-----Original Message-----
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.


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

------------------------
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."
---------------------------------------------------------------------------
--------------

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
---------------------------------------------------------------------------
-----
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.