Number Tricks Get Down to Basics

The title should read “... Get Down to Basic Operations.” Number tricks can provide young people with motivation to practice their ‘rithmetic and provide teachers the information on who needs more practice and who doesn’t! Adults are entertained and even mystified by a good number trick.

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Number Tricks Explained ~ But don't peak yet!

Addition Race

Ask your audience to give you three numbers which you will write in a column. If these are fourth-graders, ask for two-digit numbers. If they are older, increase the number of digits. This trick is scalable. Let’s say the youngsters give you 34, 89 and 62. You write those on the blackboard in a column and under those write 65, 10 and 37.

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You say that you will race them to come up with the sum of the six numbers. But as soon as they dip their heads to begin crunching, you saunter over to the board and write 297 as the total. You will always be first and always be right. But how do you pick your three numbers? That is Uncle Bob’s poser to you, and he will give you another example to assist you.

If the group gives you 78, 31, and 55, you will augment the list with 44, 68, and 21, and then you will immediately know that the total is 297 once more. But what is the secret behind this trick? Does the pair 55 and 44 help you at all?

One more poser before we move to the second trick. How does this problem scale up to larger numbers. What three numbers must follow 2355, 1766, and 3253, and what will be the magic total.

Subtraction ~ The Secret Digit

Have each person in a group write down a four-digit number of their choosing. Warn them that you will try to guess something about their particular number, so they should not do something simplistic like use the same digit four times. Ask them to make a new number by scrambling those same digits. They will subtract those two numbers to get the difference and, in that number, they will circle a secret digit. The secret digit can be any number but zero. When they tell you the other digits in the difference, you will tell them what their secret digit is.

For example, if one person chooses 7832 and subtracts 3287, s/he will get the difference 4545. If that person tells you the digits 5, 4 and 5, you will know quickly that the secret digit is 4. You merely make the total of all the digits come out to 9 or 18, as in this case, or 27.

But what is the secret behind the trick?

Multiplication Race

Ask a person to give you a number between 15 and 95. This person might be armed with a calculator, but you will attempt to beat the calculator in a multiplication race. If the person gives 34, you might stall a little for time, “34, hmm, let’s see.” Then you say without hesitation, “34 times 26 equals 884.” Ask another person for a number in range, and when they give you 79, you answer with “times 81 equals 6399.” You are wowing them now.

After several examples, someone might see how you are picking the second number. If you get 63 from someone, a number that is 60 + 3, you will choose 57 as the second factor because it is 3 less than 60. After that it takes just a second to deduct 9 from 3600, and report the product 3591. You just took the difference of the squares of 60 and 3, and it is equal to 63 x 57.

The squares you need to know are 20 x 20 through 90 x 90, which all end in double zeros, and 1 x 1 through 5 x 5 because your choice will never be more than five away from a multiple of ten. That means that the final two digits of the product will be 99, 96, 91, 84, or 75, and the first two digits are one less than a square number.

But what is the secret behind the trick?

3591 = 63 * 57 = 3600 - 9

Division Out and Back

Ask each person to write a secret three-digit number, 372 for example. Ask them to make a six-digit number by appending those same digits in order. In our example the number is now 372,372. They will now perform three divisions which are fairly simple to do by hand. Ask them to divide by 7 and advise them that there will be no remainder. Ask them to take the new number (53196, if you are dividing at home) and divide it by eleven. Again, you insist that there is no remainder, as is true for 53196 / 11. Finally, take the new quotient and divide it by 13. Ask the group if they see anything familiar about the final quotient.

What is the secret behind the trick?

June 2006

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