This Parent's Corner column will present puzzles and mathematical techniques for parents to work on with children between the ages of 7 to 12. The column will always include a fully worked-out explanation, allowing parents to explain what will sometimes seem to be mathematical magic. As the column is still experimental, we welcome your comments. Please send them to firstname.lastname@example.org. --Dennis Shasha
POLISH HAND MAGIC
I don't really know the origin of this trick but a smart and elegant Polish woman taught it to me, thus the title.
In the process of learning multiplication by single digit numbers, children have an easy time provided that one of the numbers is under 5 (for example, 2 x 3 or 4 x 7). They have much more trouble when both are 5 or above. Polish Hand Magic offers an easy way to overcome that problem.
The general method is
1. Take each number, subtract 5, and put up that many fingers.
2. Sum the fingers that are up. That's the 10s place.
3. Multiply the fingers that are down. That's the 1s place and there may be a carry.
The trick is more easily explained by a few examples. Suppose you want to multiply 9 by 7. Lets represent 9 by ||||. (that is, four fingers up and the pinkie down) and 7 by ||... (two fingers up and three down). The total number of fingers that are up is 4 + 2 = 6. The product of the fingers that are down is 3 x 1 = 3. So the answer is 63.
Here is another example: 8 x 6. 8 becomes |||.. and 6 becomes |...., so 4 fingers are up in total and 2 x 4 is 8. Thus, the answer is 48.
This even works for extreme cases like 10 x 5. 10 becomes ||||| and 5 becomes ....., so we have 5 fingers up in total and the product of the down fingers is 0 x 5 = 0. So the answer is 50.
Another interesting case is 6 x 7. 6 becomes |.... and 7 become ||..., so three fingers are up and the product of the down fingers is 4 x 3 = 12. So we get 30 + 12 = 42.
It seems magical I know. You can teach it as magic to your child, but can you yourself see what's going on? If not, read on.
WHY IT WORKS
As with many number tricks, you can understand it using simple algebra. The following explanation should be within reach for a 14-year-old.
Suppose you are multiplying Y by Z. For Y, the number of fingers up is Y - 5 and the number of fingers down is 10 - Y. Similarly, for Z the number of fingers up is Z - 5 and the number of fingers down is 10 - Z.
We sum the up fingers and use that result for the ten's place:
(10(Y-5)) + (10(Z-5)) = 10Y - 50 + 10Z - 50 = 10Y + 10Z - 100
Now we multiply the down fingers:
(10 - Y)(10 - Z) = 100 - 10Y - 10Z + (Y)(Z).
When we add those two results together, we get:
10Y + 10 Z - 100 + 100 - 10Y - 10Z + (Y)(Z).
All the terms cancel except (Y)(Z). So the method gives us the product of Y and Z.