2. Tell your partner to examine the cards, and give you the cards on which the number

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Oswald Lenard Martin
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1 Magic Cards Instructions: 1. Ask your partner to pick a whole number between 1 and 63 (and keep it secret). 2. Tell your partner to examine the cards, and give you the cards on which the number appears. (A number may occur once, twice or up to six times, so check the cards very carefully.) 3. Now you can tell your partner what the secret number was. The trick: just add the numbers in the upper left corners of the cards. Magic Cards Instructions: 1. Ask your partner to pick a whole number between 1 and 63 (and keep it secret). 2. Tell your partner to examine the cards, and give you the cards on which the number appears. (A number may occur once, twice or up to six times, so check the cards very carefully.) 3. Now you can tell your partner what the secret number was. The trick: just add the numbers in the upper left corners of the cards.

8 The 27 Card Trick Description: You fan a deck of 27 cards, so that your partner can see them. Ask your partner to memorize one of the cards and keep it secret; also, ask your partner to pick a whole number between 1 and 27, inclusive, and tell it to you. Then you deal out the cards into three piles, and ask which of the piles the the card is in; then pick up the three stacks of cards. Deal out the cards into piles a second time, ask where the card is, and pick up the three piles. Deal into piles, ask where it is, and pick up the piles of cards one last time. Now count as you deal the cards face up: the card will come up when you get to the number chosen. Instructions: 1. Subtract one from the number. 2. Convert the resulting number from base 10 to base 3. It will have a 9's digit, a 3's digit, and a 1's digit. (Each of these will be a number equal to 0, 1, or 2. These digits tell you how to pick up the cards.) 3. Deal out the cards into 3 piles: be careful to deal the cards into piles in order. Ask your partner to tell you which of the three piles the card is in after you have nished dealing. 4. The rst time: If the 1's digit is a zero, put the pile with the card on the bottom of the other two piles, before you turn the deck over; If the 1's digit is a 1, put the pile with the card between the other two piles; If the 1's digit is a 2, put the pile with the card on top of the other two piles. 5. Turn the deck over and deal the cards into piles again. Ask where the card is. 6. The second time: If the 3's digit is a zero, put the pile with the card on the bottom of the other two piles, before you turn the deck over; If the 3's digit is a 1, put the pile with the card between the other two piles; If the 3's digit is a 2, put the pile with the card on top of the other two piles. 7. Turn the deck over and deal the cards into piles one last time. Ask where the card is. 8. The last time: If the 9's digit is a zero, put the pile with the card on the bottom of the other two piles, before you turn the deck over; If the 9's digit is a 1, put the pile with the card between the other two piles; If the 9's digit is a 2, put the pile with the card on top of the other two piles. 9. Now turn the deck over and count the cards as you deal. The card you turn over when you say the number you were given should be the secret card.

9 Lesson Plan for the card tricks 1. Magic Cards (a) Do the Trick with a volunteer. (b) Teach the Students how to do it. (c) The Decimal number system: Every whole number can be written as a sum. For Example 324 = or 5280 = The numbers 1000, 100, and 10 are all powers of 10: 1000 = 10 3 ; 100 = 10 2 ; 10=10 1 and the multipliers are digits between 0 and 9, inclusive. (d) Base { 2, or Binary, notation. In the same way, we can represent numbers as sums of powers of 2. The powers of 2 are For example 2 1 =2; 2 2 =4; 2 3 =8; 2 4 =16; 2 5 =32; etc: 25 = = # # # # # so (11001) 2 represents 25 in base 2 notation. Notice that we put zeros in as placeholders for the powers that were missing. (Give another example, this time with an even number. Explain: look for the highest power of 2 which is less than or equal to the number, and subtract it from the number. Then do the same thing with the remainder.) (e) The way the magic cards work is that the number in the upper left corner of each card is a power of 2; and the numbers on the card are the numbers which have that power of 2 as part of their binary representation. So that when you add up the numbers in the upper left corners, you are just adding up the numbers in the binary representation of your number, so of course you just get your number back. 2. The 27 card trick. (a) Base 3, or ternary, notation. By the same token, we can represent any whole number as a sum of multiples of powers of 3, with multipliers which are 0, 1, or 2. (In base 10 notation, the multipliers are the digits: 0, 1, up to 9. In base 2 notation, the only multipliers are 0 and 1; in general, in any base, the multipliers are 0, 1, up to one less than the base.)

10 (b) The powers of 3 are 3 1 =3; 3 2 =9; 3 3 =27; 3 4 = 81. So if we want to represent a number in base 3, choose the largest power of 3 which is less than or equal to the number, and divide it into the number; then repeat the process with the remainders, until nally you get a remainder which 0, 1, or 2. For example, start with the number 65: 27 is the largest power of 3 which is less than or equal to 65; 27 goes into 65 twice, with a remainder of 11, so 65 = (2 27) is the largest power of 3 which is less than or equal to 11; 9 goes into 11 once with a remainder of 2; so 65 = (227)+11 = (227)+(19)+(21). Then 65 = = # # # # so (2102) 3 represents 65 in base 3 notation. Example: (48) 10 = (1210) 3. point out that the 0 at the end is needed as a placeholder. (c) Have them practice converting from base 10 to base 3 notation. Have then practice converting from base 3 to base 10 notation. (d) Have then practice the 27 card trick.

Teaching the TERNARY BASE

Features Teaching the TERNARY BASE Using a Card Trick SUHAS SAHA Any sufficiently advanced technology is indistinguishable from magic. Arthur C. Clarke, Profiles of the Future: An Inquiry Into the Limits