## Archives: Magical Math

Mathematical Magic and Number Play

*Math Gym 2012-13*

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 ... 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 (4-sided figure). 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?7. Choose any whole number and write it in words (e.g. "thirty-nine" or "one hundred twelve"). Count the letters of the word(s), and write down that total (e.g. "thirty-nine" has 10 letters). Now write

*that*number in words. Count*those*letters and write the total down. Keep repeating until you get stuck in an infinite loop with just one number. That number will always be ... oh, never mind. Why not just try it a few times and find out for yourself?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

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 + ? = ?*9. Multiply any two numbers between 6 and 10 -- on your fingers!

**Place your hands with your palms facing you, so your pinkies are next to each other. Your thumbs each represent the number 6, your index fingers stand for 7, and so on, with your pinkies representing 10. Touch the two fingers you’re multiplying together, and bend all the fingers between them while keeping your other fingers straight.***Step 1:***The 2nd digit of your answer will be the the number of fingers that are bent on your left hand***Step 2:***the number of fingers that are bent on your right hand. (If the product is bigger than 9, use the last digit only, and add 1 to the answer you get in step 3, below).***TIMES**Count***Step 3:***all*the fingers that aren't bent on both hands (including the fingers that are touching). That will be the first digit of your answer (unless you have to add 1 to it because you carried that 1 in step 2).*Example:*For 8x6, touch the middle finger of your left hand (#8) to the thumb of your right hand (#6), and bend down all the fingers between them (the 4th and 5th fingers on your left hand and all but your thumb on your right). There are 2 bent fingers on your left hand and 4 on your right, so the second digit of the answer is 2x4=8. There are 4 fingers that aren't bent, so the first digit is 4. That means 6x8=48.*Can you figure out why that works?*

*Math Gym 2011-12*

1. 12,345,679 x 9 = 111,111,111

12,345,679 x 18 = 222,222,222

12,345,679 x 27 = 333,333,333

12,345,679 x 36 = 444,444,444

12,345,679 x 45 = 555,555,555

12,345,679 x 54 = 666,666,666

12,345,679 x 63 = 777,777,777

12,345,679 x 72 = 888,888,888

Can you predict the next one?

12,345,679 x ? = ?

12,345,679 x 18 = 222,222,222

12,345,679 x 27 = 333,333,333

12,345,679 x 36 = 444,444,444

12,345,679 x 45 = 555,555,555

12,345,679 x 54 = 666,666,666

12,345,679 x 63 = 777,777,777

12,345,679 x 72 = 888,888,888

Can you predict the next one?

12,345,679 x ? = ?

2. Have a friend pick a secret number between 10 and 19 (they shouldn't tell you the number). Tell them to add the digits and subtract that sum from their original number. Tell them to focus hard on the answer. Pretend to be reading their mind. Then -- abracadabra! -- tell them what that answer is.

What will it be? Experiment with a few different secret numbers between 10 and 19 and see what happens. Why do you think it always comes out that way? Can you design a magic trick that works for secret numbers between 20 and 29? How about one that works for any whole number bigger than 9?

What will it be? Experiment with a few different secret numbers between 10 and 19 and see what happens. Why do you think it always comes out that way? Can you design a magic trick that works for secret numbers between 20 and 29? How about one that works for any whole number bigger than 9?

3. Want a trick for multiplying any single digit by 9? Hold your hands out face down in front of you and spread out your fingers. For 9x3 take your 3rd finger from the left (which is the middle finger of your left hand) and bend it down, leaving the other fingers out straight. On the left side of the bent finger there are 2 fingers, and on the right there are 7. That makes 2--7 … and 9x3=27. This trick works for all the 9 times tables up to 9x10. Can you figure out why?

4. Have a friend pick any 3-digit number. Give them a calculator and tell them to multiply it by 7, then that number by 11, then that number by 13. Meanwhile, write down the answer on a piece of paper. You’ll get it much faster than they do ! It’s just the starting number twice (so if the starting number is 456, the answer will be 456,456). Can you figure out why?

5. What’s special about 10,213,223?

It describes itself. Read the digits left to right, and it says: 1 zero; 2 ones; 3 twos; 2 threes. And that’s exactly what the number is made of! Can you find another number like that?

It describes itself. Read the digits left to right, and it says: 1 zero; 2 ones; 3 twos; 2 threes. And that’s exactly what the number is made of! Can you find another number like that?

6. The “square of” a number means that number times itself. Want to find the square of any 2-digit number ending in 5 really fast? The first two digits of the answer will be the first digit of the number you’re finding the square of, times that number plus one. So, e.g., the first two digits of 75 squared will be 7x8, or 56. The last two digits of the answer are always 25. So 75 squared is 5625. Another example: The first 2 digits of 35 squared are 3x4, or 12 – so 35 squared is 1225. Why does that work?

7. 1 x 8 + 1 = 9

12 x 8 + 2 = 98

123 x 8 + 3 = 987

1234 x 8 + 4 = 9876

12345 x 8 + 5 = 98765

123456 x 8 + 6 = 987654

1234567 x 8 + 7 = 9876543

12345678 x 8 + 8 = 98765432

Can you predict the next one?

? x 8 + ? = ?

12 x 8 + 2 = 98

123 x 8 + 3 = 987

1234 x 8 + 4 = 9876

12345 x 8 + 5 = 98765

123456 x 8 + 6 = 987654

1234567 x 8 + 7 = 9876543

12345678 x 8 + 8 = 98765432

Can you predict the next one?

? x 8 + ? = ?

8. Here’s a famous number triangle, called Pascal’s Triangle. Each number is the sum of the two numbers above it:

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1 … and so on.

It’s famous because there are many neat things about it. Here’s one: in the second row, the numbers going across add up to 2. In the third, they add up to 4 (2x2). In the fourth, they add up to 8 (2x2x2), and so on. Here’s another: start with any “1” on the left side of the triangle, and go diagonally down to the right as far as you want. Add all those numbers together. Their sum will be the number in the next row just to the left of where you stopped. Can you find other cool patterns?

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1 … and so on.

It’s famous because there are many neat things about it. Here’s one: in the second row, the numbers going across add up to 2. In the third, they add up to 4 (2x2). In the fourth, they add up to 8 (2x2x2), and so on. Here’s another: start with any “1” on the left side of the triangle, and go diagonally down to the right as far as you want. Add all those numbers together. Their sum will be the number in the next row just to the left of where you stopped. Can you find other cool patterns?

9. Want a trick to multiply any 2-digit (or bigger) number by 11? The first and last digits of the answer will be the same as the first and last digits of your starting number. The second digit of the answer will be the sum of the 1st and 2nd digits of the starting number, and each subsequent middle digit of the answer will be the sum of the starting number’s next pair of digits. (If any sums are greater than 10, carry the 1 and add it to the preceding digit of the answer.) Can you figure out why it works?

For example,

34 x 11: 3 (3+4) 4 = 374

213 x 11: 2 (2+1) (1+3) 3 = 2343

1427x11: 1 (1+4) (4+2) (2+7) 8 = 15697

For example,

34 x 11: 3 (3+4) 4 = 374

213 x 11: 2 (2+1) (1+3) 3 = 2343

1427x11: 1 (1+4) (4+2) (2+7) 8 = 15697

10.

(1) Choose a whole number.

(2) Double the number.

(3) Add 10 to your new number.

(4) Now, divide the total by 2.

(5) Finally, subtract your original number.

(6) Your answer will always be 5!

Can you figure out why?

(1) Choose a whole number.

(2) Double the number.

(3) Add 10 to your new number.

(4) Now, divide the total by 2.

(5) Finally, subtract your original number.

(6) Your answer will always be 5!

Can you figure out why?

11. 9 x 9 + 7 = 88

98 x 9 + 6 = 888

987 x 9 + 5 = 8888

9876 x 9 + 4 = 88888

98765 x 9 + 3 = 888888

987654 x 9 + 2 = 8888888

Can you predict the next one?

? x 9 + ? = ?

98 x 9 + 6 = 888

987 x 9 + 5 = 8888

9876 x 9 + 4 = 88888

98765 x 9 + 3 = 888888

987654 x 9 + 2 = 8888888

Can you predict the next one?

? x 9 + ? = ?

12. Give 3 dice to a friend and have them roll the dice where you can’t see them. Tell them to multiply the number on die #1 by 2, add 5, and multiply by 5. Then have them add that total to the number from die #2, and then multiply the result by 10. Finally, have them add the number on die #3. Ask them to tell you the total. In your head, subtract 250 from that number. The answer you get will be the numbers on the three dice (e.g. if they tell you 591, you subtract 250 to get 341 and the dice are 3, 4, and 1). Can you figure out how that worked ?

13. Challenge a friend to race you to square any number between 91 and 109. They can use a pencil and paper, but with a little practice, you can do it faster in your head! (“Squaring” a number means multiplying it by itself.)

(1) Figure out the "secret number": how far away the original number is from 100. For example, if you want to square 94, 6 is the secret number, because 94 is 6 less than 100;

(2) if your original number is less than 100, subtract the secret number from your original number; if it is more, add it, instead. That answer will be the first two or three digits of the solution. In our example, 94-6=88, so the first two digits of 94 squared will be 88;

(3) Multiply the secret number by itself to get the last two digits of the solution. In our example, 6x6=36, so the last two digits of 94 squared are 36. (If the product is only a single digit, put a zero in front of it -- so if you are multiplying 2x2, for example, the last two digits will be 04.)

(4) Combine the first two and last two digits from steps 2 and 3 above to get the solution. In our example, 94 squared is 8836. Try it out!

(1) Figure out the "secret number": how far away the original number is from 100. For example, if you want to square 94, 6 is the secret number, because 94 is 6 less than 100;

(2) if your original number is less than 100, subtract the secret number from your original number; if it is more, add it, instead. That answer will be the first two or three digits of the solution. In our example, 94-6=88, so the first two digits of 94 squared will be 88;

(3) Multiply the secret number by itself to get the last two digits of the solution. In our example, 6x6=36, so the last two digits of 94 squared are 36. (If the product is only a single digit, put a zero in front of it -- so if you are multiplying 2x2, for example, the last two digits will be 04.)

(4) Combine the first two and last two digits from steps 2 and 3 above to get the solution. In our example, 94 squared is 8836. Try it out!

14.

(1) Have your friend think of a number. That number will be called the "mystery number".

(2) Tell them to add 5 to their number, then add 2, and then subtract 3.

(3) Then have them subtract the mystery number and multiply by 100.

(4) Pretend to read their mind. Tell them the number they ended up with is 400.

Unless they made a mistake, it will be. Can you figure out why?

(1) Have your friend think of a number. That number will be called the "mystery number".

(2) Tell them to add 5 to their number, then add 2, and then subtract 3.

(3) Then have them subtract the mystery number and multiply by 100.

(4) Pretend to read their mind. Tell them the number they ended up with is 400.

Unless they made a mistake, it will be. Can you figure out why?

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

What does a snake do?

471x265 + 410699

What you say when you really need help?

(15 x 15 – 124) x 5

Why does the math teacher choose the homework?

(30,000,000 – 2457433) x 2

What does a snake do?

471x265 + 410699

What you say when you really need help?

(15 x 15 – 124) x 5

Why does the math teacher choose the homework?

(30,000,000 – 2457433) x 2

16.

(1) Think of a number and write it on a sheet of paper. Fold it up and place it in your pocket (remember it!).

(2) Ask a friend to think of a number but not tell it to you.

(3) Tell them to double the number they chose.

(4) In your head, double the number you wrote. Tell your friend to add that number to the number they have

so far, then divide the answer by 2.

(5) Then tell them to subtract the number they currently have from the one they started with (or, if the new

number is bigger, the other way around).

(6) Pull out the piece of paper in your pocket and give it to your friend.

The number on the paper will be their answer! (Can you figure out why it works?)

(1) Think of a number and write it on a sheet of paper. Fold it up and place it in your pocket (remember it!).

(2) Ask a friend to think of a number but not tell it to you.

(3) Tell them to double the number they chose.

(4) In your head, double the number you wrote. Tell your friend to add that number to the number they have

so far, then divide the answer by 2.

(5) Then tell them to subtract the number they currently have from the one they started with (or, if the new

number is bigger, the other way around).

(6) Pull out the piece of paper in your pocket and give it to your friend.

The number on the paper will be their answer! (Can you figure out why it works?)

17. Have a friend think of a 2-digit number. Have them tell you the number, then let them use a calculator to multiply it by 3, then that result by 7, then that result by 13, then that result by 37. Meanwhile, write down the answer: you’ll have it first. It’s just the original 2-digit number repeated 3 times. For example, if the original number was 78, the answer is 787878. Why?

18. 6 x 7 = 42

66 x 67 = 4422

666 x 667 = 444222

6666 x 6667 = 44442222

66666 x 66667 = 4444422222

666666 x 666667 = 444444222222

Can you predict the next one?

? X ? = ?

66 x 67 = 4422

666 x 667 = 444222

6666 x 6667 = 44442222

66666 x 66667 = 4444422222

666666 x 666667 = 444444222222

Can you predict the next one?

? X ? = ?

19. Have a friend write down any 8-digit number without showing it to you. Have them add up the digits and subtract that sum from the original number. Then have them cross out any digit they want of that answer. Have them add up the digits that remain and tell you the sum. From that sum, you can figure out what digit they crossed out. Just subtract the number they gave you from the next multiple of 9 (so, for example, if they said 22, it would be 27-22=5).

20. Ask a friend their favorite digit from 1-9. Multiply it by 12345679. Multiply that by 9. The answer will contain only their favorite digit.