From: owner-macpow@forum.swarthmore.edu

[mailto:owner-macpow@forum.swarthmore.edu]On Behalf Of stan wagon

Sent: Thursday, August 30, 2001 11:46 AM

To: Problem of the Week

Subject: Problem 938

To: PoW Enthusiasts

From: Stan Wagon, Macalester College Problem of the Week Host

Welcome back. I have been away from campus for a year and am looking

forward to a fun series of problems in the 2001-2002 season. After this

opening problem I will summarize some of my activities and publications in

the past year. Many thanks to Tom Halverson and Dan O'Loughlin for their

supervision of the PoW in my absence.

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Problem 938. Lions and Lambs

----------------------------

Suppose some lions and lambs satisfy:

1. There are at least two lions.

2. Each lion has bitten at least three lambs.

3. For any two lions, there is exactly one lamb that has been bitten

by both.

4. For any two lambs, there is at least one lion that has bitten both.

5. One of the lions has bitten six lambs.

How many lions are there? How many lambs are there?

Source: Andy Liu, Univ. of Alberta

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NEWS OF GENERAL INTEREST

------------------------

A project I have been working on for several years, "The Mathematical

Explorer", has just been released by Wolfram Research, Inc. This is an

electronic book consisting of 15 chapters that explore various themes of

modern mathematics, with an eye to computational explanations and

explorations. Topics include the four-color theorem, the Riemann

Hypothesis, calculus, Fermat's Last Theorem, pi, puzzles, unusual number

systems, square wheels, check digits, and several others. For more info.

see http://www.wolfram.com/explorer. The book is powered by Mathematica,

but it stands alone: Mathematica is NOT required to run it. The price is

$69.95. The product comes with a version of Mathematica adequate to run the

code in the product, but somewhat less than a full copy of Mathematica.

Readers who are not Mathematica users might find this product a useful

introduction to the software.

Here is part of the official WRI press release:

"The Mathematical Explorer is a stand-alone product, built

on Mathematica technology, that is aimed at mathematical hobbyists --

those who have an interest in mathematics but are not necessarily

professionally

trained mathematicians. You might think of the market as consisting of

the

sort of people who used to read Martin Gardner's 'Mathematical Games'

column in

Scientific American.

"The Mathematical Explorer includes some of the most famous subjects

in the

history of mathematics: Fermat's Last Theorem, the Riemann Hypothesis,

Escher patterns,

the mathematics of cryptography, the digits of pi, and much more. The

Mathematical

Explorer differs from previous treatments in that it is fully

interactive, encouraging

users to 'walk in the computational footsteps' of the great

mathematicians and simulate

some of their discoveries."

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--------------

In the fall, I wrote a lengthy review of three number theory books for the

"Amer. Math. Monthly" (E. Burger, P. Ribenboim, and G. Tenenbaum &
M.

Mendes France). A rewarding aspect of that was that it caused me to look at

the simplest possible heuristic argument one can give for the prime number

theorem, which states that the number of primes under x [pi(x)] is

asymptotic to x/log x. With help from experts A. Granville and C.

Pomerance, and using ideas that are pretty much all Chebyshev's, I included

in the review a quite short proof (using nothing beyond elementary

calculus) of what is known as Chebyshev's Theorem: If pi(x) is asymptotic

to c x/log x, then the constant c must be 1. In short, if logs enter the

picture (and it is easy to see that this is plausible) then the logs must

be natural logs. A copy of the review (postscript file) can be downloaded

from

http://stanwagon.com/wagon/Misc/review.ps.

I spent the fall at the Mathematical Sciences Research Institute in

Berkeley. The most interesting mathematics has been a research project with

R. Crandall (Reed College) on sums of squares. We conjectured and then

proved a nifty closed form for the constant c_k in the expression

Sum[r_k(n)2, {n, 1, N}] is asymptotic to c_k N^(k-1),

where r_k(n) is the number of representations of n as a sum of k squares

and k = 3, 4, 5, ... .

In January, our Minnesota team again took part in the Breckenridge

International Snow Sculpture Championships. We sculpted exactly what we

hoped to, but won no prize. Photos of our and other pieces (the Swiss team

that we defeated to win second place in 2000 took first place in 2001) are

at

http://stanwagon.com. The nifty thing about our 2001 project was that

Matthias Weber, an expert in minimal surfaces, was able to come up with an

equation for the complicated surface invented by sculptor Robert Longhurst.

Matthias joined our team and did soap bubble displays (minimal surfaces)

for the viewers. We plan to enter again, under the guidance of sculptress

Bathsheba Grossman, in Jan. 2002. See http://www.bathsheba.com/gallery.html

for her interesting sculptures.

I have just been informed that a paper of mine on Gaussian primess (with E.

Gethner and B. Wick) from 1998 ("Amer Math Monthly", vol. 105, 327-337)
has

been awarded the MAA's Chauvenet prize; a pleasant surprise. My book with

D. Bressoud, "A Course in Computational Number Theory", appeared in
August

2000 (Key College Press), and was named one of the top ten math books of

2000 by the American Library Association. In mid-summer another paper on

the Gaussian primes ("The Gaussian zoo", with J. Renze and B. Wick)

appeared in "Experimental Mathematics"

(http://www.expmath.org/expmath/volumes/10/10.html).

In April, I returned to Macalester for the honors thesis defence of

Sigbjorn Vik, who carried out a nice project on theoretical and practical

issues of the computational geometry problem of triangulating a polygon (or

a set of polygons). More info: sigbjorn.vik@uwc.net .

In July, E. Packel and I taught our annual "Rocky Mountain Mathematica"

course in Frisco, Colorado. We had 44 attendees: high school and college

teachers and industrial mathematicians. More info:

http://rmm.lfc.edu.

-------------------------------------

NEWS OF PERHAPS LESS GENERAL INTEREST

-------------------------------------

In late April, I and three friends did a hard ski traverse in Canada: 7

days over glaciers ending at Fairy Meadow Hut, where we had another week

skiing with friends (including recent Macalester grad Robbie Nachtwey) who

flew there by helicopter. It was a most rewarding adventure. In August, I

and two others returned to B.C. to try to climb Mt. Bryce. We failed. But

we did succeed on our secondary objective, the northwest buttress of Eagle

Peak at Rogers Pass. This took us 28 hours tent-to-tent, 8 of which were

spent on a small ledge watching the moon cross the sky. Fortunately, the

weather was good. The climb was a Grade IV, 5.7+. Photos of these

adventures are at http://stanwagon.com.

In December, I visited New Zealand, but bad weather meant our climbing

plans failed. Still, the island was fascinating, and made a very positive

impression.

In September, I did some good climbs in Yosemite, including the famous

Fairview Dome and the East Buttress of El Capitan. The latter was very hard

-- too hard for me, as I had to pull on the rope several times.

In March, I and a partner completed the 38-mile ski race from Crested Butte

to Aspen (Colo.). This is a super event, which starts at midnight. There

was 7000' of climb, and the sunrise as we crested a 12000-foot pass was

memorable. We finished 24th of 96 teams, in 11.3 hours, which is not bad.

And, back in May 2000, I and three others carried out a 17-day expedition

to Mt. Logan, Canada's highest peak. We skied to 19000 feet, but failed by

a short distance to make the West Summit. Still, it was memorable. We spent

four nights at the highest camp (17,500' -- we were almost surely the

highest people in N. America on those nights) in temps. of -40 (the fixed

point of the C and F scales). The final day was awesome as we went until 2

a.m. to ski from that high camp over an 18,500-foot pass, and all the way

to base camp at 9500'. Photos and a full report -- including a description

of the Jim Carrey story (!) -- are at stanwagon.com.

Whew...I'm tired just thinking about all this....

Stan

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This is the Problem of the Week from Stan Wagon at Macalester College

in St. Paul, Minnesota <<wagon@macalester.edu>>. This Macalester

tradition was started by the late Joe Konhauser in 1968 and

has continued unbroken since then.

These problems are intended for our undergraduates. I do not necessarily

wish to receive e-solutions unless:

(a) you have an interesting nonstandard approach to a solution;

(b) you have a variation or extension that might be worthy

of dissemination; or

(c) you have information about the history of the problem.

Of course, I encourage your problem suggestions, preferably with solutions.

Macalester students should not subscribe to this list. They should

get the printed postings every Friday.

To subscribe to this list, send e-mail to

majordomo@mathforum.com

with ONLY the following words in the message body:

subscribe macpow

To unsubscribe, e-mail from the account to be unsubscribed to

majordomo@mathforum.com

with ONLY these words in the message body:

unsubscribe macpow

The Math Forum maintains an electronic archive of our

Problem of Week at http://mathforum.com/wagon/

Solutions are posted for a limited time. The book

"Which Way Did the Bicycle Go?" by J. Konhauser, D. Velleman,

and S. Wagon is available from the Math. Assoc. of America and contains

191 problems and solutions selected from the Problems of the Week

from 1968 to 1995.

This is the Problem of the Week from Tom Halverson at Macalester College

in St. Paul, Minnesota <<halverson@macalester.edu>>. This Macalester

tradition was started by the late Joe Konhauser in 1968 and

has continued unbroken since then.

These problems are intended for our undergraduates. I do not necessarily

wish to receive e-solutions unless:

(a) you have an interesting nonstandard approach to a solution;

(b) you have a variation or extension that might be worthy

of dissemination; or

(c) you have information about the history of the problem.

Of course, I encourage your problem suggestions, preferably with solutions.

Macalester students should not subscribe to this list. They should

get the printed postings every Friday.

To subscribe to this list, send e-mail to

majordomo@mathforum.com

with ONLY the following words in the message body:

subscribe macpow

To unsubscribe, e-mail from the account to be unsubscribed to

majordomo@mathforum.com

with ONLY these words in the message body:

unsubscribe macpow

The Math Forum maintains an electronic archive of our

Problem of Week at

http://mathforum.com/wagon/

Solutions are posted for a limited time. The book

"Which Way Did the Bicycle Go?" by J. Konhauser, D. Velleman,

and S. Wagon is available from the Math. Assoc. of America and contains

191 problems and solutions selected from the Problems of the Week

from 1968 to 1995.