## Below you will see a few Number Secrets. See if you can find exceptions or contradictions. If so, you may become famous!

1. Pick a number larger that 1. Make a second number by doubling the first. Can you find a prime number in between the first and second number? For example, choose 5 as your first number. Then 5 x 2 = 10 is your second number. Can we find a prime number in between 5 and 10? Yes, how about 7?

2. Pick an even number larger than two. Can you find two prime numbers that add up to your even number? For example, pick 48. Can we find two prime numbers that add up to 48? How about 31 + 17? Yes, this works! Can you find an exception? Any even number larger than 2 where this idea does not work?

3. Now try this with any odd number or even number. Instead of finding two prime numbers you might need three instead. Does it work? Or not? Get your hands dirty. If you are stuck, try 55 first. Now, pick your own number. Does it work? How about 31 + 19 + 5?

4. Try this secret pattern!

1 x 1 = 1

11 x 11 = 121

111 x 111 = 12321 You can read these numbers forwards or backwards.

1111 x 1111 = ?

11111 x 11111 = ?

How far can you go? Will the pattern break down somewhere?

5. Can 0.9999999... ever be equal to 1?

Try this idea:

1/3 = 0.333333 forever

2/3 = 0.666666 forever.

Now, add 1/3 + 2/3. You get 1 right?

Now, also add 0.333333 forever and 0.6666666 forever. What do you get? 0.99999999 forever.

Thus, we have shown that 1.0000000 = 0.99999999 forever.

Are you convinced?

6. Looking at decimal numbers that repeat, try 7/11, maybe with your calculator.

What do you see? 0.63 63 63 63 ... forever. Cool!

Now, try 6/7. What do you see? Do the digits repeat somewhere? Yes, way out there!

0.857 142857 142857 ...

Can you find some other numbers that do this? We call these numbers Rational Numbers (or Fractions) when they can be expressed as a ratio of two numbers (zero must be excluded). Even if they have never ending decimal names. But they must follow a repeating pattern.

If these numbers with decimals do not repeat what do will call them? Can you think of any?

7. Here's another idea from Dr. Burger's lectures for Number Theory in the Great Courses series.

Take any even number and the next even number; say 2 and 4.

Add 1 to their product -- 2 X 4 = 8, then add 1 to get 9.

Now, try another pair (consecutive), say 4 and 6. Multiply them together and add 1 to the answer. What do you get?

How about 6 and 8? Multiply and add 1. What kind of number do you see? Any patterns? If you pick any two consecutive even numbers, and do this same procedure, can you predict your result ahead of doing any multiplication?

2 X 4 = 8 + 1 = 9 = 3 ^2

4 x 6 = 24 + 1 = 25 = 5 ^2

6 x 8 = 48 + 1 = 49 = 7 ^ 2

...

How about any N and its neighbor N + 2? Pick an N, and thus also N + 2. What would you predict the result to be?

Which N and what power? Do you notice anything else?

Can you make a Conjecture about your pattern? Can you make a try at proving your conjecture?

If this conjecture can be proved then it rises to the more important level of a Theorem.

8. Now, try the same procedure (or algorithm) for two consecutive Odd Numbers. Does the same pattern follow?

9. This example, was first studied by Lothar Collatz in 1937, and is called the 3N + 1 question (explained by Dr. Burger again, in Lecture 2). Dr. Collatz lived from 1910 - 1990.

Here you will generate a list of natural numbers. Pick a natural number. If your number is even, make it smaller by dividing by 2. If your number is odd make it larger by multiply by 3 and add 1 -- (thus, 3N +1).

Next, take your answer and apply the same rules: If even, divide by 2. If odd multiply by 3 and add 1.

Keep doing this and watch what happens.

Get your hands dirty and and try some examples: Start with 1. Multiply by 3 and add 1; hence, you get 4. 4 is even so divide by 2 to get 2. 2 is even so divide by 2 to get 1. And so forth. Do you see a pattern?

Choose 2 for your starting natural number and follow the same procedure. What do you see?

Choose 3 for your starting natural number and follow the same procedure. What do you see?

Choose 4 for your starting natural number and follow the same procedure. What do you see?

Choose 5 for your starting natural number and follow the same procedure. What do you see?

Choose 11 for your starting natural number and follow the same procedure. What do you see?

Are you ready to make a Conjecture?

These sequences are sometimes called "Hailstone Sequences", because like a forming hailstone, they move up and down before falling into a familiar pattern. By the way, if you choose a number like a modest 27, you will see a list of 111 numbers before you come to our seed number of 1. The largest number that occurs is our list is 9232 before our list settles down to our familiar pattern discovered above. Interesting, for sure. Dr. Burger says that mathematicians and computer scientists have tested every natural number up to 3 x 10 ^ 18. That's a 3 with 18 zeros. In all of these tests the list does eventually settle down into our pattern

(1 4 2 1 4 2 ...) However, the Collatz Conjecture remains one of the most famous open questions today in mathematics. No one has been able to prove that his conjecture (and now yours too) works in all cases.

10. Here is a problem with an interesting twist.

Write down any three digits and then write them again in the same order to make a six digit number. For example, 123 123.

Now, seven is considered to be a lucky number in some cultures. Try dividing your number by seven and see if you get any remainders. You may want to use a calculator. No?

Well, then try another lucky number -- 11. Does it divide evenly into your number? No remainders?

Finally, try the unlucky number 13. Does 13 divide evenly into your number without any remainder? Are you sure?

Maybe we just got lucky with our choice of numbers. Try another one and see what happens. Remember to pick a three digit number, and then rewrite those same three digits again in the same order after the first three digits, making a six digit number. What happened?

Why? How can we explain this? Hint: What do have to multiple 123 by to get 123 123? What are the prime factors of this number?

11. One more like the one above? OK. Pick a two digit number, say 47. Now repeat these two digits twice more in the same order. Thus, we have 47 47 47. Now try to divide your six digit number by 3. By 7, by 13, and finally by 37. What happens? Cool huh?

12. Find how magic number called the Kaprekar Constant -- namely 6 1 7 4, randomly appears.

Click on the video below to see how this constant magically appears.

2. Pick an even number larger than two. Can you find two prime numbers that add up to your even number? For example, pick 48. Can we find two prime numbers that add up to 48? How about 31 + 17? Yes, this works! Can you find an exception? Any even number larger than 2 where this idea does not work?

3. Now try this with any odd number or even number. Instead of finding two prime numbers you might need three instead. Does it work? Or not? Get your hands dirty. If you are stuck, try 55 first. Now, pick your own number. Does it work? How about 31 + 19 + 5?

4. Try this secret pattern!

1 x 1 = 1

11 x 11 = 121

111 x 111 = 12321 You can read these numbers forwards or backwards.

1111 x 1111 = ?

11111 x 11111 = ?

How far can you go? Will the pattern break down somewhere?

5. Can 0.9999999... ever be equal to 1?

Try this idea:

1/3 = 0.333333 forever

2/3 = 0.666666 forever.

Now, add 1/3 + 2/3. You get 1 right?

Now, also add 0.333333 forever and 0.6666666 forever. What do you get? 0.99999999 forever.

Thus, we have shown that 1.0000000 = 0.99999999 forever.

Are you convinced?

6. Looking at decimal numbers that repeat, try 7/11, maybe with your calculator.

What do you see? 0.63 63 63 63 ... forever. Cool!

Now, try 6/7. What do you see? Do the digits repeat somewhere? Yes, way out there!

0.857 142857 142857 ...

Can you find some other numbers that do this? We call these numbers Rational Numbers (or Fractions) when they can be expressed as a ratio of two numbers (zero must be excluded). Even if they have never ending decimal names. But they must follow a repeating pattern.

If these numbers with decimals do not repeat what do will call them? Can you think of any?

7. Here's another idea from Dr. Burger's lectures for Number Theory in the Great Courses series.

Take any even number and the next even number; say 2 and 4.

Add 1 to their product -- 2 X 4 = 8, then add 1 to get 9.

Now, try another pair (consecutive), say 4 and 6. Multiply them together and add 1 to the answer. What do you get?

How about 6 and 8? Multiply and add 1. What kind of number do you see? Any patterns? If you pick any two consecutive even numbers, and do this same procedure, can you predict your result ahead of doing any multiplication?

2 X 4 = 8 + 1 = 9 = 3 ^2

4 x 6 = 24 + 1 = 25 = 5 ^2

6 x 8 = 48 + 1 = 49 = 7 ^ 2

...

How about any N and its neighbor N + 2? Pick an N, and thus also N + 2. What would you predict the result to be?

Which N and what power? Do you notice anything else?

Can you make a Conjecture about your pattern? Can you make a try at proving your conjecture?

If this conjecture can be proved then it rises to the more important level of a Theorem.

8. Now, try the same procedure (or algorithm) for two consecutive Odd Numbers. Does the same pattern follow?

9. This example, was first studied by Lothar Collatz in 1937, and is called the 3N + 1 question (explained by Dr. Burger again, in Lecture 2). Dr. Collatz lived from 1910 - 1990.

Here you will generate a list of natural numbers. Pick a natural number. If your number is even, make it smaller by dividing by 2. If your number is odd make it larger by multiply by 3 and add 1 -- (thus, 3N +1).

Next, take your answer and apply the same rules: If even, divide by 2. If odd multiply by 3 and add 1.

Keep doing this and watch what happens.

Get your hands dirty and and try some examples: Start with 1. Multiply by 3 and add 1; hence, you get 4. 4 is even so divide by 2 to get 2. 2 is even so divide by 2 to get 1. And so forth. Do you see a pattern?

Choose 2 for your starting natural number and follow the same procedure. What do you see?

Choose 3 for your starting natural number and follow the same procedure. What do you see?

Choose 4 for your starting natural number and follow the same procedure. What do you see?

Choose 5 for your starting natural number and follow the same procedure. What do you see?

Choose 11 for your starting natural number and follow the same procedure. What do you see?

Are you ready to make a Conjecture?

These sequences are sometimes called "Hailstone Sequences", because like a forming hailstone, they move up and down before falling into a familiar pattern. By the way, if you choose a number like a modest 27, you will see a list of 111 numbers before you come to our seed number of 1. The largest number that occurs is our list is 9232 before our list settles down to our familiar pattern discovered above. Interesting, for sure. Dr. Burger says that mathematicians and computer scientists have tested every natural number up to 3 x 10 ^ 18. That's a 3 with 18 zeros. In all of these tests the list does eventually settle down into our pattern

(1 4 2 1 4 2 ...) However, the Collatz Conjecture remains one of the most famous open questions today in mathematics. No one has been able to prove that his conjecture (and now yours too) works in all cases.

10. Here is a problem with an interesting twist.

Write down any three digits and then write them again in the same order to make a six digit number. For example, 123 123.

Now, seven is considered to be a lucky number in some cultures. Try dividing your number by seven and see if you get any remainders. You may want to use a calculator. No?

Well, then try another lucky number -- 11. Does it divide evenly into your number? No remainders?

Finally, try the unlucky number 13. Does 13 divide evenly into your number without any remainder? Are you sure?

Maybe we just got lucky with our choice of numbers. Try another one and see what happens. Remember to pick a three digit number, and then rewrite those same three digits again in the same order after the first three digits, making a six digit number. What happened?

Why? How can we explain this? Hint: What do have to multiple 123 by to get 123 123? What are the prime factors of this number?

11. One more like the one above? OK. Pick a two digit number, say 47. Now repeat these two digits twice more in the same order. Thus, we have 47 47 47. Now try to divide your six digit number by 3. By 7, by 13, and finally by 37. What happens? Cool huh?

12. Find how magic number called the Kaprekar Constant -- namely 6 1 7 4, randomly appears.

Click on the video below to see how this constant magically appears.

13. Here is a number puzzle submitted by a grade 6 student. Pick a number between 1 and 10. Double it. Add 10. Divide by 2. Now, subtract your original number. We can guess your answer! Is it 5?

14. Another puzzle submitted by a grade 6 student. Pick a natural number 1 to 9 . Multiply by 3. Add 3 Multiply by 3 again. Add the two digits together. What do you get?

14. Another puzzle submitted by a grade 6 student. Pick a natural number 1 to 9 . Multiply by 3. Add 3 Multiply by 3 again. Add the two digits together. What do you get?