MATHEMATICAL AMAZEMENTS and SURPRISES

FASCINATING FIGURES AND NOTEWORTHY NUMBERSBy ALFRED S. POSAMENTIER INGMAR LEHMANN

Prometheus Books

Copyright © 2009 Alfred S. Posamentier and Ingmar Lehmann
All right reserved.
ISBN: 978-1-59102-723-2

Contents

PREFACE…………………………………………………………..7Chapter 1. Amazing Number Properties and Relationships…………………9Chapter 2. Arithmetic Entertainments and Novelties…………………….55Chapter 3. Arithmetic Loops…………………………………………97Chapter 4. Geometry Surprises……………………………………….147Chapter 5. Mathematical Nuggets: Amazing, but True!……………………197AFTERWORD by Herbert A. Hauptman…………………………………….249ACKNOWLEDGMENTS……………………………………………………253APPENDIX. List of Triangular, Square, and Cubic Number…………………255BIBLIOGRAPHY………………………………………………………259INDEX…………………………………………………………….261

Chapter One

AMAZING NUMBER PROPERTIES and RELATIONSHIPS

For the most part, numbers play a major role in our lives, allowing us to quantify and to create order. Yet numbers can be seen also for their own sake. They may harbor unusual properties, exhibit some hidden beauty, or just allow us to marvel at their inherent nature. In this chapter, we will investigate some of the attractive features that numbers can offer. We hope to amaze you with lots of unusual surprises.

There have been many ways to write numbers over the past millennia. The Egyptians, for example, used hieroglyphics to express numbers, and the Romans had a number system that is still used today, such as for chapter numbers or cornerstone dates. Yet these have proved to be far too cumbersome and inconvenient, and so when Leonardo of Pisa-better known today as Fibonacci (ca. 1175-1240)-introduced the “Indian numerals” 9, 8, 7, 6, 5, 4, 3, 2, 1, and 0 in the first words of his monumental book Liber Abaci, written in 1202, we had the use of these numerals for the first time in Western civilization. As we investigate the properties of numbers, we will use this standard base-10 notation.

In this chapter, we will introduce you to many beautiful number arrangements, number properties, number relationships, and interesting processes that we can use to analyze numbers. We will occasionally show a pattern and, in the absence of proof, ask you to accept the results, since we do not want to break the flow of the book with proofs. Yet, by the same token, we want you to be aware that not all “apparent patterns” of number relationships hold true for all cases. To give you just one illustration of a number pattern that appears to lead to a general result-but does not-we will consider the question of the French mathematician Alphonse de Polignac’s (1817-1890) Conjecture:

Every odd number greater than 1 can be expressed as the sum of a power of 2 and a prime number.

If we inspect the first few cases, we find that this appears to be a true statement. However, as you will see from the list in figure 1-1, it holds true for the odd numbers from 3 through 125 and then is not true for 127, after which it continues to hold true again for a while.

Perhaps you can find the next number that fails de Polignac’s Conjecture. Remember, though, when we present you with a pattern, we will assure you that it will hold true for all cases.

In 1849, Alphonse de Polignac proposed another conjecture that has not been proved or disproved to date. It is as follows:

There are infinitely many cases of two consecutive prime numbers with a difference of some even number n.

For example, suppose we let n = 2. There are consecutive prime number pairs whose difference is 2, such as (3, 5), (11, 13), (17, 19), and so on. Note, we still have not established if this conjecture is true or false.

THE AMAZING ZERO SUMS

When you use a calculator to determine the following sum, you will find it to be zero.

[123,789.sup.2] + [561,945.sup.2] + [642,864.sup.2] – [242,868.sup.2] – [761,943.sup.2] – [323,787.sup.2] = 0

This may be nice, since we have the squares of large numbers and they seem to show no particular pattern. Yet when we begin to manipulate these numbers in a very orderly fashion, the zero sum amazingly remains.

1. In the first case, we will delete the hundred-thousands place (the left-most digit) from each number:

[23,789.sup.2] + [61,945.sup.2] + [42,864.sup.2] – [42,868.sup.2] – [61,943.sup.2] – [23,787.sup.2], and the sum remains 0.

We repeat this process by deleting the left – most digit of each number from each of the next few and look at the results:

[3,789.sup.2] + [1,945.sup.2] + [2,864.sup.2] – [2,868.sup.2] – [1,943.sup.2] – [3,787.sup.2] = 0

[789.sup.2] + [945.sup.2] + [864.sup.2] – [868.sup.2] – [943.sup.2] – [787.sup.2] = 0

[89.sup.2] + [45.sup.2] + [64.sup.2] – [68.sup.2] – [43.sup.2] – [87.sup.2] = 0

[9.sup.2] + [5.sup.2] + [4.sup.2] – [8.sup.2] – [3.sup.2] – [7.sup.2] = 0

2. We now will follow an analogous process, but this time we’ll delete the units digit (the right-most digit) from each of the numbers, and again amazingly we see each time that the resulting sum is zero:

[123,789.sup.2] + [561,945.sup.2] + [642,864.sup.2] – [242,868.sup.2] – [761,943.sup.2] – [323,787.sup.2] = 0

[12,378.sup.2] + [56,194.sup.2] + [64,286.sup.2] – [24,286.sup.2] – [76,194.sup.2] – [32,378.sup.2] = 0

[1,237.sup.2] + [5,619.sup.2] + [6,428.sup.2] – [2,428.sup.2] – [7,619.sup.2] – [3,237.sup.2] = 0

[123.sup.2] + [561.sup.2] + [642.sup.2] – [242.sup.2] – [761.sup.2] – [323.sup.2] = 0

[12.sup.2] + [56.sup.2] + [64.sup.2] – [24.sup.2] – [76.sup.2] – [32.sup.2] = 0

[1.sup.2] + [5.sup.2] + [6.sup.2] – [2.sup.2] – [7.sup.2] – [3.sup.2] = 0

3. We will now combine the two types of deletions into one by removing the right and left digits from each number and, yes, again we retain zero sums!

[123,789.sup.2] + [561,945.sup.2] + [642,864.sup.2] – [242,868.sup.2] – [761,943.sup.2] – [323,787.sup.2] = 0

[2,378.sup.2] + [6,194.sup.2] + [4,286.sup.2] – [4,286.sup.2] – [6,194.sup.2] – [2,378.sup.2] = 0

[37.sup.2] + [19.sup.2] + [28.sup.2] – [28.sup.2] – [19.sup.2] – [37.sup.2] = 0

This is not to be taken for granted; it is a truly amazing number relationship.

A Most Unusual Number

In this book, we are not concerned about a number’s mystical connections; our interest in numbers is purely mathematical. Yet in some societies for curious reasons some numbers symbolize good luck and others bad luck. For example, an inordinate number of children were born in China on August 8, 2008, and over seventeen thousand couples were married in Beijing on that date. It was also the date on which the twenty-ninth Summer Olympics opened in Beijing at 8:08:08 PM. Why that date? For the Chinese, 8 is a lucky number, and that date can be written as 08-08-08. What makes 8 so special is open for interpretation. Mathematically speaking, 8 is a perfect cube: 8=[2.sup.3]; and 8 is the only cube that is one less than a perfect square, 9. Also, 8, the sixth Fibonacci number, is the only Fibonacci number (other than 1) that is a perfect cube.

On the other hand, the number 666, popularly known through its biblical association, is often referred to as the “number of the beast” and signifies bad luck omens. Yet again, we are only concerned about this number’s mathematical properties, many of which are quite amazing. To begin with, the number 666 is obviously a palindrome-that is, it is a number that reads the same in both directions. Yet if we were to write this number in Roman numerals-666 = DCLXVI-we find that all the numerals less than 1,000 are used and in descending order!

The number 666 just happens to be the sum of the first thirty-six numbers:

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + … + 30 + 31 + 32 + 33 + 34 + 35 + 36 = 666

Not only is the sum 666, but since the sum of initial consecutive natural numbers always yields a triangular number, 666 is a triangular number. Surely, we know that 36 is a square number (that is, [6.sup.2]). Therefore, for 666, we can say that a square number of initial natural numbers has given us a triangular number.

To further demonstrate the uniqueness of the number 666, consider the first seven prime numbers: 2, 3, 5, 7, 11, 13, 17. If we take the square of each of them and then find their sum, yes, we arrive at 666.

[2.sup.2] + [3.sup.2] + [5.sup.2] + [7.sup.2] + [11.sup.2] + [13.sup.2] + [17.sup.2] = 4 + 9 + 25 + 49 + 121 + 169 + 289 = 666

Another amazing fact about this strange number, 666, is that the sum of its digits (6 + 6 + 6) is equal to the sum of the digits of its prime factors. That is, since we have in prime factored form 666 = 23337, the sum of the digits, 2 + 3 + 3 + 3 + 7, is equal to 6 + 6 + 6.

It is also curious that the sum of two consecutive palindromic prime numbers, 313 + 353, is equal to 666.

The number 666 is equal to the sum of the digits of its 47th power, and is also equal to the sum of the digits of its 51st power. That is,

[666.sup.47] = 5049969684420796753173148798405564772941516295265408188117632668 936540446616033068653028889892718859670297563286219594665904733 45856

[666.sup.51] = 99354075759138594033426351134129598072385863746943100899712069 131346071328296758253023455821491848096074897283890063763421569 407683599029436416

You can check this by finding that the sum of the digits of each of the above large numbers is equal to 666.

The number 666 is equal to the sum of the cubes of the digits of its square, plus the digits of its cube. This means that if we find the square and the cube of 666:

[666.sup.2] = 443,556

[666.sup.3] = 295,408,296

and then take the sum of the cubes of the digits of the square of 666:

[4.sup.3] + [4.sup.3] + [3.sup.3] + [5.sup.3] + [5.sup.3] + [6.sup.3] = 621

and add it to the sum of the digits of cube of 666:

2 + 9 + 5 + 4 + 0 + 8 + 2 + 9 + 6 = 45

we then get 621 + 45 = 666.

Another peculiarity of 666 is that if we consider the prime factors of each of the two consecutive numbers 20,772,199 and 20,772,200, each of the sums of their prime factors is 666. That is:

20,772,199 = 741157461 -> 7 + 41 + 157 + 461 = 666

20,772,200 = 22255283367 -> 2 + 2 + 2 + 5 + 5 + 283 + 367 = 666

A strange occurrence of the number 666 is when we try to insert + signs into the sequence of numbers 1, 2, 3, 4, 5, 6, 7, 8, 9.

Here are two ways to do it:

1 + 2 + 3 + 4 + 567 + 89 = 666 or 123 + 456 + 78 + 9 = 666

If we consider the reverse sequence 9, 8, 7, 6, 5, 4, 3, 2, 1, we can do it as follows:

9 + 87 + 6 + 543 + 21 = 666

The value of [pi] is known to about 1.24 trillion places, where the decimal digits follow no discernable pattern. Yet, strangely enough, when you take the sum of the first 144 decimal places, you find it to be 666.

[pi] [approximately equal to] 3.14159265358979323846264338327950288419716939 9375105820974944592307816406286208998628034825342117067982148086513282 30664709384609550582231725359

The sum of the digits is: 1 + 4 + 1 + 5 + 9 + 2 + 6 + 5 + 3 + 5 + 8 + 9 + 7 + 9 + 3 + 2 + 3 + 8 + 4 + 6 + 2 + 6 + 4 + 3 + 3 + 8 + 3 + 2 + 7 + 9 + 5 + 0 + 2 + 8 + 8 + 4 + 1 + 9 + 7 + 1 + 6 + 9 + 3 + 9 + 9 + 3 + 7 + 5 + 1 + 0 + 5 + 8 + 2 + 0 + 9 + 7 + 4 + 9 + 4 + 4 + 5 + 9 + 2 + 3 + 0 + 7 + 8 + 1 + 6 + 4 + 0 + 6 + 2 + 8 + 6 + 2 + 0 + 8 + 9 + 9 + 8 + 6 + 2 + 8 + 0 + 3 + 4 + 8 + 2 + 5 + 3 + 4 + 2 + 1 + 1 + 7 + 0 + 6 + 7 + 9 + 8 + 2 + 1 + 4 + 8 + 0 + 8 + 6 + 5 + 1 + 3 + 2 + 8 + 2 + 3 + 0 + 6 + 6 + 4 + 7 + 0 + 9 + 3 + 8 + 4 + 4 + 6 + 0 + 9 + 5 + 5 + 0 + 5 + 8 + 2 + 2 + 3 + 1 + 7 + 2 + 5 + 3 + 5 + 9 = 666.

This remarkable-and sometimes “unlucky”-number seems to almost have a boundless array of number “coincidences” embedded within, such as that the sum of the numbers on a roulette wheel is 666.

Here are a few delectable number relationships that lead to 666:

666 = [1.sup.5] – [2.sup.6] + [3.sup.6]

666 = (6 + 6 + 6) + ([6.sup.3] + [6.sup.3] + [6.sup.3])

666 = ([6.sup.4] – [6.sup.4] + [6.sup.4]) – ( [6.sup.3] + [6.sup.3] + [6.sup.3]) + (6 + 6 + 6)

666 = [5.sup.3] + [6.sup.3] + [7.sup.3] – (6 + 6 + 6)

666 = [2.sup.1][3.sup.2][2.sup.3][3.sup.4]

We can even generate 666 by representing each of its three digits in terms of 1, 2, and 3:

6 = 1 + 2 + 3 6 = 123

6 = square root of [1.sup.3] + [2.sup.3] + [3.sup.3]

Therefore, 666 = (100)(1 + 2 + 3) + (10)(123) + (square root of [1.sup.3] + [2.sup.3] + [3.sup.3])

The number 666 is also related to the Fibonacci numbers in a variety of ways. Consider the following, where [F.sub.n] is the nth Fibonacci number:

[F.sub.1] – [F.sub.9] + [F.sub.11] + [F.sub.15] = 1 + 1 + 27 + 125 + 512 = 666

and when you inspect the subscripts, you get:

1 – 9 + 11 + 15 = 6 + 6 + 6

Similarly, for the cubes of the Fibonacci numbers:

[F.sub.1.sup.3] + [F.sub.2.sup.3] + [F.sub.4.sup.3] + [F.sub.5.sup.3] + [F.sub.6.sup.3] = 1 + 1 + 27 + 125 + 512 = 666

and now the subscripts give us:

1 + 2 + 4 + 5 + 6 = 6 + 6 + 6

Our fascination with the number 666 is just to exhibit the beauty that lies in much of mathematics. Exploring the recreational side of mathematics is an enjoyable by-product of the important role it plays in all of scientific exploration and discovery.

Palindromic Numbers

The number 666, as we mentioned earlier, is a palindrome. So we can use this number as a springboard to the next of our fascinating numerical amusements-where mathematics parallels amusing word games. A palindrome in mathematics is a number, such as 666 or 123,321, that reads the same in either direction. A palindrome can also be a word, phrase, or sentence that reads the same in both directions. Here are a few amusing palindromes:

EVE RADAR REVIVER ROTATOR LEPERS REPEL MADAM I’M ADAM STEP NOT ON PETS DO GEESE SEE GOD PULL UP IF I PULL UP NO LEMONS, NO MELON DENNIS AND EDNA SINNED ABLE WAS I ERE I SAW ELBA A MAN, A PLAN, A CANAL, PANAMA A SANTA LIVED AS A DEVIL AT NASA SUMS ARE NOT SET AS A TEST ON ERASMUS ON A CLOVER, IF ALIVE, ERUPTS A VAST, PURE EVIL; A FIRE VOLCANO

There is a well-known Latin palindromic sentence that stems from the second century CE and has an additional amazing property. It reads: “Sator arepo tenet opera rotas,” which commonly translates to “Arepo the sower holds the wheels at work.” When the letters are placed in a five-by-five square arrangement (see figure 1-2), you can read the sentence in all directions. This is quite astonishing!

Palindromic numbers or numerical expressions can lead us to consider that dates can be a source for some symmetric inspection. For example, the year 2002 is a palindrome, as is 1991.9 There were several dates in October 2001 that appeared as palindromes when written in the American style: 10/1/01, 10/22/01, and others. Europeans had the ultimate palindromic moment at 8:02 PM on February 20, 2002, since they would have written it as 20:02, 20.02.2002.

Looking further, the first four powers of 11 are palindromic numbers:

[11.sup.0] = 1

[11.sup.1] = 11

[11.sup.2] = 121

[11.sup.3] = 1,331

[11.sup.4] = 14,641

A palindromic number can either be a prime number or a composite number. For example, 151 is a prime palindrome and 171 is a composite palindrome, since 171 = 3319. Yet, with the exception of 11, a palindromic prime must have an odd number of digits.

It is interesting to see how a palindromic number can be generated from other given numbers. All you need to do is to continually add a number to its reversal (i.e., the number written in the reverse order of digits) until you arrive at a palindrome.

For example, a palindrome can be reached with a single addition, such as with the starting number 23:

23 + 32 = 55, a palindrome

Or it might take two steps, such as with the starting number 75: 75 + 57 = 132 and 132 + 231 = 363, a palindrome

Or it might take three steps, such as with the starting number 86: 86 + 68 = 154, 154 + 451 = 605, and 605 + 506 = 1,111, a palindrome

The starting number 97 will require six steps to reach a palindrome, as you can see here: 97 + 79 = 176, 176 + 671 = 847, 847 + 748 = 1,595, 1,595 + 5,951 = 7,546, 7,546 + 6,457 = 14,003, and 14,003 + 30,041 = 44,044

The number 98 will require twenty – four steps to reach a palindrome.

Be cautioned about using the starting number 196; this one has not yet been shown to produce a palindrome number-even with more than three million reversal additions. We still do not know if this one will ever reach a palindrome. If you were to try to apply this procedure on 196, you would eventually-at the sixteenth addition-reach the number 227,574,622, which you would also reach at the fifteenth step of the attempt to get a palindrome from the starting number 788. This would then tell you that applying the procedure to the number 788 has also never been shown to reach a palindrome. As a matter of fact, among the first 100,000 natural numbers, there are 5,996 numbers for which we have not yet been able to show that the procedure of reversal additions will lead to a palindrome. Some of these are 196; 691; 788; 887; 1,675; 5,761; 6,347; and 7,436.

(Continues…)

Excerpted from MATHEMATICAL AMAZEMENTS and SURPRISESby ALFRED S. POSAMENTIER INGMAR LEHMANN Copyright © 2009 by Alfred S. Posamentier and Ingmar Lehmann. Excerpted by permission.
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