Material added 18 November 2003
Material added 10 November 2003
My third column for maa.org talks about matrix
revolutions and groups. I finally made a really nice picture
of all the Archimedean points, it's near the end. One link I
eventually took out referred to Nova's The Elegant Universe,
which discusses string theory. Near the middle of the show,
everything boiled down to a formula by Euler, which caused me to break
out laughing (everything seems to lead back to Euler). He also
developed rotation matrices, but didn't use a matrix form. Since
I wanted to focus on matrices, I pulled the reference.
Historically, flu
epidemics have a coincidental tendency to start during periods of massive solar activity.
Livio Zucca has put together a page on all
(?) the equilateral
pentagons that can tile the plane.
For those of you that don't have $10000
for Euler's Opera Omnia,
there is now an Euler
website.
Bob
Abbott: The big rollingblockmaze craze is over, but maybe you'll
want to mention that I just added five more of these
mazes to my site. These are mazes I created for GAMES and they
appeared in the magazine over the last couple of years. They're
pretty good, especially my rollingslab maze.
The latest Games magazine is worth a look,
among other things, you'll find the Games 100,
a new maze by Robert Abbott, and Robert Lodge's "Ultimate
Calculatrivia".
Material added 3 November 2003
My second column for maa.org concerns the
Möbius Function. As a puzzle of the week, consider the numbers
that are products of two different primes. In the Möbius
function, these would have value positive one. There are three distinct
digits that can be arranged to make
product of two distinct primes. In fact, all six arrangements give
the product of two distinct primes. Can you find the smallest of these
numbers? Answers and Solvers.
Erich Friedman's Math Magic discusses
various Square
Packings this month.
The weekly DottoDot puzzles at monkeyingaround.com
are rather nice. Walter Pullen has a page describe maze types and
solving algorithms. Brian Berg is an expert at stacking cards. Jim Morey
has made a java
applet that interactively connects all the Archimedean solids.
Although Euler never traveled there, he
made the Bridges of Königsberg famous with the following
question: Is it possible to make a tour so that one passes
just once over all the bridges over the river Preger in Königsberg?
Jan Kåhre is a mathematician that has recently visited
Königsberg, and he has found a solution for the problem.
There is an audio paradox involving
continually rising tones. These are know as Shepard Tones.
You can here them at the pages of Scott
Flinn, noah.org,
and exploratorium.edu.
Material
added 25 October 2003
I've started doing columns for maa.org. My first one is Paterson's
Worms Revisited. Slashdot
discusses it.
The NKS website, wolframscience.com, has been
redesigned, and a forum
has been added, along with a PDF
list of open problems. I like my Number
Systems notebook for Mathematica on the Summer School
portion.
The Borromean Rings,
often seen in relation to a discontinued
brand of beer, can be made by linking cubes together. You
should have no trouble gluing the faces of cubes together to get 3
rings that are linked in the same way as the Borromean Rings, so that
they fit into a side5 cube. Fitting them into a side4 cube is
also possible. Can anyone figure out how? Answer and solvers. I solved this
using
the Livecube system.
Stephen Collin has written a nice program
for Penrose Rhombs.
Johan de Ruiter has a page of interesting Pathfinder puzzles.
Material added 18 October 2003
Various interesting contests have
started. 1.
Al Zimmermann regularly sponsors
interesting contests. His latest is Squares, where one must minimize
the number of squares in a grid. The top solver gets
$500. 2. The 12th
World Puzzle Championship has finished in Papendal, the
Netherlands. Ulrich Voigt and WeiHwa Huang finished 1st and 2nd.
Other information is available wpc.puzzles.com
and Puzzelsport.
3. Cihan Altay: "PQRST 07 Puzzle Competition
starts on October 25th Saturday at 20:00 (GMT+02). You'll have one week
to solve and rate 10 puzzles." 4.
Denis Borris found a way to divide a
square into 8 dissimilar integersided right triangles. He
believes he has the smallest possible solution. If anyone can
beat his solution, he will send a $2 Canadian Toonie coin to whoever
has the smallest square. I'd also like to see the best solution
for 7
dissimilar integersided right triangles making a square. For
this, I
will send the best solution 7 different state quarters. Send
Answer. 5. The
NPR puzzle of the week is numerical: 71 42 12 83 54.
Fractal
Maze, by Mark
J P Wolf (mark.wolf at cuw dot edu)
"Traditional mazes can always be
solved by brute force methods. Every path that is not the correct
path is either a dead end or a loop of finite length, so given enough
time, one can explore every path of the maze. Fractal mazes,
however, have wrongway paths that are infinitely long, making them
neither dead ends or loops; they are much harder to recognize as the
wrong way. The fractal maze is fractal because it has identical
copies of itself embedded within itself, which can be entered. In
the maze below, you must enter them to solve the maze. Begin at
the MINUS and make your way to the PLUS. When you enter a smaller
copy of the maze, be sure to record the letter name of that copy, as
you will have to leave this copy on the way out. You must exit
out of each nested copy of the maze that you have entered into, leaving
in the reverse order that you entered them in (for example: enter A,
enter B, enter C, exit C, exit B, exit A). Think of it as a
series of nested boxes. If there is no exit path leaving the
nested copy, you have reached a dead end. The eight pins on each
of the sides of the maze represent these connections to the outside of
each copy (obviously, you cannot go outside of the main maze
itself. Watch your entrances and exits, and go from MINUS to
PLUS. This is from a work in progress, 100 Enigmatic Puzzles. Added: There are three different solutions to the
fractal maze. 1. The easy one, which does not go very deep,
begins by entering copy C of the maze. 2. A harder solution begins by
entering copy E of the maze, and goes the deepest. 3. The hardest
solution, which begins by entering copy A of the maze, has the most
steps." Yogy's
Fractal Maze Simulator. Answers
and Solvers. Yogy Namara's
Solution.Yogy's bigger solution.
Click on the image for a
larger version. Robert
Abbott: "This maze embodies a mindblowing concept!"]
George Orwell, 1984: "The Ministry of Truth 
Minitrue, in Newspeak  was startlingly different from any other
object in sight. It was an enormous pyramidal structure of glittering
white concrete, soaring up, terrace after terrace, 300 metres into the
air." Did you know that North Korea built a 330 meter
tall pyramid in white concrete, the Ryugyong
Hotel? It is an unfinished, unused shell, due partially to
using substandard
concrete. Ironically, Pyongyang contains many ministries.
A more successful pyramid is the Luxor Hotel in Las
Vegas. Now, to the puzzle:
A pyramid timer was recently built by Oskar van Deventer. "A
3sided Hourglass interprets binary in three phases: reset, preparation
and run. In the reset phase, all sand is collected in one compartment.
Next is the preparation phase, when the user has to decide several
times whether to turn the Hourglass left or right. Every time, the
sands distributes equally over the two lower compartment. The operation
instruction shows the example where the user chooses turning
rightleftrightrightleft, using an 8minute Hourglass. In this
example the user ends up with 3.5 minutes of sand in the right
compartment, which enables him to boil his egg in 3.5 minutes. In
theory, all time intervals between 0 and 8 minutes can be achieved with
this hourglass, doubling its time resolution after every turn. Question
to the reader: how to use the Threelegged Hourglass to boil an egg
exactly 2.25 minutes?" Answer and Solvers.
Jeremy Galvagni's 3bulb Solution. Jeremy Galvagni's tetra solution.
Multibulb hourglasses by Oskar van Deventer. Click on the first
image for greater detail.
Material
added 12 October 2003
Dividing a square into smaller squares is
often called the Mrs. Perkins'
Quilt problem. I mentioned a while ago that Erich Friedman found a
way to divide a side67
square into 20 smaller squares all with doubledigit sides. Can
you find his solution? Answer. The only
solver of this puzzle was Lance Gay. Lance followed up by finding
better solutions for squares of size 88 to 90. Robert Wainwright
sent me better answers for sizes 53 and 91. All of these new
solutions were graciously accepted by Richard Guy, for the next edition
of Unsolved Problems in Geometry.
John Conway let me know that Dudeney named the puzzle. Geoffrey
Morley sent in transcriptions of Duijvestijn's solutions. Erich
Friedman sent me this list of best
solutions (Antony Boucher and Lance Gay filled in the blanks). The
current best known solutions are as follows: {1

1}, {4
 2}, {6
 3}, {7  4}, {8  5}, {9  6,7}, {10  8,9},
{11 
1013}, {12

1417}, {13  1823}, {14
 2429}, {15 
3039,41}, {16  40,4253}, {17  5470}, {18  7191}, {19 
92108}. Lance Gay: "Since seeing your writeup on
mathpuzzle.com last month, I have written some Mrs. Perkins quilt
search software to look for solutions. For fun, I added a routine to
print out the bitmaps of solutions. I run my searches on a pair of 1.8
GHz MSWindows machines. It appears that computers have gotten faster
since people have last done some serious searching. For now, I have
completed all searching for sides up to 100. I am now concentrating on
larger squares (such as attempting to beat the order20 side154
square). My search algorithm is not exhaustive so the trick is reducing
the search space to the most fruitful areas." John Conway: "You
might add the two 1950s papers both called "Mrs Perkins's Quilt", one
by me and one by G.B.Trustrum, in Proc. Camb. Phil. Soc. I gave
the answers for low n, and an upper bound of order n^(1/3) for general
n, which Trustrum improved to order log(n). Since there's an
obvious logarithmic lower bound, all that remains is to find the best
constant. I don't know if there's been any progress on that
problem since those two papers." Antony Boucher sent me some
corrections for the list, and also sent solving code.
New quilts discovered by Lance Gay.
Jorge Rezende has put together a
page on magic
polyhedra.
A recent Astronomy
Picture of the Day showed a hurricane next to a galaxy. Both are
governed by the laws of logarithmic spirals.
Material
added 7 October 2003
Last year, I had a great time with Theo
Gray as he won an Ignobel
Prize. This year, there was another interesting batch of
winners. Theo Gray also helped to design a little product called Mathematica,
made in ChampaignUrbana. This morning, I learned about a whole
host of Nobel Prize winners that are neighbors of mine  1. Paul
Lauterbur, Nobel Prize in Medicine, 2. Anthony J
Leggett, Nobel Prize in Physics, 3. Alexei Abrikosov,
Nobel Prize in Physics. I guess I live in the genius capital
of the world. If you look at that last photo, you'll see that
Alexei is a Mathematica
user. I'll try to get a copy of that coollooking notebook from
him for our Infocenter.
To heighten the odd coincidences in math,
physics, and Theo, Theo sent me a link to a version of Escher's
Relativity in Lego a few days ago. A gorgeous construction,
well worth a look. Lego isn't all that great of a building
material for polyforms, though. If you would like to study
polycubes, you should visit livecube.com.
A 6x6x6 brick of cleverly configurable cubes is $30. I've been
wanting to try the 25 Lpentominoes in a 5x5x5 cube for awhile, and now
I can finally do it.
Previous Puzzles of the week are here.
