## Friday, March 26, 2010

### Divide by Parts

**Divide 110 into two parts so that one will be 150 percent of the other. What are the 2 numbers?**

### Is A Guilty?

### Who is in the picture?

### Order Finder

Isaac and Albert were excitedly describing the result of the Third Annual International Science Fair Extravaganza in Sweden. There were three contestants, Louis, Rene, and Johannes. Isaac reported that Louis won the fair, while Rene came in second. Albert, on the other hand, reported that Johannes won the fair, while Louis came in second.

In fact, neither Isaac nor Albert had given a correct report of the results of the science fair. Each of them had given one correct statement and one false statement. What was the actual placing of the three contestants?

### Bulbs

How would you identify which switch corresponds to which light bulb, if you are only allowed one trip upstairs?

### Immortal wars

## Friday, March 5, 2010

### The 11 Rule

To multiply any two digit number by 11:

- For this example we will use 54.

- Separate the two digits in you mind (5__4).

- Notice the hole between them!

- Add the 5 and the 4 together (5+4=9)

- Put the resulting 9 in the hole 594. That's it! 11 x 54=594

The only thing tricky to remember is that if the result of the addition is greater than 9, you only put the "ones" digit in the hole and carry the "tens" digit from the addition. For example 11 x 57 ... 5__7 ... 5+7=12 ... put the 2 in the hole and add the 1 from the 12 to the 5 in to get 6 for a result of 627 ... 11 x 57 = 627

Practice it on paper first!

### Squaring a 2-digit number ending in 8

- Choose a 2-digit number ending in 8.
- The last digit of the answer is always 4: _ _ _ 4
- Multiply the first digit by 6 and add 6 (keep the

carry): _ _ X _ - Multiply the first digit by the next consecutive

number and add the carry: the product is the first

two digits: XX _ _.

#### Example:

- If the number is
**78**: - The last digit of the answer is 4: _ _ _ 4
- Multiply the first digit (7) by 6 and add 6 (keep the

carry): 7 × 6 = 42, 42 + 6 = 48; the next digit of the

answer is 8 (keep carry 4): _ _ 8 4 - Multiply the first digit (7) by the next number (8)

and add the carry (4):

7 × 8 = 56, 56 + 4 = 60 (the first two digits): 6 0 _ _ - So
**78 × 78 = 6084**.

** See the pattern?**

- For 38 × 38
- The last digit of the answer is 4: _ _ _ 4
- Multiply the first digit (3) by 6 and add 6 (keep the

carry): 3 × 6 = 18, 18 + 6 = 24; the next digit of the

answer is 4 (keep carry 2): _ _ 4 4 - Multiply the first digit (3) by the next number (4)

and add the carry (2):

3 × 4 = 12, 12 + 2 = 14 (the first two digits): 1 4 _ _ - So
**38 × 38 = 1444**

Learn the pattern,practice other examples, and you will be a whiz at giving these squares.

### Squaring a 2-digit number ending in 7

- Choose a 2-digit number ending in 7.
- The last digit of the answer is always 9: _ _ _ 9
- Multiply the first digit by 4 and add 4

(keep the carry): _ _ X _ - Multiply the first digit by the next consecutive number and

add the carry: the product is the first two digits:

XX _ _.

#### Example:

- If the number is
**47**: - The last digit of the answer is 9: _ _ _ 9
- Multiply the first digit (4) by 4 and add 4

(keep the carry): 4 × 4 = 16, 16 + 4 = 20; the next

digit of the answer is 0 (keep carry 2): _ _ 0 9 - Multiply the first digit (4) by the next number (5)

and add the carry (2):

4 × 5 = 20, 20 + 2 = 22 (the first two digits): 2 2 _ _ - So
**47 × 47 = 2209**.

** See the pattern?**

- For 67 × 67
- The last digit of the answer is 9: _ _ _ 9
- Multiply the first digit (6) by 4 and add 4 (keep the

carry): 4 × 6 = 24, 24 + 4 = 28; the next digit of the

answer is 0 (keep carry 2): _ _ 8 9 - Multiply the first digit (6) by the next number (7)

and add the carry (2):

6 × 7 = 42, 42 + 2 = 44 (the first two digits): 4 4 _ _ - So
**67 × 67 = 4489**.

### Squaring a 2-digit number ending in 6

- Choose a 2-digit number ending in 6.
- Square the second digit (keep the carry): the last digit

of the answer is always 6: _ _ _ 6 - Multiply the first digit by 2 and add the carry

(keep the carry): _ _ X _ - Multiply the first digit by the next consecutive number and

add the carry: the product is the first two digits:

XX _ _.

#### Example:

- If the number is
**46**, square the second digit :

6 × 6 = 36; the last digit of the answer is 6

(keep carry 3): _ _ _ 6 - Multiply the first digit (4) by 2 and add the carry

(keep the carry): 2 × 4 = 8, 8 + 3 = 11; the next digit

of the answer is 1: _ _ 1 6 - Multiply the first digit (4) by the next number (5)

and add the carry: 4 × 5 = 20, 20 + 1 = 21

(the first two digits): 2 1 _ _ - So
**46 × 46 = 2116**.

** See the pattern?**

- For 76 × 76, square 6 and keep the carry (3):

6 × 6 = 36; the last digit of the answer is 6: _ _ _ 6 - Multiply the first digit (7) by 2 and add the carry:

2 × 7 = 14, 14 + 3 = 17; the next digit of the answer

is 7 (keep carry 1): _ _ 7 6 - Multiply the first digit (7) by the next number (8)

and add the carry: 7 × 8 = 56, 56 + 1 = 57

(the first two digits: 5 7 _ _ - So
**76 × 76 = 5776**.

### Squaring a 2-digit number ending in 5

- Choose a 2-digit number ending in 5.
- Multiply the first digit by the next consecutive number.
- The product is the first two digits: XX _ _.
- The last part of the answer is always 25: _ _ 2 5.

#### Example:

- If the number is
**35**, 3 × 4 = 12 (first digit

times next number). 1 2 _ _ - The last part of the answer is always 25: _ _ 2 5.
- So
**35 × 35 = 1225**.

** See the pattern?**

- For 65 × 65, 6 × 7 = 42 (first digit

times next number): 4 2 _ _. - The last part of the answer is always 25: _ _ 2 5.
- So
**65 × 65 = 4225**.

## Thursday, March 4, 2010

### Folded sheet of paper

### Painted cubes

### Squaring a 2-digit number ending in 5

- Choose a 2-digit number ending in 5.
- Multiply the first digit by the next consecutive number.
- The product is the first two digits: XX _ _.
- The last part of the answer is always 25: _ _ 2 5.

#### Example:

- If the number is
**35**, 3 × 4 = 12 (first digit

times next number). 1 2 _ _ - The last part of the answer is always 25: _ _ 2 5.
- So
**35 × 35 = 1225**.

** See the pattern?**

- For 65 × 65, 6 × 7 = 42 (first digit

times next number): 4 2 _ _. - The last part of the answer is always 25: _ _ 2 5.
- So
**65 × 65 = 4225**.

### Squaring a 2-digit number ending in 4

- Take a 2-digit number ending in 4.
- Square the 4; the last digit is 6: _ _ _ 6

(keep carry, 1.) - Multiply the first digit by 8 and add the carry (1);

the 2nd number will be the next to the last digit:

_ _ X 6 (keep carry). - Square the first digit and add the carry: X X _ _.

#### Example:

- If the number is
**34**, 4 × 4 = 16 (keep carry, 1);

the last digit is _ _ _ 6. - 8 × 3 = 24 (multiply the first digit by 8), 24 + 1 = 25

(add the carry):

the next digit is 5: _ _ 5 6. (Keep carry, 2.) - Square the first digit and add the carry, 2: 1 1 5 6.
- So
**34 × 34 = 1156**.

** See the pattern?**

- For 84 × 84, 4 × 4 = 16 (keep carry, 1);

the last digit is _ _ _ 6. - 8 × 8 = 64 (multiply the first digit by 8),

64 + 1 = 65 (add the carry):

the next digit is 5: _ _ 5 6. (Keep carry, 6.) - Square the first digit and add the carry, 6: 7 0 5 6.
- So
**84 × 84 = 7056**.

### Three simultaneous equations

Find all positive real solutions of the simultaneous equations:

- x + y
^{2}+ z^{3}= 3 - y + z
^{2}+ x^{3}= 3 - z + x
^{2}+ y^{3}= 3

### Fermatt's squares

^{p−1}− 1)/p is always an integer if p is an odd prime. For what values of p is x a perfect square?

### Compute this series?

Compute the infinite product

[sin(x) cos(x/2)]^{1/2} · [sin(x/2) cos(x/4)]^{1/4} · [sin(x/4) cos(x/8)]^{1/8} · ... ,

where 0 x 2.

### Squaring a 2-digit number ending in 4

- Take a 2-digit number ending in 4.
- Square the 4; the last digit is 6: _ _ _ 6

(keep carry, 1.) - Multiply the first digit by 8 and add the carry (1);

the 2nd number will be the next to the last digit:

_ _ X 6 (keep carry). - Square the first digit and add the carry: X X _ _.

#### Example:

- If the number is
**34**, 4 × 4 = 16 (keep carry, 1);

the last digit is _ _ _ 6. - 8 × 3 = 24 (multiply the first digit by 8), 24 + 1 = 25

(add the carry):

the next digit is 5: _ _ 5 6. (Keep carry, 2.) - Square the first digit and add the carry, 2: 1 1 5 6.
- So
**34 × 34 = 1156**.

** See the pattern?**

- For 84 × 84, 4 × 4 = 16 (keep carry, 1);

the last digit is _ _ _ 6. - 8 × 8 = 64 (multiply the first digit by 8),

64 + 1 = 65 (add the carry):

the next digit is 5: _ _ 5 6. (Keep carry, 6.) - Square the first digit and add the carry, 6: 7 0 5 6.
- So
**84 × 84 = 7056**.

### Squaring a 2-digit number ending in 3

- Take a 2-digit number ending in 3.
- The last digit will be _ _ _ 9.
- Multiply the first digit by 6: the 2nd number will be

the next to the last digit: _ _ X 9. - Square the first digit and add the number carried from

the previous step: X X _ _.

#### Example:

- If the number is
**43**, the last digit is _ _ _ 9. - 6 × 4 = 24 (six times the first digit): _ _ 4 9.
- 4 × 4 = 16 (square the first digit), 16 + 2 = 18

(add carry): 1 8 4 9. - So
**43 × 43 = 1849**.

** See the pattern?**

- For 83 × 83, the last digit is _ _ _ 9.
- 6 × 8 = 48 (six times the first digit): _ _ 8 9.
- 8 × 8 = 64 (square the first digit), 64 + 4 = 68

(add carry): 6 8 8 9. - So
**83 × 83 = 6889**.

### Squaring a 2-digit number ending in 2

- Take a 2-digit number ending in 2.
- The last digit will be _ _ _ 4.
- Multiply the first digit by 4: the 2nd number will be

the next to the last digit: _ _ X 4. - Square the first digit and add the number carried from

the previous step: X X _ _.

#### Example:

- If the number is
**52**, the last digit is _ _ _ 4. - 4 × 5 = 20 (four times the first digit): _ _ 0 4.
- 5 × 5 = 25 (square the first digit), 25 + 2 = 27 (add carry): 2 7 0 4.
- So
**52 × 52 = 2704**.

** See the pattern?**

- For 82 × 82, the last digit is _ _ _ 4.
- 4 × 8 = 32 (four times the first digit): _ _ 2 4.
- 8 × 8 = 64 (square the first digit), 64 + 3 = 67 (add carry): 6 7 2 4.
- So
**82 × 82 = 6724**.

### Squaring a 2-digit number ending in 1

- Take a 2-digit number ending in 1.
- Subtract 1 from the number.
- Square the difference.
- Add the difference twice to its square.
- Add 1.

#### Example:

- If the number is
**41**, subtract 1: 41 - 1 = 40. - 40 × 40 = 1600 (square the difference).
- 1600 + 40 + 40 = 1680 (add the difference twice

to its square). - 1680 + 1 = 1681 (add 1).
- So
**41 × 41 = 1681**.

** See the pattern?**

- For 71 × 71, subtract 1: 71 - 1 = 70.
- 70 × 70 = 4900 (square the difference).
- 4900 + 70 + 70 = 5040 (add the difference twice

to its square). - 5040 + 1 = 5041 (add 1).
- So
**71 × 71 = 5041**.

### Squaring a 2-digit number beginning with 9

- Take a 2-digit number beginning with 9.
- Subtract it from 100.
- Subtract the difference from the original number:

this is the first part of the answer. - Square the difference: this is the last part of the answer.

#### Example:

- If the number is
**96**, subtract: 100 - 96 = 4, 96 - 4 = 92. - The first part of the answer is 92 _ _ .
- Take the first difference (4) and square it: 4 × 4 = 16.
- The last part of the answer is _ _ 16.
- So
**96 × 96 = 9216**.

** See the pattern?**

- For 98 × 98, subtract: 100 - 98 = 2, 98 - 2 = 96.
- The first part of the answer is 96 _ _.
- Take the first difference (2) and square it: 2 × 2 = 4.
- The last part of the answer is _ _ 04.
- So
**98 × 98 = 9604**.

### Squaring a 2-digit number beginning with 5

- Take a 2-digit number beginning with 5.
- Square the first digit.
- Add this number to the second number to find the first part of the answer.
- Square the second digit: this is the last part of the answer.

#### Example:

- If the number is
**58**, multiply 5 × 5 = 25 (square the first digit). - 25 + 8 = 33 (25 plus second digit).
- The first part of the answer is 33 3 3 _ _
- 8 × 8 = 64 (square second digit).
- The last part of the answer is 64 _ _ 6 4
- So
**58 × 58 = 3364**.

** See the pattern?**

- For 53 × 53, multiply 5 × 5 = 25 (square the first digit).
- 25 + 3 = 28 (25 plus second digit).
- The first part of the answer is 28 2 8 _ _
- 3 × 3 = 9 (square second digit).
- The last part of the answer is 09 _ _ 0 9
- So
**53 × 53 = 2809**.

### Squaring a 2-digit number beginning with 1

- Take a 2-digit number beginning with 1.
- Square the second digit

(keep the carry) _ _ X - Multiply the second digit by 2 and

add the carry (keep the carry) _ X _ - The first digit is one

(plus the carry) X _ _

#### Example:

- If the number is
**16**, square the second digit:

6 × 6 = 36 _ _ 6 - Multiply the second digit by 2 and

add the carry: 2 × 6 + 3 = 15 _ 5 _ - The first digit is one plus the carry:

1 + 1 = 2 2 _ _ - So
**16 × 16 = 256**.

** See the pattern?**

- For 19 × 19, square the second digit:

9 × 9 = 81 _ _ 1 - Multiply the second digit by 2 and

add the carry: 2 × 9 + 8 = 26 _ 6 _ - The first digit is one plus the carry:

1 + 2 = 3 3 _ _ - So
**19 × 19 = 361**.