Addition Puzzle

You'll need to print this out (use ctrl-P on your computer).

I Wonder Why - 34

Take the numbers 1 through 16 and write them into a 4 x 4 grid like so:

1	2	3	4
5	6	7	8
9	10	11	12
13	14	15	16

Now choose a random number in the grid and circle it. Mark out the rest of the numbers in the same row and column.
Do this three times total and you should have one number left. Circle it.

Sum the four numbers you have circled. Why does the sum always equal 34?

Answer: By choosing this way we are picking 4 numbers that do not share rows or columns. The sum of the first column is 1+5+9+13 = 28, but by forcing picks 1, 2 and 3 columns to the right for each row the sum is forced to be 28+1+2+3 = 34.

Odd and Even Multiplication

Take any two odd numbers, like 5 and 7. Multiply them together. 5 X 7 = 35
Now some more odd numbers and multiply them together.
What can you say about multiplying two odd numbers?
Answer: When you multiply two odd numbers the answer is always an odd number.

Now take two even numbers, like 4 and 8. Multiply them together. 4 X 8 = 32
Now take some more even numbers and multiply them together.
What can you say about multiplying two even numbers?
Answer: When you multiply two even numbers the answer is always an even number.

Now take an even and an odd number, like 5 and 8. Multiply them together. 5 X 8 = 40
Now take some more even and odd pairs of numbers and multiply them together.
What can you say about multiplying an odd and even number?
Answer: The answer is always even.

If multiplication works like this with odd and even pairs, what can you say about multiplying more that two numbers together?
Look at the following multiplications and see if you can see a pattern:
2 X 3 X 4 X 5 = 120                        E O E O = E
2 X 3 X 4 X 5 X 7 = 840                 E O E O O = E
2 X 3 X 2 X 3 X 5 X 3 = 540          E O E O O O = E
2 X 3 X 3 X 5 X 3 = 270                 E O O O O = E
3 X 5 X 3 = 45                                 O O O = O

Answer: Any even number in the multiplication will make the answer even.

Sums of Consecutive Numbers

Alice has been thinking about sums of consecutive numbers. Here is part of her working out:


9 = 4 + 5 and 2 + 3+ 4
10 = 1 + 2 + 3 + 4
11 = 5 + 6
12 = 3 + 4 + 5
13 = 6 + 7
14 = 2 + 3 + 4 + 5
15 = 7 + 8 and 4 + 5 + 6 and 1 + 2 + 3 + 4 + 5

Rice looked over Alice's shoulder:
"I wonder if we could write every number as the sum of consecutive numbers?"
"Some numbers can be written in more than one way! I wonder which ones?"
"9 , 12 and 15 can all be written using three consecutive numbers. I wonder if all multiples of 3 can be written in this way?"

Let's see if you can answer some of the questions.

Getting Started:
Start by trying some simple cases; perhaps start by exploring what happens when you add two or three consecutive numbers.

1 + 2 + 3 = 6
2 + 3 + 4 = 9
3 + 4 + 5 = 12

Can you explain why the total has gone up by 3 ?
Let us know what you find out. I'll put your explanations with your name on the bottom as answers.

Classic Number Guessing

This trick will impress even your math teacher (but she may already know it).

• Think of a number.
• Double it.
• Add 10 .
• Halve it.
• Take away you original number.
• Is your answer 5?

Try this with a different starting number. Did you get a different result?
Why does this happen?

Write the answer on a piece of paper without letting anybody see it and seal it in an envelope. Have somebody hold the envelope and at the end ask them to open it and reveal the number you wrote at the beginning. Wow, Magic!

You can also try this with a different number than 10 to be added.
What is your result then?

Answer: You can add any number instead of 10 and the final answer will be 1/2 of that number.

Magic 123

Start with a number and count the number E of even and the number O of odd digits. Let's say you pick the number 7397418. E=2 and O=5,

Write them down next to each other following by their sum E + O. So the new number is 257.

Treat the result as a new number and continue the process. So 257 becomes 123. 123 becomes 123.

In this example, iterations converge very rapidly. Moreover, in just a few steps they reach the number 123 which has exactly 3 digits of which 1 is even and two are odd. Therefore, applying the computations to 123 produces the number 123 itself, such that further iterations become really mindless. Of course we can always start with another number.

How to Add or Subtract Two Fractions by the Butterfly Method



The butterfly method is used in Japan to teach addition or subtraction of two fractions. Japanese schools teach this method and encourage students to "forget" the butterfly drawing as they get better using the method.

To add or subtract fractions the butterfly way,

1. Write the fractions side-by-side as usual and draw two wings along the diagonals made by the numerator of one fraction and the denominator of the other fraction and draw an antenna on each wing.

2. As suggested by the wings, that look like a multiplication sign, multiply the numbers in each wing and put the product in the antenna for the wing.

3. Think or say: “This poor butterfly needs a body.” To give it a body, connect the bottom parts of the wings with a body-like loop and multiply the two denominators it connects, putting the product inside the body.

4. Add or subtract the numbers in the antennae in keeping with what is being done to the fractions and put the result over the number in the body.

5. If necessary, reduce or simplify the result.

How to Tell Which Fraction is Largest

The secret is to cross-multiply the two fractions. Cross-multiplication is a handy math skill to know. You can use it for a quite few different purposes. You are going to use it to find out which of two fractions is greater.

To cross-multiply two fractions:

1. Multiply the numerator of the first fraction by the denominator of the second fraction and jot down the answer.
2. Multiply the numerator of the second fraction by the denominator of the first fraction and jot down the answer.

For example, suppose you have these two fractions:

 3              4

---  and    ---

 7              9

When you cross-multiply, you get these two numbers: 3 X 9 = 27 and 4 X 7  = 28

Make sure you start with the numerator of the first fraction. To find out which of two fractions is larger, place the two numbers you get, in order, under the two fractions. The larger number is always under the larger fraction.

 3              4

---            ---

 7              9

27            28

so 4/9 is larger than 3/7 (or 4/9 > 3/7).

What do you do if you have more than 2 fractions?

Answer: You take them two at a time.

For example, we have the fractions 5/13, 7/17, and 8/19. Which is the largest?

Take the first two and cross multiply:

 5                                7

---             and           ---

13                              17

5 X 17 = 85               7 X 13 = 91

so 7/17 is larger than 5/13.

Now take the larger of these two, 7/17 and compare it with the third, 8/19.  

 7                                8

---             and           ---

17                              19

7 X 19=133              8 X 17=136

so 8/19 is larger than 7/17 is larger than 5/13.