Wednesday, April 16, 2014

Over the course of many years, I have come across mathematical puzzles and tricks that can baffle even the best mathematical minds. But what most of these tricks come down to is simple algebra. Here's some math tricks that you may have encountered. but couldn't figure out. Enjoy the art behind the mathematical magic presented in the next several paragraphs.

Here's some mathematical "magic" you can perform with a single person or many people in an audience. Pick a person and ask that person to pick any single or double digit number. Then ask the person to do the following, double the number, add 14, divide by two and subtract the original number. You will astound most people when you ask, "Is the number you are now thinking of seven?" Try this yourself and you'll see that the answer will always be 7.

There reason is the simple algebra behind this. The number chosen is x, double the number to give you 2x. Add 14 to get 2x + 14. Divide by 2 to get x + 7. Now subtract the original number x, to get x + 7 - x = 7. This will work no matter what number is used. The answer will always be half the number you tell the person to add in the second step of the problem.

Here's another trick in which the answer will always be 1,089. Write down any three-digit number in decreasing order of digits. For example, 875 and 942 would both work. Let's use 875. Reverse the number and subtract from the original number. Therefore we have 875 - 578 = 297. Add this result to the reverse of itself. So we add 297 to 792 and we get 1,089. Try this with any three-digit number in which the numbers are from largest to smallest.

For the next trick, give someone a piece of paper and a pencil or pen. Have the person write the numbers 1 through 10 down the left hand side. Ask the person to pick two numbers between 1 and 20 and write them on lines 1 and 2. Next have the person write the sum of lines 1 and 2 and write in line 3, the sum of lines 2 and 3 and write in line 4 and so on. After the card is filled, ask the person to show the card. You should quickly be able to tell the person the sum of all 10 numbers quicker than he or she could add them on a calculator.

For example, if the person starts with the numbers 9 and 2, you can quickly tell that the sum of the numbers is 671. If the numbers chosen are 3 and 19, you can quickly say the sum is 1,837. How can this be done so fast? SImply multiply the number in line 7 by 11. If you want to check this out to show that it works, Get a card and go through the entire process. You can also astound by asking the person to divide the number in line 10 by the number in line 9. You will instantly be able to tell the person that the first three digits in that result is 1.61.

How does this work? The first result is simply based on the fact that you assign the value in line 1 the variable x and the value in line 2 the variable y. Line 3 is then x + y, line 4 is x + 2y, line 5 is 2x + 3y, line 6 is 3x + 5y, line 7 is 5x + 8y. When you get through and add all 10 lines you get 55x + 88y. That is 11 times 5x + 8y, which is line 7.

Knowing that line 10 divided by line 9 is around 1.61 is a bit more complex, involving four variables and adding fractions badly (adding numerators and denominators), showing that the value falls between two original fractions. It's not important to know how every result is arrived, but you can amaze friends by having this answer ahead of time by writing 1.61 on the back of the card.

Some of these tricks I have used over the years with my students, while the last method I obtained form the book, Secrets of Mental Math, by Arthur Benjamin and Michael Shermer.

Look for a few more mental math tricks in upcoming articles.

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