20?

Mathematical purist's request: Show these techniques to kids when they're young, but make them do the proofs when they've learned some algebra.