# Math Tricks for All Ages

### This web page is devoted to the

*incredibly boffo*

idea that math can be fun!

### Try these tricks:

- Easy Magic Addition
- Magic Square #15
- Magic Square #34
- Make Your Own Magic Square
- Upside Down Magic Square
- Anti-magic Square
- Win Bets With This Square
- Magic Card Trick
- Lightning Calculator
- Fun Number Tables
- Did You Know...?
- A Trick for This Year
- Where is the String?
- Humongous Magic Square

## Here are some interesting links:

- A Reading list of Tricky Math Books, most of which I have used for this site.
- Learn about the original computer: The Abacus (http://www.ee.ryerson.ca:8080/~elf/abacus/)
- Play a Math-Chase Game (http://dev.eyecon.com/marcia) -- for one or two players. (If you're using Netscape,
**Do Not Scroll down the page while this loads**. - Play at Shoot Balls (http://www.fi.uu.nl/wisweb/en/applets/bollen/Welcome.html).
- Play Flippo 24 (http://www.fi.uu.nl/wisweb/en/applets/bollen/Welcome.html).
- Test your knowledge of the multiplication tables (http://www.fi.uu.nl/wisweb/en/applets/tafels/Welcome.html)
- Try your hand at estimation (http://www.fi.uu.nl/wisweb/en/applets/bollen/Welcome.html).
- Explore Geometryin a fun and interactive way.
- Try the Tower of Hanoi Puzzle (http://www.eng.auburn.edu/~fwushan/Hanoi1.html).
- See what a Spriographis (http://www.mainstrike.com/mstservices/handy/Spiro/).
- See what a Mandelbrot setis (http://www.franceway.com/java/fractale/mandel_b.htm).
*If you want more math challenges*try the new PBS MATHLINE MATH CHALLENGESsite. Try it, you'll like it. (But remember we were first.)

## Magic Addition Trick #1

*Amaze the peons with this one.* It's simple. It's effective. It gets them every time.

- Ask your mark to
*pick three (3)*between 1 and 9.**different**numbers - Tell him or her (or her or him) to write the three numbers down next to each other, largest first and smallest last, to form a single 3-digit number. Tell him/her not to tell you what the numbers are.
- Next have her or him form a new 3-digit number by reversing the digits, putting the smallest first and the largest last. And write this number right underneath the first number.
- Now have him or her subtract the lower (and smaller) 3-digit number from the upper (and larger) 3-digit number. Tell them not to tell you what the result is.
- Now you have a choice of wrap-ups:
- Ask your friend to add up the three digits of the number that results from subtracting the smaller from the larger 3-digit number. Then amaze him or her by teling them what the sum of those three numbers is.
*The sum of the three digit answer will always be 18!* - Tell your friend that if she or he will tell you what the first OR last digit of the answer is, you will tell her or him what the other two digits are. This is possible because
*the middle digit will always be 9, and the other two digits will always sum to 9!*So to get the digit other than the middle one (which is 9) and other than the digit that your friend tells you, just subtract the digit your friend tells you from 9, and that is the unknown digit.

- Ask your friend to add up the three digits of the number that results from subtracting the smaller from the larger 3-digit number. Then amaze him or her by teling them what the sum of those three numbers is.

## Magic Square #15

Every row and column sums to 15 in this magic square. So do both diagonals!

8 | 3 | 4 |

1 | 5 | 9 |

6 | 7 | 2 |

## Magic Square #34

Every row and column sums to 34 in this magic square. So do both diagonals!

1 | 15 | 14 | 4 |

12 | 6 | 7 | 9 |

8 | 10 | 11 | 5 |

13 | 3 | 2 | 16 |

## A Recipe for Your Own 3 X 3 Magic Square

Here's a recipe for making your own 3 X 3 magic number square. This recipe and both of the above two magic squares comes from one heck of a great book called, *Mathematics for the Million*, by Lancelot Hogben, published by Norton and Company. I highly recommend it. You don't need much math at all to get into the adventure of numbers told in this classic book.

Some necessary rules and definitions:

- Let the letters
*a*,*b*, and*c*stand for integers (that is, whole numbers). - Always choose
*a*so that it larger than the sum of*b*and*c*. That is,*a*>*b*+*c*. This guarantees no entries in the magic square is a negative number. - Do not let 2 X
*b*=*c*. This quarantees you won't get the same number in different cells. - Using the formulas in the table below, you can make magic squares where the sum of the rows, columns, and diagonals are equal to 3 X whatever
*a*is.

a + c |
a + b - c |
a - b |

a - b - c |
a |
a + b + c |

a + b |
a - b + c |
a - c |

To create the first Magic Square #15 above, you let *a* be equal to 5, let *b* be equal to 3, and let *c* be equal to 1. Here are some others:

*a*= 6,*b*= 3,*c*= 2*a*= 6,*b*= 3,*c*= 1*a*= 7,*b*= 3,*c*= 2*a*= 7,*b*= 4,*c*= 2*a*= 8,*b*= 6,*c*= 1*a*= 8,*b*= 5,*c*= 2*a*= 8,*b*= 4,*c*= 3

Try making up some of your own.

## Upside Down Magic Square

Here's a magic square that not only adds up to 264 in all directions, but it does it even when it's upside down! If you don't believe me, look at it while you are standing on your head! (Or, just copy it out and turn it upside down.)

96 | 11 | 89 | 68 |

88 | 69 | 91 | 16 |

61 | 86 | 18 | 99 |

19 | 98 | 66 | 81 |

## Anti-magic Square

Here's a magic square with as many *different* totals as possible.

5 | 1 | 3 |

4 | 2 | 6 |

8 | 7 | 9 |

This table produces 8 *different* totals.

## Win Bets with this Magic Square

OK, here's a neat way to win bets with a magic square. Call a friend on the phone. Have him or her get a pencil and paper and bring it to the phone, so he or she can write down numbers from 1 to 9. Tell your friend that you will take turns calling out numbers from 1 to 9. Neither one of you can repeat a number that the other one calls out. Both of you then write down the numbers 1 to 9. Then when your friend says one of the numbers he or she draws a circle around that number, and so do you. When you say a number, you draw a square around that number, and so does your friend. The winner is whoever is the first one to get three numbers that add up exactly to 15.

Say you go first and you call out 8. Your friend might call out 6. You then call out 2. Your friend calls out 5, and you call out 4. Your friend calls out 7, and you call out 3. Then you tell your friend that you have just won because you called out 8, 3, and 4, which add up to 15.

Your friend will want to play again. So this time you can bet him you'll win, with the condition that in case of a draw (where you use up the numbers 1 to 9 without either of you getting a 15 total) nobody owes anything.

If you know the trick, you will never lose, and will probably will most times.

**The tricks** Actually the trick is based on both tic-tac-toe and a magic square. The magic square look like this:

8 | 1 | 6 |

3 | 5 | 7 |

4 | 9 | 2 |

Because this is a magic square, every row and every column and every diagonal adds up to 15. So if you've got this square in front of you with your friend on the phone, you can put an X in the squares of the number you call out, and an O in the squares of the numbers your friend calls out. Then, just like in tic-tac-toe, you try to get three X's in a row, because those will always add up to 15.

So in the example above, when you call out 8, you put an X in the upper left corner. When your friend says 6, you put an ) in the upper right corner. And so on.

## Mathematical Card Trick

You need an ordinary pack of cards for this wower. No fancy shuffling is required. Just follow these easy steps:

- Shuffle the cards to mix them thoroughly.
- Deal out 36 cards into a pile.
- Ask a friend to pick one of the 36 cards, look at it and memorize it, and then put it back in the pile without letting you see it.
- Shuffle the 36 cards.
- Lay the 36 cards out in 6 rows of 6 cards each. Be sure to deal the top row from left to right. Then deal the second row beneath it from left to right. And so on with each succeeding row laid out underneath the one before.
- Ask your friend to look at the cards and tell you which
**row**the chosen card is in. Remember what number the row is. - Carefully pick the cards up
*in the same order you laid them down*. So the first card on the left of the top row is on top of the stack, and the last card on the right of the bottom row is on the bottom of the stack. - Now lay the cards down in 6 rows of 6 cards each, but this time
*spread the card one column at a time*. Instead of proceeding from one row to the next, proceed from one column to the next. Lay the first six cards in a*column*from top to bottom on the far left. Then lay out the next six cards in a second column of six cards just to the right of the first column of six cards. Continue doing this until you have 6 columns of 6 cards each (which looks the same as 6 rows of 6 cards each -- because it*is*the same). - Once again ask your friend which
*row*contains the chosen card. - When you friend tells you which row the card is in, you can say what the exact chosen card is. How? If your friend said the card was in row 2 the first time, and in row 5 the second time, then the chosen card is the one in the second column of the fifth row. This is because the way you arrange the cards, what were rows the first time around become columns the second time around.

## Lightning Calculator

Here's a trick to wow them everytime! Have someone write down their Social Security number. Then have them rewrite it so that it is all scrambled up. (If they don't have a Social Security number, have them write down any 9 digits between 1 and 9.) If there are any zeroes, have them change them to any other number between 1 and 9. Then have them copy their nine numbers, in the same order, right next to the orginal nine numbers. This will give them a number with 18 digits in it, with the first half the same as the second half. Next change the *second* digit to a 7, and change the *eleventh* digit (this will be the same number as the second digit but in the second nine digits) to a 7 also. Then bet them that you can tell them what is left after dividing the number by 7 faster than they can figure it out by hand. The answer is 0 -- 7 divides into this new number exactly with nothing left over!

## Fun Number Tables

The following fun tables are from one of my favorite books of all time, *Recreations in the Theory of Numbers*, by Albert H. Beiler, published by Dover Publications. This book actually explains the mathematical reasons these tricks work.

## Did You Know...?

Each and every 2-digit number that ends with a 9 is the sum of the multiple of the two digits plus the sum of the 2 digits. Thus, for example, 29= (2 X 9) + (2 + 9). 2 X 9 = 18. 2 + 9 = 11. 18 + 11 = 29.

40 is a unique number because when written as "forty" it is the only number whose letters are in alphabetical order.

A **prime** number is an integer greater than 1 that cannot be divided evenly by any other integer but itself (and 1). 2, 3, 5, 7, 11, 13, and 17 are examples of prime numbers.

139 and 149 are the first consecutive primes differing by 10.

69 is the only number whose square and cube between them use all of the digits 0 to 9 once each:

69^{2} = 4761 and 69^{3} = 328,509.

One pound of iron contains an estimated 4,891,500,000,000,000,000,000,000 atoms.

There are some 318,979,564,000 possible ways of playing the first four moves on each side in a game of chess.

The earth travels over one and a half million miles every day.

There are 2,500,000 rivets in the Eiffel Tower.

If all of the blood vessels in the human body were laid end to end, they would stretch for 100,000 miles.

## A Math Trick for This Year

This one will supposedly only work in 1998, but actually one change will let it work for any year.

1. Pick the number of days a week that you would like to go out (1-7).

2. Multiply this number by 2.

3. Add 5.

4. Multiply the new total by 50.

5. In 1998, if you have already had your birthday this year, add 1748. If not, add 1747. In 1999, just add 1 to these two numbers (so add 1749 if you already had your birthday, and add 1748 if you haven't). In 2000, the number change to 1749 and 1748. And so on.

6. Subtract the four digit year that you were born (19XX).

Results:

You should have a three-digit number.

The first digit of this number was the number of days you want to go out each week (1-7).

The last two digits are your age.

(Thanks for passing this one on to me, Judy.)

## Where is the String?

The next time you are with a group of people, and you want to impress them with your psychic powers, try this. Number everyone in the group from 1 to however many there are. Get a piece of string, and tell them to tie it on someone's finger while you leave the room or turn your back. Then say you can tell them not only who has it, but which hand and which finger it is on, if they will just do some easy math for you and tell you the answers. Then ask one of them to answer the following questions:

1. Multiply the number of the person with the string by 2.

2. Add 3.

3. Multiply the result by 5.

4. If the string is on the right hand add 8.

If the string is on the left hand add 9.

5. Multiply by 10.

6. Add the number of the finger (the thumb = 1).

7. Add 2.

Ask them to tell you the answer. Then mentally subtract 222. The remainder gives the answer, beginning with the right-hand digit of the answer.

For example, suppose the string is on the third finger of the left hand of Player #6:

1. Multiply by 2 = 12.

2. Add 3 = 15.

3. Multiply by 5 = 75.

4. Since the string is on the left hand, add 9 = 84.

5. Multiply by 10 = 840.

6. Add the number of the finger (3) = 843.

7. Add 2 = 845.

Now mentally subtract 222 = 623. The right-hand digit (3) tells you the string is on the third finger. The middle digit tells it is on the left hand (the right hand would = 1). The left-hand digit tells you it is Player #6 who has the string.

By the way, when the number of the person is over 9, you will get a FOUR-digit number, and the TWO left-hand digits will be the number of the Player.

What is the Secret?

(This is from a great book called, *Giant Book of Puzzles & Games,* by Sheila Anne Barry. Published by Sterling Publishing Co., Inc., NY, 1978, recently reissued in paperback.)

Stay tuned for more Math Tricks. They will be added from time to time, so be sure to check in again.