
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?.
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
AFRL/WS
(937) 255-4782
John.sparks@wpafb.af.mil
Wright-Patterson
Educational Outreach
The Air Force Research Laboratory (AFRL)Numbers, Puzzles,
and Curios
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!
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!
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!
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?
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 949All 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.
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?
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.
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?
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!
2000!
Many Numbers andTwo QuestionsIs the number 2000 abundant or deficient?
13
5
11
7
3
2
Are prime numbers abundant or deficient?
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.
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 9The magic total is 15. In how many different ways do the rows, columns and diagonals sum to 15?
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 16What is the magic total? In how many different ways do the rows, columns and diagonals sum to this total?
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?
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 = ?
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, 1769In 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.
Create a 3 by 3 magic square using the prime numbers 5, 17, 29, 47, 59, 71, 89, 101, and 113.
3
The Divisibility Test by 3If 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!
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
Behold a Great Mystery!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!
Y
Two Limericks for theAlgebra and Math TimidX 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.
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!
50(2n + 5) + 1750 - 1947
We have just made an algebraic sentence!
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.
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 RomeSalve, Anno
Millenium Duo
MM
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.
Connect the 9 dots using four straight line segments
Without backtracking. Crossovers are permitted.
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?
“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
Answer: 27 feet
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.
the Sand
A
D
B
C
Observer’s
Initial Point
Observer’s
Final Point
E
Thales (640-560 B.C.) and Offshore Boat DistanceProcess: 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.
A
B
The Pythagorean TheoremPythagoras (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.
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!
Nile
Syene
(Aswan)
Eratosthenes (275-194 B.C.) Measures the EarthEratosthenes 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.
Shadow
Tower at Alexandria
sun
Well at Syene
7.20
Mirror
The Trigonometry Behind Eratosthenes’ Method7.20/3600 = 500 miles/X
Solving for X,
X =25,000 miles