Numbers, Puzzles,and Curios

John C. SparksAFRL/WS

(937) 255-4782John.sparks@wpafb.af.milWright-Patterson

Educational Outreach

The Air Force Research Laboratory (AFRL)

1 + 2 = 4

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!

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!

For Starters, Can YouFind the Error?

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.

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.

Let’s Examine the Year Whose Number is 2000

Welcome2000!

Welcome2000!

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!

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!

Challenge: How Big Can You Make the Number!

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.

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?

What is Kaprekar’s Process?

Let’s Cycle the Numbers517, 263, and 949

5171) 751 - 157 = 5942) 954 - 459 = 4953) 954 - 459 = 495

5171) 751 - 157 = 5942) 954 - 459 = 4953) 954 - 459 = 495

2631) 632 - 236 = 3962) 963 - 369 = 5943) 954 - 459 = 4954) 954 - 459 = 495

2631) 632 - 236 = 3962) 963 - 369 = 5943) 954 - 459 = 4954) 954 - 459 = 495

9491) 994 - 499 = 5452) 554 - 455 = 0993) 990 - 099 = 8914) 981 - 189 = 7925) 972 - 279 = 6936) 963 - 369 = 5947) 954 - 459 = 4958) 954 - 459 = 495

9491) 994 - 499 = 5452) 554 - 455 = 0993) 990 - 099 = 8914) 981 - 189 = 7925) 972 - 279 = 6936) 963 - 369 = 5947) 954 - 459 = 4958) 954 - 459 = 495

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.

1947 (my birth year)1) 9741 - 1479 = 82622) 8622 - 2268 = 63543) 6543 - 3456 = 30874) 8730 - 0378 = 83525) 8532 - 2358 = 61746) 7641 - 1467 = 6174

1947 (my birth year)1) 9741 - 1479 = 82622) 8622 - 2268 = 63543) 6543 - 3456 = 30874) 8730 - 0378 = 83525) 8532 - 2358 = 61746) 7641 - 1467 = 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?

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

numbers?

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

numbers?

For 4 Digits, Kaprekar’s Magic Constant is 6174

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.

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.

What are Perfect Numbers?

Meet the First Seven Perfect Numbers

6: known to the Greeks28: known to the Greeks496: known to the Greeks8128: known to the Greeks33550336: recorded in medieval manuscript8589869056: Cataldi found in 1588137438691328: Cataldi found in 1588

Challenge: Can you show that496 is a perfect number?

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!

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!

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!

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!

What are Abundant and Deficient Numbers?

Many Numbers andTwo Questions

Welcome2000!

Welcome2000!

Is the number 2000 abundant or

deficient?

Are prime numbers abundant or deficient?

2

5 7 11

13

3

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.

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.

220 = 1+2+4+71+142all 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?

•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?

Some of the First Friendly Pairs to be Discovered

49

2

35

7

81

6

Those FascinatingMagic Squares!

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

sum to 15?

4 9 2

3 5 7

8 1 6

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

Jesus.

This 3 by 3 Magic SquareUses the Numbers 1 to 9

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?

16 3 2 13

5 10 11 8

9 6 7 12

4 15 14 1

In 1514, Albrecht Durer

created an engraving

named Melancholia in

which this magic square

appeared.

Albrecht Durer’s Engraving

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?

1 15 6 12

8 10 3 13

11 5 16 2

14 4 9 7

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 Magic Square has More Awesome Properties!

12 13 1 8

6 3 15 10

7 2 14 11

9 16 4 5

123 + 33 + 143 + 53 = 4624 = 682

and93 + 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 = ?

Benjamin Franklin’s 8 by 8Magic Square, 1769

52 61 4 13

14 3 62 51

53 60 5 12

11 6 59 54

20 29 36 45

46 35 30 19

21 28 37 44

43 38 27 22

55 58 7 10

9 8 57 56

50 63 2 15

16 1 64 49

23 26 39 42

41 40 25 24

18 31 34 47

48 33 32 17

In this square, only the

horizontal and vertical rows sum to the

same quantity.What is

Franklin’s Magic Total?

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

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

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.

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

divisible 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!

3 3

The Divisibility Test by 3

Pick a number from 1 to 9 Multiply the number by 2 Add 5 Multiply the result by 50 Have you had your birthday this year?

If yes, add 1751If no, add 1750

Subtract the four-digit year that you were born

Behold a Great Mystery!

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!

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!

753

The Tough Question is,Why Does it Work?

This questionmust be

answeredusing

algebra!

This questionmust be

answeredusing

algebra!n means a

number!

X is a pronoun like “me”,But more of an “it” than a “he”.So why sit afraidWhen that letter is made,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 cruelBy all whom those letters do vex!

Two Limericks for theAlgebra and Math Timid

X

Y

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) Have you had your birthday this year? no Add 1750: 50(2n + 5) + 1750 Subtract the four-digit year that you were born

Algebra Solves the Mystery!

50(2n + 5) + 1750 - 1947We have just made an algebraic sentence!

Simplifying the Sentence &Making it Easier to Read!

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.

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.

50(2n + 5) + 1750 - 1947100n + 250 + 1750 - 1947100n + 2000 - 1947100n + 53

The original “magical gibberish” reduces to nothing more than 100n

plus my age!

Step 1Step 1

Step 2Step 2

Step 3Step 3

3435

40747

135

136

Some Miscellaneous Number “Curios”

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

Carl Gauss Learning Arithmetic at Age 5!

Take a Look at TheseUnusual Numbers!

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

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

136: 13 + 33 + 63 = 244 and 23 + 43 + 43 = 136169: 169 = 132 and 961 = 312

135: 135 = 11 + 32 + 53 175: 175 = 11 + 72 + 53

407: 407 = 43 + 03 + 73

371: 371 = 33 + 73 + 13

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?

Also, Take a Look atThese Unusual Numbers!

1634: 1634 = 14 + 64 + 34 + 44

2620: 2620 and 2924 are friendly.

4913: 4913 =173 and 4 + 9 + 1 + 3 = 179240: 9240 has 64 divisors. Can you find them all?

3435: 3435 = 33 + 44 + 33 + 55

2025: 2025 = 452 and 20 + 25 = 45

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

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

What About 666 Which isRoman Numeral DCLXVI?

666 = 16 - 26 + 36

666 = 2 x 3 x 3 x 37 and 6 + 6 + 6 = 2 + 3 + 3 + 3 + 7666 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 = 295408296and ( 43 + 43 + 53 + 53 + 63 ) +( 2 + 9 + 5 + 4 + 0 + 8 + 2 + 9 + 6 ) = 666!

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

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

Salve, AnnoMillenium Duo

MM

Salve, AnnoMillenium Duo

MMThe 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.

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

Puzzles for Everyone: Logic to Calculus!

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 segmentsWithout 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?

How Many SquaresAre in This Figure?

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.

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

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.

55

55

8

Steel beam being rolled from the first corridor into

the second corridor

Answer: 27 feet

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.

Some Ancient Geometry for Advanced Students

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.

Observer’sInitial Point

Stake inthe Sand

Observer’sFinal Point

BC

A

D

E

The Pythagorean Theorem

C

B

A

For a right triangle with legs A and B and

Hypotenuse C,A2 + B2 = C2.

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!

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!

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

C

B

A

CB

A

CB

A

C

B

A

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

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

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.

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.

Alexandria

Syene(Aswan)

Nile

The Distance from Alexandria to Syene is about 500 miles.

The Trigonometry Behind Eratosthenes’ Method

sun7.20

7.20

Tower at Alexandria

Well at Syene

Shadow

Mirror

7.20/3600 = 500 miles/XSolving for X,

X =25,000 miles

7.20/3600 = 500 miles/XSolving for X,

X =25,000 miles

Four Easy Pieces: Where Did the Little Gap Go?

The Tangram Paradox: Where is the Area?

The Twelve Pentominoes: Cut Them Out and Play!

Thank You!

The End33 + 43 + 53 = 63