Amaze the Kids, and Yourself, with Some Nifty Math Tricks

SCIENCE JOURNAL
By SHARON BEGLEY

Halfway through school vacation, kids who miss factoring polynomials are doing the next best thing: e-mailing each other math tricks. I got the latest one courtesy of Lauren, a soon-to-be seventh-grader who lives on New York's Long Island, via her uncle George. For those of you without your own resident teenager, it goes like this (the reader is advised that this week's column is best enjoyed with pencil and calculator):

Pick the number of times a week you would like to [fill in the blank]. The G-rated version is "eat chocolate." The R-rated version is left to the reader's discretion. The basic idea is, try for something between one and 10. Then multiply by 2, supposedly because in the first step you were too abstemious.

Add 5.

Multiply by 50.

If you've already had your birthday this year, add 1752; if not, add 1751. Last, subtract your four-digit year of birth.

Presto! You have a three-digit number: The first digit is the number you originally chose, and the last two are your age.

Like much number magic, this one reduces nicely to simple algebra, revealing some neat properties of numbers, notes Mike Breen of the American Mathematical Society.

Call the chosen number X. Doubling it gives you 2X. Adding 5 yields 2X + 5. Multiplying by 50 gives you 50 (2X + 5) which, by the distributive property of multiplication, equals 100X + 250.

Now it gets cute. Multiplying X by 100 amounts to shifting the original number two decimal places to the left. So at this point, the first four steps have amounted to moving the chosen number to the hundreds place and adding 250. When we add 1752 or 1751, we've got this: 100X + 250 + 1752 (if you've had your birthday this year) or 100X + 250 + 1751 (if this year's birthday is still to come). Group the last two numbers, and you have 100X + 2002 or 100X + 2001. Now it's becoming clear: All of the operations have simply moved the original number to the hundreds place and added the current year (for those who have had their birthday) or the previous one.

In fact, any manipulations that get you to "100X + current year" will produce the "magic" result each time.

Subtracting your birth year means 100X + 2002 - 1972, for instance -- but the final subtraction is exactly how to calculate your age, knowing the current year and your birth year. The subtraction yields 100X + age. You're left with your original number, X, in the hundreds place, so if you're born in 1972 but before today you've got X30. X is the number you chose; the final two digits, in this case 30, are your age.

Other number tricks might get you through a tedious car trip with the kids -- and, says Mr. Breen, illustrate interesting properties of numbers.

Tell your passengers that you have memorized the first word of every page in their current Harry Potter. To prove it, ask them to pick any three-digit number except a triplet like 444. Then reverse the digits: 781 would become 187. Subtract smaller from larger (781 - 187 = 594). Now tell them the first word on page 594 (yes, you do have to memorize the first words on a handful of pages).

The trick is that the difference between any two three-digit numbers XYZ and ZYX is a multiple of 99. So no matter what number is chosen, the manipulation can yield only 99, 198, 297, 396, etc. These are the pages you have to study.

You can also impress kids and influence spouses by speedily squaring any number ending in 5, such as 85. The last two digits are always 25. To get the first digits, multiply the first digit by one more than itself (8 times 9 in this case, or 72). Put that in front of the 25. So 85 squared equals 7225. And 125 squared equals 15,625, because 12 times 13 equals 156.

Why? The original number can be expressed as 10x + 5; this is just a general algebraic expression for a number ending in 5. Squaring that gives you 100x2 + 100x + 25, or 100x(x+1) + 25. There's that 25 at the end. On the left is the number x times one more than itself, x + 1, moved to the left by multiplying by 100. To make this work, the reader is advised to ask the straight man to choose a two- or three-digit number, or the multiplication gets messy.

For your last trick, tell if a number is divisible by 11. Start in the ones place on the far right, and add every other digit (for 87934, add 4 + 9 + 8 to get 21). Now add the remaining digits (3 + 7, or 10). Subtract the two (21 - 10, or 11). If the result is divisible by 11, so is the original number[3 + 4][6 + 7 + 3], 87934.

Next time your children ask what algebra is good for, tell them, "Magic."

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