MAGIC SQUARES

This box of numbers is a magic square. The numbers in any row, straight or diagonal, add up to 15.

You can make it look different by turning the square like this.

Or using it in a mirror image, like this.

Memorize one of these magic squares.

To do your trick, draw an empty nine-box square on a blackboard or paper and ask your audience to call out numbers from 1 to 9 in any order. As they are called out, put them into the correct box. Of course, only you know where to put each number to make a magic square!

When all of the boxes are filled, ask your audience to add up the rows of numbers, and they will see that all the rows add up to 15.

Here is another square where the rows add up to 24. You can memorize this one too.

As above, ask your audience to call out numbers from 4 to 12. Write each number in the correct box of the square as it is called out. When the square is filled, ask your audience to add up the rows. Every row adds up to 24!

Your patter might go something like this: "When my mother was pregnant with me, she won a lot of money at a bingo game—enough to pay the doctor for my delivery. This made such an impression on her, that I was born with numbers buzzing through my head. Here is a square with nine empty boxes, etc."

If you really want to make an impression, here is a 25-box square where all the rows add up to 65! If you can memorize it, great. Otherwise you can write it on a tiny piece of paper and hide it in the palm of your hand with a bit of clear tape.

In this square, the horizontal, vertical and two long diagonal rows all add up to 65.

As before, draw your empty magic square on the blackboard and call for numbers from one to twenty-five in random order.

To further implant in the minds of your audience that you are indeed a wizard, repeat the performance, but in your mind first give the square a quarter turn or two and fill in the boxes accordingly. It will look like a completely different magic square which you are able to make to add up to 65 again.

CALENDAR TRICK

You can take any page of a calendar and mark off a square four numbers across and four numbers up and down. The sum of the numbers in the upper left and lower right corners will always equal the sum of the remaining two corners. In the illustration it is 36.

Do this: Circle any number inside the square and cross out all the other numbers in the same row across and the same up-and-down row. Now circle a number you haven't marked yet and do the same thing. Continue until all the numbers in the marked square are either circled or crossed out. The sum of the circled numbers will be the same as the sum of the four corner numbers—in this case 72.

To do this trick, ask someone in your audience to mark off a 4-x-4 square on a calendar page while you watch. As this is done, be sure to see the numbers in either set of opposite corners. In your head, add them and multiply by two. Write the total on a piece of paper, fold it several times, and hand it to one of your audience for safekeeping.

Now tell the person holding the calendar page to hide it from your view and then to circle numbers and cross out, as described above. When all the numbers have been crossed out and circled, tell someone to add up all the circled numbers; then open your piece of paper, and show everyone that you had predicted the correct answer, in this case—72

Remember, you must be sure to see one set of corner numbers!

Your patter can go something like this: "I have magic telepathic perception which enables me to see into the immediate future. I am therefore able to foresee the numbers you will circle and know their total even before you circle them. There is no way you can squirm out of this situation, because I will have perceived it before you do it."

MAGIC ADDITION BOXES

This is a very easy trick to do. It is a trick where you show a square made of many boxes; each box has a number in it, as in the illustration, and is large enough for a penny to fit in it and cover the number.

Ask your audience to cover any number with a penny and cross out all the numbers in line with it in the rows across and down, just as in the Calendar Trick on page 15. Have them repeat this out of your sight until all the numbers are either covered or crossed out. Then announce the sum of all the covered numbers—in this case, 52. When the pennies are removed and the numbers added, everyone will see that you were right!

If you study the square in the illustration for a minute, you will see that it is really an addition square. Each number inside the square is really the sum of the two numbers outside the two rows—across and up and down— that number is in.

You can build such a square with any number of boxes. We chose to use a 5-x-5 square, and we chose 52 as our key number. Any two-digit number will work. First make the square. To make it easy to remember the key number for the square you have built, put the digits of the key number in the upper left and lower right boxes. In our illustration the 5 and the 2 are so placed. Then, in pencil, put numbers outside the square that add up to those digits—these numbers are circled in the illustration. The 2 plus the 3 equal the 5, and the 0 and 2 equal 2. Now write other numbers on the outside of the square in pencil. Any numbers will do, as long as they all add up to your key number—52 for this square. In our illustration we have chosen the 8, 6, and 9, and the 10, 7, and 5. Now fill in the boxes by adding up the outside numbers. The 8 and the 3 equal 11, 6 + 3 = 9, 9 + 3 = 12, 0 + 3 = 3. This completes the top row in our illustration. Complete the rest of the square. Now erase all the penciled numbers outside the square and you are ready for your trick. Prepare several such squares, each one different, so that your presentation will have variety and mystery.

SUBTRACTION TRICK

Tell your audience that you are not only a mathematical genius, but that you can also read their minds.

Tell someone to think of a three-digit number where the three digits are all different, reverse it, and subtract the smaller from the larger number. Let them use paper and pencil. Ask for the last digit of the answer, and then immediately announce the whole answer.

It's as easy as pie. When the smaller number is subtracted from the larger, the middle digit is always 9, and the two outside digits always add up to 9. So, given the last digit, you can quickly determine the first digit, and you know the middle one is 9, so you have the whole answer. Here's an example.

Number thought of 451

Reversed                    –154

Subtracted                   297

If the last number is a nine, the first must be a zero, since you know that the first and last number must add up to nine. The answer in such a case is 099.

TAKE A NUMBER

This is a "take-a-number" trick that is easy to do, yet mystifying. It is simple enough for your audience to do without pencil and paper. The variations are endless. You can change it each time you do it to keep your audience off balance.

Tell your audience to think of a number, triple it, add 18, divide by 3, subtract the original number. Tell them that the answer is 6. Several people can do it at once, and each may think of a different number, yet the answer will always be 6.

Variation: The "add" number can be any number divisible by 3. Then the answer will always be of the "add" number.

Variation: Think of a number, double it, add 18, take the half (divide by 2), subtract the original number. The answer will be 9, which is half the "add" number.

How this works can be seen if you examine the process closely. You have taken a number and multiplied it by three and then divided it by three again (forgetting the added number for a moment). So you're back where you started, except for the added number. When you subtract the first number, what's left is nothing except a third of the added number. The number you multiply by must always be the same as the one you divide by, and the number you add must be an even multiple of it.

MAGIC NUMBERS

The following pages describe a few magic numbers. They are all strange, and have strange things happen to them. You can have lots of fun with your friends if you memorize them and play with them.

Tell your audience that when you were very little you had several sets of blocks with numbers on them. You used to build castles and walls with them. Frequently, when they fell down, they were arranged in strange and magic combinations. Following are some.

Pass out paper and pencil and tell your audience to add, and then to multiply, the following pairs of numbers: 9 and 9, 24 and 3, 47 and 2, 497 and 2. Tell them to examine their answers and see the magic relationships between the answers. The addition answers and the multiplication answers have their digits reversed.

This magic number is 37.

Ask your audience to multiply this number by 1, by 2, by 3, etc., through 9. Then multiply each answer by 3, and see the magic sequence of answers.

The answers will be 111, 222, 333, 444, etc.

This magic number is 12345679. Notice there is no 8 in it.

Tell your audience to multiply this number by 1, 2, 3, 4, etc. Then, since nine is the largest digit and also has magic properties, tell them to multiply the answers by nine. They will be astounded by their answers. The answers will by 111111111, 222222222, 333333333, 444444444, etc.

Another magic number is 15873. Tell them that this time instead of 9, you will use the magic number, 7. Why is 7 magic? Well, there are 7 openings in the head, 7 days of the week; the sum of opposite faces of dice is 7, the Biblical span of life is 7 X 10. Tell them to multiply the number by 1, 2, 3, 4, etc., and then multiply the answers by 7.

The answers will be 111111, 222222, 333333, 444444, etc.

Now tell your audience to take the strings of ones, twos, threes, etc. and multiply each string by nine and see the magic results. Tell them to note the sum of the first and last digit of these answers.

Magic Number 142857

Pass out pencil and paper and ask each person, in turn, to multiply this number by 2, 3, 4, 5, and 6. Tell the next three people, if there are that many, to multiply it by 7, 14, and 21 respectively.

When the answers are ready, tell your audience to add the digits up in their answers. Each person will get the same total, 27, except the last three. They will get twice 27, or 54.

Next point out other magic features of the number. The first digits of all the answers are in ascending order, which you would expect since the number gets bigger as you multiply by a larger number. But all those first digits are contained in the magic number itself! All this is shown in the chart.

The next magic feature of this number is that all the answers have all the same digits in the same order as the original number! If you arrange them in a circle, you can see this magic fact.

Your patter might go something like this: "Once in a dream I saw a large truck go by. It had these numbers on its wheels: 1, 4, 2, 8, 5, 7, and they were going round and round. It made me dizzy, and when I awoke they were still going around in my head, so I wrote them down in a circle and found that they made a really magic number. See all the things it can do?"

Now, if you multiply the magic number by multipliers larger than 6 you will see an interesting phenomenon. When the number is multiplied by 7, the answer is a string of 9's. When the number is multiplied by multiples of 7—14, 21, 28, etc.—the answer is also a string of 9's except that the first and last digits add up to 9.

But when the number is multiplied by other multipliers, as shown, the answers still show the original magic number with the digits in the same order, except that one of the digits is broken into two numbers and equals the sum of those numbers. For instance, when the number is multiplied by 8, the answer in the wheel is 1, 4, 2, 8, 5, 7 (6 + 1).