The Air Force Research Laboratory (AFRL)

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John C. Sparks AFRL/WS (937) 255-4782 John.sparks@wpafb.af.mil. Wright-Patterson Educational Outreach. The Air Force Research Laboratory (AFRL). Numbers, Puzzles, and Curios. 1 + 2 = 4. For Starters, Can You Find the Error?.

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The Air Force Research Laboratory (AFRL)

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John C. Sparks

AFRL/WS

(937) 255-4782

John.sparks@wpafb.af.mil

Wright-Patterson

Educational Outreach

The Air Force Research Laboratory (AFRL)

Numbers, Puzzles,

and Curios

1 + 2 = 4

For Starters, Can YouFind the Error?

There is a second arithmetic error somewhere else in this presentation! The wizards will give a prize to the first three students who find this second error!

Let’s Examine the Year Whose Number is 2000

2000 = 1x24x53

Notice that the first five digits can each be used exactly once to form the number 2000. The digits 4 and 3 are called exponents and indicate the number of times that we should multiply the digit to the immediate left. Example: 24 means 2x2x2x2.

Welcome

2000!

Challenge: How Big Can You Make the Number!

Using each of the digits 1, 2, 3, and 4 just once, what is the biggest number that you can make? You can add, subtract, multiply, and divide your digits. You may also raise to a power. Is your number bigger than 2000? Unless you are an arithmetic whiz, you might want to use a hand-held calculator to figure this problem out!

What is Kaprekar’s Process?

Take any three-digit number whose digits are not all the same. Rearrange the digits twice in order to make the largest and smallest numbers possible. Subtract the smaller number from the larger. Repeat. This is called Kaprekar’s process.

What is so special about 495?

517

1) 751 - 157 = 594

2) 954 - 459 = 495

3) 954 - 459 = 495

263

1) 632 - 236 = 396

2) 963 - 369 = 594

3) 954 - 459 = 495

4) 954 - 459 = 495

949

1) 994 - 499 = 545

2) 554 - 455 = 099

3) 990 - 099 = 891

4) 981 - 189 = 792

5) 972 - 279 = 693

6) 963 - 369 = 594

7) 954 - 459 = 495

8) 954 - 459 = 495

Let’s Cycle the Numbers517, 263, and 949

All three numbers stop at 495! 495 is called the Kaprekar constant. This magic constant works for any three-digit number having at least two different digits.

For 4 Digits, Kaprekar’s Magic Constant is 6174

A Challenge!

Pick the birth year of someone you know like your mother, father, grandparent, aunt, uncle, or a good friend. How many repeats of Kaprekar’s process does it take to reach the magic constant of 6174?

1947 (my birth year)

1) 9741 - 1479 = 8262

2) 8622 - 2268 = 6354

3) 6543 - 3456 = 3087

4) 8730 - 0378 = 8352

5) 8532 - 2358 = 6174

6) 7641 - 1467 = 6174

Additional challenge: can you figure out if there is a Kaprekar constant for two-digit numbers?

What are Perfect Numbers?

A perfect number is a number equal to the sum of all divisors excluding itself. All divisors of a number smaller than the number are called proper divisors.

6 is perfect because 6 = 1 + 2 + 3.

28 is perfect because 28 = 1 + 2 + 4 + 7 + 14.

Meet the First Seven Perfect Numbers

6: known to the Greeks

28: known to the Greeks

496: known to the Greeks

8128: known to the Greeks

33550336: recorded in medieval manuscript

8589869056: Cataldi found in 1588

137438691328: Cataldi found in 1588

Challenge: Can you show that

496 is a perfect number?

What are Abundant and Deficient Numbers?

An abundant number is a number where the sum of all proper divisors is greater than the number itself. For my sister’s birth year of 1950, there are 21 proper divisors (I think!) 1, 2, 3, 5, 6, 10, 13, 25, 26, 30, 39, 50, 65, 75, 78, 130, 150, 195, 325, 650, and 975. The sum of these numbers is 2853 > 1950. Therefore, 1950 is abundant!

A deficient number is a number where the sum of all its proper divisors is less than the number itself. For my birth year of 1947, there are three proper divisors 1, 3, and 649. They sum to 653 < 1947. Therefore, 1947 is deficient!

Welcome

2000!

Many Numbers andTwo Questions

Is the number 2000 abundant or deficient?

13

5

11

7

3

2

Are prime numbers abundant or deficient?

220 = 1+2+4+71+142

all of the proper divisors of 284

284 =1+2+4+5+10+11+20+22+44+55+110

all of the proper divisors of 220

What are Friendly Numbers?

A pair of numbers is called friendly if each number in the pair is the sum of all proper divisors of the other number. 220 and 284, known by the Greeks, are the first and smallest friendly pair.

Some of the First Friendly Pairs to be Discovered
• 220 & 284: known to the Greeks
• 1184 & 1210: discovered by Paganini at age 16 in 1866
• 17,163 & 18,416: discovered by Fermat in 1636
• 9,363,584 & 9,437,056: discovered by Descartes in 1638
• Fact: Over 1000 pairs of friendly numbers are now known!

Challenge: Can you show that

1184 and 1210 are friendly?

The Chinese knew of this 3 by 3 magic square 1000 years before the birth of Jesus.

4

3

8

9

5

1

2

7

6

This 3 by 3 Magic SquareUses the Numbers 1 to 9

The magic total is 15. In how many different ways do the rows, columns and diagonals sum to 15?

16

5

9

4

3

10

6

15

7

11

2

14

8

13

1

12

In 1514, Albrecht Durer created an engraving named Melancholia in which this magic square appeared.

This 4 by 4 Magic Square Uses the Numbers 1 to 16

What is the magic total? In how many different ways do the rows, columns and diagonals sum to this total?

1

8

11

14

15

10

5

4

16

3

6

9

13

12

7

2

A perfect square is a magic square where every 2 by 2 block and the corners of every 3 by 3 and 4 by 4 block also sum to the magic total.

This 4 by 4 Magic Square Is also a Perfect Square!

In how many different ways do the rows, columns, diagonals, 2 by 2 blocks, and 3 by 3 blocks sum to the magic total?

7

12

6

9

13

3

2

16

4

14

15

1

8

10

5

11

This Magic Square has More Awesome Properties!

123 + 33 + 143 + 53 = 4624 = 682

and

93 + 23 + 153 + 83 = 4624 = 682

Verify that the sums of the squares of the numbers in the 1st and 4th rows are equal. Verify that the sums of the squares of the numbers in the 2nd and 3rd rows are also equal. Is there a similar property shared by the four columns?

122 +132 + 12 + 82 = ?

and

92 + 162 + 42 + 52 = ?

18

9

52

50

48

14

55

20

23

53

21

46

43

11

16

41

29

61

3

31

60

40

6

35

63

1

28

58

33

38

26

8

4

59

5

36

30

62

37

7

39

27

64

57

34

25

2

32

22

42

17

44

15

19

49

45

12

54

24

51

13

47

56

10

Benjamin Franklin’s 8 by 8Magic Square, 1769

In this square, only the horizontal and vertical rows sum to the same quantity.

What is Franklin’s Magic Total?

Question: Can We Make a 2 by 2 Magic Square?

Go ahead and try it! Use the numbers 1 to 4.

Try this Teaser at Home!

Create a 3 by 3 magic square using the prime numbers 5, 17, 29, 47, 59, 71, 89, 101, and 113.

3

3

The Divisibility Test by 3

If the sum of the digits of a number is divisible by 3, then the number itself is divisible by 3.

Example

For 147, 972, 1 + 4 + 7 + 9 + 7 + 2 =39

which is divisible by 3.

Therefore, 147,972 should be divisible by 3.

Let’s see: 147,972 / 3 = 49,324!

Pick a number from 1 to 9

Multiply the number by 2

Multiply the result by 50

Subtract the four-digit year that you were born

Behold a Great Mystery!

753

Go Ahead and Try it!I Did; See Below!

You should have a three-digit number. The first digit is your original number; the next two digits are your age. It really works, and 2001 is the only year it will ever work!

The Tough Question is,Why Does it Work?

This question

must be

using algebra!

n means a number!

X

Y

Two Limericks for theAlgebra and Math Timid

X is a pronoun like “me”,

But more of an “it” than a “he”.

So why sit afraid

For a number is all it can be.

Twin variables come, Y and X,

As frightfully mean as T-Rex.

You’ll find them at school,

Unknowns labeled cruel

By all whom those letters do vex!

Algebra Solves the Mystery!
• Pick a number from 1 to 9: call the number n
• Multiply the number by 2: we have 2n
• Add 5: now we have 2n + 5
• Multiply the result by 50: 50(2n + 5)
• Add 1750: 50(2n + 5) + 1750
• Subtract the four-digit year that you were born

50(2n + 5) + 1750 - 1947

We have just made an algebraic sentence!

Simplifying the Sentence &Making it Easier to Read!

50(2n + 5) + 1750 - 1947

100n + 250 + 1750 - 1947

Step 1

100n + 2000 - 1947

Step 2

100n + 53

The original “magical gibberish” reduces to nothing more than 100n plus my age!

Step 3

Now, suppose I had picked 8 for my n. Then my final number would have been 852. As you can see, the first digit is my original pick; the second digit, my age.

Carl Gauss Learning Arithmetic at Age 5!

Add the counting numbers 1 through 100 without using your calculator. When told by his teacher to do the same, Carl Gauss (1777-1855), at age 5, correctly completed the task within one minute.

1, 2, 3, 4, 5 … 96, 97, 98, 99, 100

499: 499 = 497 + 2 and 497 x 2 = 994

407: 407 = 43 + 03 + 73

371: 371 = 33 + 73 + 13

47: 47 + 2 = 49 and 47 x 2 = 94

135: 135 = 11 + 32 + 53

175: 175 = 11 + 72 + 53

136: 13 + 33 + 63 = 244 and 23 + 43 + 43 = 136

169: 169= 132 and 961 = 312

567: 5672 = 321489. Not counting the exponent 2, this equality uses each of the digits just once. The only other number that does this is 854. Can you show that this fascinating result is true for 854?

Take a Look at TheseUnusual Numbers!

504: 504 = 12 x 42 and 504 = 21 x 24

1634: 1634 = 14 + 64 + 34 + 44

2025: 2025 = 452 and 20 + 25 = 45

2620: 2620 and 2924 are friendly.

3435: 3435 = 33 + 44 + 33 + 55

4913: 4913 =173 and 4 + 9 + 1 + 3 = 17

9240: 9240 has 64 divisors. Can you find them all?

54,748: 54,748= 55 + 45 + 75 + 45 + 85

Also, Take a Look atThese Unusual Numbers!

666 = 6 + 6 + 6 + 63 + 63 + 63

666 = 16 - 26 + 36

666 = 22 + 32 + 52 + 72 + 112 + 132 + 172

666 = 2 x 3 x 3 x 37 and 6 + 6 + 6 = 2 + 3 + 3 + 3 + 7

666 is a “Smith number” since the sum of its digits is equal to the sum of the digits of its prime factors.

6662 = 443556 and 6663 = 295408296

and ( 43 + 43 + 53 + 53 + 63 ) +

( 2 + 9 + 5 + 4 + 0 + 8 + 2 + 9 + 6 ) = 666!

What About 666 Which isRoman Numeral DCLXVI?

The earliest inscription in Europe containing a very large number is on the Columna Rostrata, a monument erected in the Roman Forum to commemorate the victory of 260 BC over the Carthaginians. C, the symbol for 100,000 was repeated 23 times for a total of 2,300,000.

A Big Number fromAncient Rome

Salve, Anno

Millenium Duo

MM

The Farmer, Wolf, Goat, and Cabbage Problem

A farmer and his goat, wolf, and cabbage come to a river that they wish to cross. There is a boat, but it only has room for two, and the farmer is the only one that can row. However, if the farmer leaves the shore in order to row, the goat will eat the cabbage, and the wolf will eat the goat. Devise a minimum number of crossings so that all concerned make it across the river safely.

The Four Line Connect

Connect the 9 dots using four straight line segments

Without backtracking. Crossovers are permitted.

Two Fathers andTwo Sons

There are two fathers and two sons on a boat. Each person caught one fish. None of the fish were thrown back. Three fish were caught. How is it possible?

A Question on the Microsoft Employment Exam

“U2” has a concert that starts in 17 minutes, and they must all cross a bridge to get there. All four men begin on the same side of the bridge. You must devise a plan to help the group get to the other side on time! The additional constraints are many! It is night. There is but one flashlight. A maximum of two people can cross at one time. Any party who crosses, either 1 or 2 people, must have the flashlight with them. The flashlight must be walked back and forth; it cannot be throw, etc. Each band member walks at a different speed. A pair must walk together at the rate of the slower man’s pace. The rates are: Bono--1 minute to cross, Edge--2 minutes to cross, Adam--5 minutes to cross, Larry--10 minutes to cross.

Two workmen at a construction site are rolling steel beams down a corridor 8 feet wide that opens into a second corridor feet wide. What is the length of the longest beam that can be rolled into the second corridor? Assume that the second corridor is perpendicular to the first corridor and that the beam is of negligible thickness.

Steel beam being rolled from the first corridor into the second corridor

The Infamous Girder Problem:A Real Calculus Meat-grinder!

Fact: This problem started to appear in calculus texts circa 1900. It is famous because of how it thoroughly integrates plane geometry, algebra, and differential calculus.

Stake in

the Sand

A

D

B

C

Observer’s

Initial Point

Observer’s

Final Point

E

Thales (640-560 B.C.) and Offshore Boat Distance

Process: sight the vessel straight offshore per line AB. Walk the distance BC and drive a tall stake. Walk an equal distance CD. Walk a distance DE until the stake covers the boat in a line of sight. Since triangles ABC and CDE are congruent, AB equals DE.

C

A

B

The Pythagorean Theorem

Pythagoras (569-500 B.C.) was born on the island of Samos in Greece. He did much traveling throughout Egypt learning mathematics. This famous theorem was known in practice by the Babylonians at least 1400 years before Pythagoras!

For a right triangle with legs A and B and Hypotenuse C,

A2 + B2 = C2.

B

A

A

C

B

C

C

B

C

A

A

B

An Old Proof from ChinaCirca 1000 B.C.

Proof:

(A+B)2 = C2 + 4(1/2)AB

A2 + 2AB + B2 = C2 + 2AB

:: A2 + B2 = C2

Fact: Today there are over 300 known proofs of the Pythagorean theorem!

Alexandria

Nile

Syene

(Aswan)

Eratosthenes (275-194 B.C.) Measures the Earth

Eratosthenes was the Director of the Alexandrian Library who came up with an ingenious method for determining the circumference of the earth. He made three assumptions: the earth was round, sunrays reached the earth as parallel beams, and Alexandria and Syene fell on the same meridian.

The Distance from Alexandria to Syene is about 500 miles.

7.20

Tower at Alexandria

sun

Well at Syene

7.20

Mirror

The Trigonometry Behind Eratosthenes’ Method

7.20/3600 = 500 miles/X

Solving for X,

X =25,000 miles

Thank You!

The End

33 + 43 + 53 = 63