## Math Horizons World Wide Web Treasure Hunt Results from November 2000

The ballots have been counted and recounted, no chad has been left unturned.  The winners of the  Math Horizons World Wide Web Treasure Hunt are:  Erica Voolich of the Solomon Schechter Day School,  Scott Hunter and Susan Strickland of St. Mary's College of Maryland, and Kim Groshong of Ashland University.  Erica and Scott were the only entrants to answer all 15 questions correctly.  Kim was one of several to correctly answer 14 of 15, her submission is included below so that you can see the correct answers.  Each of these three has been sent one of the coveted (and handsome) Math Horizons t-shirts.

Honorable Mention (at least 13 correct answers) to:
Kristine Harootunian, St. Lawrence University
Catherine Timmins, Aycock Middle School
Solomon Willis, Shelby, NC
Ali Bukhari, Meriden, CT
Sara Wood, St. Lawrence University
Amanda Febey, St. Olaf College
Josh Harris, Greensboro, NC
Lynelle Weldon, Andrews University
Gretchen Koch, St. Lawrence University
Charlotte Knotts-Zides’s Calculus class, Wofford College
Vivek Bachhawat, St. Lawrence University.

Many thanks to all for playing.

Steve Kennedy

1.     How many 0’s are in the First Million Digits of Pi?

Source
:  “Here are the first million digits of pi. The distribution of digits is as follows:
digit 0: 125505”

Editor’s Note:  This was the most common error (and the only one Kimberly, and many others, made).  Many people quoted Professor Bailey’s website without really thinking about it.  Please note that pi appears to be “normal,” that is all digits occur with equal frequency, so one should expect about 100,000 zeroes in any million digit block.  In fact, there are exactly 99,959.  See

If we assume that pi is normal (this is not known), then the probability that about one-eighth of the first million digits of its decimal expansion are zeroes would be incredibly small.  Note also that the digits reported by Bailey add up to way more than a million.

2.     What was the average starting salary in 1999 for math graduates with a bachelor’s degree?

Source:   According to 1999 survey by National Association of College and Employers, starting salary for those with B.S. average about \$37,300”

Website

Editor’s Note:  Lots of different answers to this one ranging all the way up to \$62,000.  Check out:

3.      By what nickname is Leonardo Pisano better known?

Answer:  Leonardo Pisano is also known as the famous Fibonacci

Source:  “Who was Fibonacci?  The "greatest European mathematician of the middle ages", his full name was Leonardo of Pisa, or Leonardo Pisano in Italian since he was born in Pisa (Italy), the city with the famous Leaning Tower, about 1175 AD. He called himself Fibonacci [pronounced fib-on-arch-ee or fee-bur-narch-ee] short for filius Bonacci which means son of Bonacci. Since Fibonacci in Latin is "filius Bonacci" and means "the son of Bonacci", two early writers on Fibonacci (Boncompagni and Milanesi) regard Bonacci as the family name so that Fib-Bonacci is like the English names of Robin-son or John-son. Fibonacci himself wrote both "Bonacci" and "Bonaccii" as well as "Bonacij"! Others think Bonacci may be a kind of nick-name meaning "lucky son" (literally, "son of good fortune").”

4.     What is the name of the sum of the reciprocals of twin primes and what does it have to do with the Pentium chip?

Answer:  Brun’s Sum, also called Brun’s constant, is the name of the sum of reciprocals of twin primes.  Dr. Thomas Nicely determined this constant to be approximately equal to 1.902165778+2.1X10^-9.  While searching for a more accurate Brun’s Constant, Dr. Nicely discovered the Pentium bug, which happened to be noted after more than a million PC’s had already been distributed with the faulty processor.  It cost Intel 475 million dollars to rectify the situation.

Source:  “Brun's sum is the sum of the reciprocals of the twin primes… Dr. Thomas Nicely is a math professor at Lynchburg University in Virginia.  He is one of the leaders in the search for Twin Primes and a more and more accurate Brun's Constant, and it was during his research that he discovered the "Pentium Bug" in 1995, I believe. He shouldn't be famous for that, though - his ideas are fantastic, and his Twin Prime research is way over my head. I like the simpler ideas.”

Website

5.     Of the 250 jobs ranked recently by a job-rating publication, how many of the top ten jobs were math or computer science related?

Answer:  9 out of the top 10

Source:  “Nine of the top 10 jobs were in computer or math-related fields, with Web site managers at the top of the heap.”

Editor’s Note:  Lots of  different acceptable answers here, too.  I received answers ranging from 6 to 11!

6.     What is the value of pi suggested by the Egyptian Rhind Papyrus?

Source:  “In the Egyptian Rhind Papyrus, which is dated about 1650 BC, there is good evidence for 4(8/9)2 = 3.16 as a value for pi”

7.     How many hours of computer time did Appel and Haken use for the final proof of the Four-Color Theorem?

Source:  “…the four-color map problem was solved (more or less) using a computer by two prairie geniuses at the University of Illinois at Champaign-Urbana, Wolfgang Haken and Kenneth Appel.   The four-color map problem, as all mathematically hip personages know, is to determine whether there is any map that requires the use of more than four different colors if you want to avoid having adjacent regions be the same color. A matter of no great consequence, you might think, but this is the sort of thing that fascinates math aficionados--in this case for well over a century. Haken and Appel proved that (as was widely suspected) four colors are all you ever need.  Cecil would be pleased to reproduce H&A's proof here, except that it took 1,200 computer hours and a zillion cubic yards of printout paper to do, so you're just going to have to take my word for it. Basically what the computer did was check out all the possible map combinations by trial and error.

8.     How many digits are in the largest know prime?

Answer:  2,098,960 digits in the 38th Mersenne prime

Source:  “As of 1 June 1999, the largest know prime is the 2,098,960 digit Mersenne prime 2^6972593-1”

9.     Who said “there is no permanent place in the world for ugly mathematics?”

Source:  “Hardy, Godfrey H. (1877 - 1947)

The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.

A Mathematician's Apology, London, Cambridge University Press, 1941.”

10.  Who is offering a million dollar prize for the solution of a mathematical conjecture formulated in 1742 and why?

Answer:  Faber and Faber today issues a \$1,000,000 challenge to prove Goldbach's Conjecture.  The contest is to publicize a new book as well as drawing interest in finding a solution to this famous problem.

Source:  “Goldbach's Conjecture was first stated in 1742 in a letter written by Christian Goldbach to the great Swiss mathematician Leonard Euler. The Conjecture is popularly represented as the conjecture that every even number greater than two is the sum of two primes.  Although Euler spent much time trying to prove it, he never succeeded. For the  next 250 years, other mathematicians would struggle in similar fashion. The proof has not been found to this day, and Goldbach's Conjecture is acknowledged to be one of the most notoriously difficult problems in all of mathematics.  On 20 March 2000, Faber and Faber are publishing Uncle Petros and Goldbach's Conjecture, the wonderful and already acclaimed novel by Apostolos Doxiadis. It has been described by John Nash, Nobel Prize Winner as 'a fascinating picture of how a mathematician could fall into a mental trap by devoting his efforts to a too difficult problem' and by George Steiner as 'deeply generous. It allows the lay-reader lucid access to intrinsically closed worlds.'  To celebrate publication, we are offering a prize of \$1million to any person who can prove Goldbach's Conjecture within the next two years*  This challenge is issued in conjunction with Bloomsbury Publishing, USA, the book's American publisher.  For further information on the publicity concerning the challenge, please call Judith Hillmore on 0171 465 7554 or e-mail her at judith.hillmore@faber.co.uk  Details of how to enter are available with the Rules of the Challenge, or on the Faber and Faber website.

*In the event that no satisfactory proof of Goldbach's Conjecture is offered in accordance with the Rules of the Challenge, the reward will not be awarded. No book purchase required.”

Website

11.             What is lucky about the number 2187 and why does Martin Gardner care?

Answer:  Every even number is the sum of two lucky numbers.  Lucky numbers are an unusual sequence identified by Stanislaw Ulam around 1955.  2187 is a lucky number and Martin Gardner’s childhood home had this house number.  This lucky number has many remarkable properties identified by Gardner.

Source:  “Around 1955,  the mathematician Stanislaw Ulam (1909-1984) identified a particular sequence made up of what he called "lucky numbers," and mathematicians     have been playing with them ever since.

Starting with a list of integers, including 1, the first step is to cross out every second number: 2, 4, 6, 8, and so on, leaving only the odd integers. The second integer not crossed out is 3. Cross out every third number not yet eliminated. This gets rid of 5, 11, 17, 23, and so on. The third surviving number from the left is 7; cross out every seventh integer not yet eliminated: 19, 39,.... Now, the fourth number from the beginning is 9. Cross out every ninth number not yet eliminated, starting with 27.        This particular sieving process yields certain numbers that permanently escape getting killed. That’s why Ulam called them "lucky." See the table below for a list of lucky numbers less than 200.

1 3 7 9 13 15 21 25 31 33 37 43 49 51 63 67 69 73 75 79 87 93 99 105

111 115 127 129 133 135 141 151 159 163 169 171 189 193 195

What’s remarkable is that the "luckies," though generated by a sieve based entirely on a number’s position in an ordered list, share many properties with primes. For example, there are 25 primes less than 100 and 23 luckies less than 100. Indeed, it turns out that primes and luckies come up about equally often within given ranges of integers. The distances between successive primes and the distances between successive luckies also keep increasing as the numbers increase. In addition, the number of twin primes -- primes that differ by 2 -- is close to the number of twin luckies.

Perhaps the most famous unsolved problem involving primes is the Goldbach conjecture, which states that every even number greater than 2 is the sum of two primes. Luckies are featured in a similar conjecture, also unsolved: Every even number is the sum of two luckies. Computer searches   have reached at least 100,000 without finding an exception. Martin Gardner describes many more features of lucky numbers in a delightful article in a recent issue of The Mathematical Intelligencer. "There is a classic proof by Euclid that there is an infinity of primes," he writes. "Although it is easy to show there is also an infinity of lucky numbers, the question of whether an infinite number of  luckies are primes remains, as far as I know, unproved."

How did the topic of lucky numbers happen to come up? The house where Gardner grew up in Tulsa, Okla., had the address 2187 S. Owasso. "Of course I never forgot this number," he says. It also happens to be one of the lucky numbers. Gardner’s imaginary friend, the noted numerologist Dr. Irving Joshua Matrix, can readily find additional remarkable properties associated with that number. Exchange the last two digits of 2187 to make 2178, multiply by 4, and you get 8712, the second number backwards. Take 2187 from 9999 and the result is 7812, the number in reverse. Moreover, the first four digits of the constant e, 2718, and the number of cubic inches in a cubic foot, 12^3 = 1728, are each permutations of 2187!”

12.            What is special about the sequence, 5,8,15,77,125, 714, and 948,…?

Answer:  This sequence contains the first element in the pair of Ruth-Aaron numbers, which are the sum of prime divisors of n= sum of prime divisors of n+1

Source:  “Playing with Ruth-Aaron Pairs

On April 8, 1974, Henry (Hank) Aaron hit his 715th major league home run, surpassing the previous mark of 714 career home runs long held by baseball great Babe Ruth. Understandably, the event received considerable coverage in newspapers and magazines and on television.

However, those reports invariably overlooked the mathematical aspects of that achievement, particularly the curious properties of the two numbers 714 and 715. It took the efforts of mathematicians Carol Nelson, David E. Penney, and Carl Pomerance at the University of Georgia to call attention to this facet.

Notice that 714 = 2 x 3 x 7 x 17 and 715 = 5 x 11 x 13; so 714 x 715 = 2 x 3 x 5 x 7 x 11 x 13 x 17.  In other words, the product of the two consecutive whole numbers 714 and 715 is equal to the product of the first seven prime numbers!

Pomerance and his colleagues wondered whether there were other pairs of consecutive numbers whose product is also the product of the first k primes. The first few instances are easy to find: 1 and 2 (1 x 2 = 2), 2 and 3 (2 x 3 = 2 x 3), 5 and 6 (5 x 6 = 2 x 3 x 5), 14 and 15 (14 x 15 = 2 x 3 x 5 x 7), and 714 and 715. The mathematicians then used a computer to search for such pairs, going as far as products of the first 3,049 primes (numbers up to 6,021 digits long). They found no more examples in that range.

Footnote: On April 26, 1974, Aaron hit his 15th grand slam home run, breaking the old National League record of 14.

And there's more. Notice that the sum of the prime factors of 714 is 2 + 3 + 7 + 17 = 29, and the sum of the prime factors of 715 is 5 + 11 + 13 = 29. How often do two consecutive numbers have prime factors that add up to the same total?

Pomerance and his coworkers conducted another computer search, looking for such pairs up to a value of 20,000.

Here are the first few examples:

Numbers, Sums
5, 6           5 = 2 + 3
8, 9           2 + 2 + 2 = 3 + 3
15, 16       3 + 5 = 2 + 2 + 2 + 2
77, 78       7 + 11 = 2 + 3 + 13
125, 126   5 + 5 + 5 = 2 + 3 + 3 + 7
714, 715   29
948, 949   86

Pomerance called these pairs Aaron numbers, and he speculated that such pairs become less frequent as their size increases. However, he didn't have a mathematical proof quantifying their scarcity.”

13.             What famous mathematical event occurred on the morning of August 8, 1900?

Answer:  David Hilbert delivered A lecture before the International Congress of Mathematicians in Paris on this date.  In this lecture David Hilbert, proposed his now famous 23 problems.

2 Sources and websites:

1.  The 1998 International Congress Mathematicians in Berlin was a "best approximation" to the first centennial of David Hilbert's hugely influential set of twenty-three problems, presented at the International Congress of Mathematicians in Paris, in August 1900.

2.      In German,

Der unten folgende Text des Hilbertschen Vortrags vom 8. August 1900 erschien erstmals in den Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen,

English translation: That below following text of the Hilbertschen lecture of the 8. August 1900 appeared in the news for the first time the royal society of the sciences to Göttingen,

14.             How many solids exist whose faces are identical and regular, and whose vertices each have the same number of faces incident on them?  (These solids need not be Platonic:  Platonic solids have the added requirement of being convex)

Source:  “All mathematicians are familiar with the Platonic solids: the tetrahedron, the cube, the octahedron. The dodecahedron, and the icosahedron. These are the five convex solids all of whose faces are identical regular polygons.  Considerably less well known are the solids obtained when the above conditions are relaxed…On the other hand, if the faces are required to be identical regular polygons, but the solid is not required to be convex, we obtain the four Kepler-Poinsot polyhedra.”

15.            Who was the first woman in the U.S. to earn a PhD in mathematics?