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Magical Math (2012-13)

Mathematical Magic and Number Play

1. Line up the digits one through 8 in a vertical column, with the 1 on top and the others, in order, beneath it. Just to the right of that column, starting next to the 1, vertically line up the digits 8 through 1 in descending order. Look: you’ve made the 9 times tables! Why does that work?

2. Have a friend put a dime in one hand and a penny in the other (not showing you which one is where). Have them multiply the value of what’s in their left hand by 4, 6, or 8. Have them multiply the value of what’s in their right hand by 3, 5, or 7. Have them add those numbers together and tell you the total. If the total is even, tell them the penny is in their left hand. Otherwise it’s in their right. Why does that work?

3. Take any positive integer. If it’s odd, multiply by 3 and add 1. If it’s even, divide by 2. Use those same rules on the new number you end up with ... and keep repeating in that way (some starting numbers will need more repetitions than others) … and no matter which number you started with, eventually you’ll get to the number 1.

4. Want to know if a number is divisible by 3? Add the digits. If

**that**number is divisible by 3, so is the original number (if you still have a large number and aren't sure whether it's divisible by 3, add the digits of the*new*number and check again ... and keep repeating until the result is small enough that you know whether it's divisible by 3). The same rule works to check whether a number is divisible by 9: if you add the digits and the sum is divisible by 9, so is the original number.5. Draw any quadrilateral. Connect the midpoints of each side to the midpoints of the adjacent sides. You’ll always draw a figure whose opposite sides are parallel (a parallelogram) -- no matter what the starting quadrilateral looked like.

6. Draw a 3x3 box around any nine days on a calendar page. Ask a friend to use a calculator to find what all nine dates add up to. In the meantime, write down the answer on a piece of paper. You’ll have the answer faster than they will! Just look at the number in the middle of the box and multiply it by 9

*. (Not an expert multiplier yet? No problem. Just multiply by 10 by adding a zero at the end of the number, and then subtract the original number.)*Can you figure out why that works?6. Choose a number and write it as a word. Count the letters and write that total down (e.g. thirty-nine has 10 letters). Write

*that*number of letters as a word. Count those and write that total down. Keep repeating -- eventually you’ll always end up with 4.8.

12 x 9 + 3 = 111

123 x 9 + 4 = 1111

1234 x 9 + 5 = 11111

12345 x 9 + 6 = 111111

123456 x 9 + 7 = 1111111

1234567 x 9 + 8 = 11111111

12 x 9 + 3 = 111

123 x 9 + 4 = 1111

1234 x 9 + 5 = 11111

12345 x 9 + 6 = 111111

123456 x 9 + 7 = 1111111

1234567 x 9 + 8 = 11111111

*Can you figure out the next one?**? x 9 + ? = ?*8. To multiply any two numbers between 6 and 10 on your fingers: Put your hands palms up in front of you. Finger numbers go 6 7 8 9 10; 10 9 8 7 6. Touch the two fingers you’re multiplying together and put down all the fingers in between them. The first digit of the answer is the sum of all the fingers that are up. The second digit is the product of the fingers down on your left hand times the fingers down on your right. If the product is bigger than 9, carry the one and add it to the first digit.

10.

Have a friend write a 5-digit number. You quickly write another under it, then they write another under that, then you, then they. You immediately can write down the sum of all 5. When it’s your turn, make sure that each digit in your number, when added to the one above it, will total 9. To get the total, put a 2 in front of the last number your friend wrote, and subtract 2 from the last digit.

Have a friend write a 5-digit number. You quickly write another under it, then they write another under that, then you, then they. You immediately can write down the sum of all 5. When it’s your turn, make sure that each digit in your number, when added to the one above it, will total 9. To get the total, put a 2 in front of the last number your friend wrote, and subtract 2 from the last digit.

11. Cut out any triangle from a piece of cardboard. Ask someone with one try to balance the cardboard on a tip of a pencil. You can do it on the first try. Draw a line from the middle of each side to the opposite point. Where those three lines meet, draw a dot. Put the pencil there, and it will balance perfectly. (That point is called the “centroid.”)

12. Have a friend choose any 3-digit number where the units and hundreds digits are different but not to tell you the number. Tell them to reverse the digits and subtract the smaller from the larger. Then have them reverse the digits of the difference and add the last two numbers. Pretend to read their mind, then tell them the number they ended up with is 1089.

13. Want to know quickly if a number is divisible by 4? Just look at the last two digits. If

**that**number is divisible by 4, so is the original number.14. Have a friend think of a number less than 10. Tell them to double it and add 8, and then divide the new number by 2 and subtract the original number. The answer will always be 4. You can change the trick around each time and predict different answers … the answer will always be half of the number you asked them to add to their original number.

15. Did you know your calculator knows English? Type in the following problems and turn the calculator upside down to read the answers.

How did the mathematician feel when he had too much pi?

1234- 463

Joe ran out of math problems to do in his free time. What does he do about it?

(851 squared) – 143667

Katie said all numbers that are divisible by 3 are odd. What do you think of that?

(23 to the fifth power) – 1118998

How did the mathematician feel when he had too much pi?

1234- 463

Joe ran out of math problems to do in his free time. What does he do about it?

(851 squared) – 143667

Katie said all numbers that are divisible by 3 are odd. What do you think of that?

(23 to the fifth power) – 1118998

16.

142857 x 1 = 142857

142857 x 2 = 285714

142857 x 3 = 428571

142857 x 4 = 571428

142857 x 5 = 714285

Do you notice something about those answers? Look at the digits. Can you predict the answer if you multiply it by 6? Because of that pattern, it's called a "cyclic number." A different interesting thing happens when you multiply it by 7, 4, or 21. Can you predict what will happen if you multiply it by 28?

142857 x 1 = 142857

142857 x 2 = 285714

142857 x 3 = 428571

142857 x 4 = 571428

142857 x 5 = 714285

Do you notice something about those answers? Look at the digits. Can you predict the answer if you multiply it by 6? Because of that pattern, it's called a "cyclic number." A different interesting thing happens when you multiply it by 7, 4, or 21. Can you predict what will happen if you multiply it by 28?

17. A "palindrome" is a number (or word) that reads the same forward and backwards (like "racecar" or "1991"). Most 2-4 digit numbers will become palindromes with this trick: Choose a starting number that’s not a palindrome. Reverse its digits and add that new number to your original one. If that sum is not a palindrome, add it to the reverse of

**that**number. Keep repeating until the answer is a palindrome. With many numbers you will get a palindrome quickly with this method; a few take longer. The number 196 is a stubborn exception: mathematicians have done thousands of additions of the reverse number but still haven’t ended up with a palindrome!18. Have a friend stack three dice into a tower and hide it from view. Ask them to tell you just what number is on top of the tower, and tell them that you'll magically tell them the sum of all the hidden faces of the tower (the bottom of the top die and the top and bottom of the two underneath). How can you do it? Need a hint? Look carefully at a die. What do the opposite sides always add up to?

19. 1089 * 1 = 1089

1089 * 9 = 9801

1089 * 2 = 2178

1089 * 8 = 8712

1089 * 3 = 3267

1089 * 7 = 7623

1089 * 4 = 4356

Can you predict (without calculating) what 1089 x 6 will be? How about 1089 x 5? The same pattern works with 2178 – which happens to be twice 1089! Check it out!

1089 * 9 = 9801

1089 * 2 = 2178

1089 * 8 = 8712

1089 * 3 = 3267

1089 * 7 = 7623

1089 * 4 = 4356

Can you predict (without calculating) what 1089 x 6 will be? How about 1089 x 5? The same pattern works with 2178 – which happens to be twice 1089! Check it out!

20. Put 12 playing cards on a table: 5 face up, 7 face down. Tell a friend they have to close their eyes and you will shuffle the cards. Without opening their eyes, they need to sort the cards into two piles of any size, flipping over any that they want, so that each pile has the same number of face-up cards.

When they fail, you can take a turn and dazzle them with your magic.

When they fail, you can take a turn and dazzle them with your magic.

*With your eyes closed, separate them into two piles of 5 and 7 cards each. Flip over every card in the smaller pile. How could that possibly work? Because of math! Let's call the number of cards that ended up face-up in the big pile U. Since you started with 5 cards face up altogether, the smaller pile then has 5-U cards face up. Since there are a total 5 cards in the small pile, 5 minus the number of heads in that pile is the number of tails. That means the smaller pile has 5-(5-U) tails, which is the same as U tails. Flip them all over and the small pile will have U heads, same as the big pile.*