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The Air Force Research
Laboratory (AFRL)
Numbers, Puzzles,
and Curios
John C. Sparks
Wright-Patterson
Educational Outreach
AFRL/WS
(937) 255-4782
[email protected]
For Starters, Can You
Find 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!
1+2=4
Let’s Examine the Year
Whose Number is 2000
2000 =
4
3
1x2 x5
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.
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!
Welcome
2000!
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?
Let’s Cycle the Numbers
517, 263, and 949
517
263
1) 751 - 157 = 594
2) 954 - 459 = 495
3) 954 - 459 = 495
1) 632 - 236 = 396
2) 963 - 369 = 594
3) 954 - 459 = 495
4) 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.
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
For 4 Digits, Kaprekar’s
Magic Constant is 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
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?
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!
Many Numbers and
Two Questions
13
5 7 11 Welcome
3
2000!
Is the number 2000
abundant or deficient?
2
Are prime numbers
abundant or deficient?
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.
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
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?
Those Fascinating
Magic Squares!
This 3 by 3 Magic Square
Uses the Numbers 1 to 9
The Chinese
knew of this 3 by
3 magic square
1000 years before
the birth of
Jesus.
4
9
2
3
5
7
8
1
6
The magic total is 15. In how many different ways
do the rows, columns and diagonals sum to 15?
This 4 by 4 Magic Square
Uses the Numbers 1 to 16
In 1514, Albrecht
Durer created an
engraving named
Melancholia in
which this magic
square appeared.
16
3
2
13
5
10 11
8
9
6
7
12
4
15 14
1
What is the magic total? In how many different ways
do the rows, columns and diagonals sum to this total?
Albrecht Durer’s Engraving
This 4 by 4 Magic Square
Is also a Perfect Square!
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.
1
15
6
12
8
10
3
13
11
5
16 2
14
4
9
7
In how many different ways do the rows,
columns, diagonals, 2 by 2 blocks, and 3
by 3 blocks sum to the magic total?
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?
12 13
1
8
6
3
15 10
7
2
14 11
9
16
4
5
122 +132 + 12 + 82 = ?
and
92 + 162 + 42 + 52 = ?
Benjamin Franklin’s 8 by 8
Magic Square, 1769
In this square, only
the horizontal and
vertical rows sum
to the same
quantity.
What is Franklin’s
Magic Total?
52 61
4 13 20 29 36 45
14
3
62 51 46 35 30 19
53 60
5 12 21 28 37 44
11
6
59 54 43 38 27 22
55 58
7 10 23 26 39 42
9
8
57 56 41 40 25 24
50 63
2 15 18 31 34 47
16
64 49 48 33 32 17
1
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.
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.
3
Example
3
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!
Behold a Great Mystery!
 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 1751
If no, add 1750
 Subtract
the four-digit year that you were born
Go Ahead and Try it!
I Did; See Below!
753
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?
n
This question
must be
answered
using
algebra!
means a number!
Two Limericks for the
Algebra and Math Timid
X is a pronoun like “me”,
But more of an “it” than a “he”.
So why sit afraid
When that letter is made,
For a number is all it can be.
Y
X
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)
 Have you had your birthday this year? no
 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!
Step 1
Step 2
Step 3
50(2n + 5) + 1750 - 1947
100n + 250 + 1750 - 1947
100n + 2000 - 1947
The original “magical gibberish” reduces to
100n + 53
nothing more than 100n plus 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.
Some Miscellaneous
Number “Curios”
3435
47
407
135
136
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
Take a Look at These
Unusual Numbers!
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?
Also, Take a Look at
These Unusual 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
What About 666 Which is
Roman Numeral DCLXVI?
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!
A Big Number from
Ancient Rome
Salve, Anno
Millenium Duo
MM
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.
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 segments
Without backtracking. Crossovers are permitted.
Two Fathers and
Two 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 Squares
Are 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 5 5 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
5 5
8
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.
Answer: 27 feet
Some Ancient Geometry for
Advanced Students
Thales (640-560 B.C.) and
Offshore Boat Distance
Stake in
the Sand
D
A
B
C
Observer’s
Initial Point
E
Observer’s
Final Point
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.
The Pythagorean Theorem
A
For a right triangle with Pythagoras (569-500 B.C.) was
born on the island of Samos in
legs A and B and
Greece. He did much traveling
Hypotenuse C,
throughout Egypt learning
A2 + B2 = C2.
mathematics. This famous
theorem was known in practice
by the Babylonians at least
1400 years before Pythagoras!
B
An Old Proof from China
Circa 1000 B.C.
B
A
A
Proof:
(A+B)2 = C2 + 4(1/2)AB
B
A2 + 2AB + B2 = C2 + 2AB
:: A2 + B2 = C2
B
A
B
A
Fact: Today there are over
300 known proofs of the
Pythagorean theorem!
Eratosthenes (275-194
B.C.) Measures the Earth
Alexandria
Nile
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.
Syene
(Aswan)
The Distance from
Alexandria to Syene
is about 500 miles.
The Trigonometry Behind
Eratosthenes’ Method
7.20
Shadow
Tower at Alexandria
Well at Syene
7.20
Mirror
7.20/3600 = 500 miles/X
Solving for X,
X =25,000 miles
sun
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 End
33 + 43 + 53 = 63