Mathematical Magic Tricks T. Christine Stevens, American Mathematical Society

<stevensc@slu.edu> Project NExT workshop, Chicago, Illinois, 7/25/17

Here are some magic tricks that I have used with students who are learning to write proofs. There are some for which I don’t remember the specific sources, but I list here some good sources for mathematical magic tricks:

Colm Mulcahy’s “Card Colm” column on the MAA website <cardcolm-maa.blogspot.com>

Old issues of Math Horizons (#5 is adapted from an article entitled “An ESPeriment with Cards,” by Colm Mulcahy, in the February, 2007, issue)

Edward Burger, Extending the Frontiers of Mathematics: Inquiries into proof and argumentation, Key College/Springer, 2007 (#1 and #2 are adapted from this book, pp. 6-7).

Arthur Benjamin and Jennifer Quinn, Proofs that Really Count, MAA, 2003 (#10 is adapted from this book, pp. 30-31).

Mathematically-based puzzle books that are aimed at middle school students (e.g. Cool Math, by Christy Maganzini, Prince Stern Sloan, 1997 – #6 appears on p. 83 of this book)

The history of mathematics (#7 is an old method of multiplication, sometimes called “the medieval peasant’s method” or “mediation and duplation”)

The website <www.cut-the-knot.org> and other sites that turn up when you google phrases like “mathematical magic tricks”

1) A card trick: Take an ordinary deck of 52 playing cards. Shuffle it thoroughly, and thendivide it into two stacks of 26 cards each. Count the number of black cards in the firststack and the number of red cards in the second stack. Magically, these numbers are thesame!a. Surprise your friends by showing them this trick.b. Explain why this trick always works.

2) A coin trick: Some pennies are spread out on a table. Each coin is lying either heads upor tails up. Unfortunately, you are blindfolded, and you cannot see the coins or thetable. You can feel your way around the table and count the coins, if you wish. Butyour touch isn’t sensitive enough to tell you whether a coin is lying heads up or tails up.

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Someone tells you how many coins are lying heads up. You choose that many coins, turn each of them over, and move them to the side. You now have two groups of coins (the ones that you moved, and the ones that you did not move). Magically, the number of coins lying heads up in the first group is the same as the number of coins lying heads up in the second group.

Carefully explain why this trick works. If you use any symbols in your proof (x, H, n, etc.), be sure to say what they represent.

3) A number trick: You will need your calculator for this trick, which uses your seven-digittelephone number. For example, my office phone number is 977-2436.

Take the first three digits of your telephone number and multiply by 80. [So I wouldmultiply 977 by 80.] Then add 1 to the result. Now multiply by 250. Then add tothe result the last four digits of your telephone number. [So I would add 2436.] Againadd the last four digits of your telephone number to the result. Subtract 250, and thendivide by 2. Magically, the final result is your telephone number!a. Practice this trick on some of your friends.b. Figure out why this trick works.

4) Pick any natural number less than 100, and write it out in English. [For example, 89 =eighty nine.] Count the number of characters (including spaces) that you used to writeyour number, thus getting a new number. [In our example, you would get 11.] Nowwrite out that number in English and count the characters, getting another new number.Perform this process five times. Magically, you will get the number four!a. Practice this trick with a variety of starting numbers.b. Why does this trick work?

[This trick relates to recursively defined sequences and also to logical paradoxes, such as “the smallest number that cannot be described in English in less than 100 characters.”]

5) Take two or three decks of cards that can be easily shuffled together. For this trick, youwill need the aces, twos, threes, fours, and fives (of all suits) from these decks. Arrangethese cards into a stack, in sets of five, in the following order: ace, two, three, four, five;ace, two, three, four, five; etc. Now shuffle the cards overhand. Then count off anynumber of cards from the stack, placing them face down, making a new stack [The nextshuffle will be easier if you count off approximately half the cards.]. Shuffle the newstack into the original stack, using a standard two-handed shuffle. Note that you haveshuffled the cards twice (once overhand, and once two-handed). Now turn the cardsover, one by one. Magically, despite the shuffling, the top five cards will be an ace, atwo, a three, a four, and a five, in some order; the next five cards will also be an ace, atwo, a three, a four, and a five, in some order; and so on, to the bottom of the deck.a. Practice this trick on your friends.

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b. Try to figure out why this trick works.6) A dice trick: Ask a friend to do the following: Take three dice and roll them. Add up

the total amount shown by the three dice. Pick up one of the dice, look on its bottom,and add that number to your total. Roll that same die, and add the resulting number tothe total. Leave the dice as they are. Now you, as magician, do the following: Add thetotal currently shown on the three dice, and add 7 to that total. Magically, your total isthe same as your friend’s final total! Explain why this trick works.

7) A magical multiplication trick: Choose two natural numbers that you want to multiplytogether, and write each of them at the top of a column. For example, if you want tomultiply 37 by 59, write

it 37 59

Now divide 37 by 2, discarding the remainder, and multiply 59 by 2. This gives you two rows:

37 59 18 118

Keep going until you get 1 in the left hand column:

37 59 18 118

9 236 4 472 2 944 1 1888

Now cross off all the rows that have even integers in the left-hand column. Then add up all the remaining numbers in the right-hand column. Magically, you will get the product of the two numbers you started with! (In our example, you would get 59+236+1888=2183.)

Show this magical multiplication trick to your friends, and try to figure out why it works.

8) A three-card trick: For this trick you need only three cards: an ace (which we will treat asa “one”), a two, and a three. Line them up in increasing order, left to right, facing up.Turn your back so that you can’t see the cards. Ask a friend to choose one of the cardsand remember which number it is (If your friend has a faulty memory, you can ask thefriend to write the number on a piece of paper and hide the paper from you.). Tell thefriend to pick up the chosen card and turn it over (in place). Then tell the friend toswitch the places of the other two cards and turn them over, too.

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Now pick up the cards so that the rightmost card is on top, the middle card is in the middle, and the leftmost card is on the bottom, keeping them face down. Without looking at the cards, move the top card to the bottom of the stack. Repeat, until you have done this exactly ten times. Keep the stack face down and lay the cards out, face up, putting the top card in the middle, the second card on the right, and the bottom card on the left.

Magically, exactly one of the following will be true: the three will be on the left, the two will be in the middle, or the one will be on the right. Whichever one it is, that is the card your friend picked!

Practice this trick on your friends, and try to figure out why it works.

Note: You can add some extra “hype” to the trick by laying the cards out face down at the end. Then ask your friend to guess which card is the one he or she picked, and turn that card over. If your friend picks the left card and it’s a three, or if your friend picks the middle card and it’s a two, or if your friend picks the right card and it’s a one, say “Congratulations! You picked your card.” Otherwise, say, “I’m sorry. You guessed wrong.” Then turn over the other two cards and use the rule given above to determine which card your friend chose at the beginning. Pick it up and say, “Here is your card.”

9) Another magical multiplication trick: For this trick, your friend will need a calculator.Ask your friend to choose a 3-digit number but not tell you what it is. Tell your friendto multiply that number by 4167, but not tell you the answer. Then tell your friend topick one of the non-zero digits in the product and keep it a secret. Finally, ask yourfriend to read the other digits of the product (including any repeated digits) in anyrandom order. You now mentally add up the digits that your friend read out, andsubtract the sum from the next strictly higher multiple of 9. Magically, the result will bethe missing digit that your friend picked!

For example, if your friend picked 217, the product would be 904,239. Your friendmight pick 4 as the secret digit and then read out 2, 9, 0, 9, 3. The sum of these digits is23, and the next higher multiple of 9 is 27. So the missing digit is 27-23=4.

a. Practice this trick on your friends.b. Explain why this trick always works.c. Why must your friend pick a non-zero digit as the secret digit?

Note: You can vary this trick by replacing 4167 with any number that is divisible by 9.

10) A Fibonacci trick: Make a table with ten rows, and label them 1 through 10. In the firsttwo rows, put any positive integers that you like. In the third row, put the sum of rows 1and 2. In the fourth row, put the sum of rows 2 and 3. Keep on in this way, until you

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have completed the table. Now divide the number in the last row by the number in the ninth row. Magically, the result (to two decimal places) is always 1.61.

Explain why this trick works. [Hint: Start by showing that the quotient you computed is always greater than $21/13$ and less than $34/21$.] If you use any variables in your explanation, be sure to say what they represent.