Fair Dice, Iamonds,Eternity, Mazes, Atom Stability, Dissections, Kites&Bricks,Triangles, Retrograde, Pentagons, Prize Problems, BOOKS, and Partridges.
added 29 Jun 2003
This week, I'm at the NKS2003
conference. You can see the Wolfram Atlas of Simple Programs,
or obtain a free
of the just-released Mathematica
5. Theo Gray helped launch the Powermac with Steve Jobs,
who was at the Mathematica launch 15 years ago.
My favorite new thing so far is Metatron
by Bathsheba Grossman.
Basically, it's a sculpture based on the cuboctahedron,
but with enough ribbons, curves, and loops to make casting almost
impossible. The very
interesting part is that this sculpture can now be printed in bronze. Other
3D-prints are nice, such as George Hart's Deep
Structure, but that's made with starch. Bathsheba was given
access to the first bronze printer, and I really like the result.
[Update -- Bathsheba is an expert on putting mathematical 3-D pointsets
into crystal. If you have a pointset you think would look
interesting, you should contact her.]
Propp Circles (which he calls the
rotor-router model) were invented by Jim Propp. Basically,
traffic cops are being dropped on a grid, and no two can stay on the
same spot. Once a traffic cop has found a spot, she directs
traffic north, then east, then south, then west. After 750,000
cops have found a spot to stand, you obtain the following pattern,
where the color indicates which direction the cop is pointing. So
far, no-one has looked at the 3D case (Busy week for both myself and
Jim). Is it spherical?
The Propp Circle (rotor-router model)
added 20 Jun 2003
I rather like the Math Magic
challenge this month ... making a number square so that all the
across and down numbers have a common divisor. The greatest
common divisor for 3x3 squares using 0-9 without repetition is
17. Can you find it? Answer and
Bryce sent me an appropriate puzzle:
In honor of Rowling's new book, I've compiled a short list. Below are
five clues, consisting of the letters in "PHOENIX" in various orders.
They were extracted from five words or phrases without changing their
order in that word or phrase. (For example, EOHINXP would be a clue for
HIENPOX, HONPEXI (2 words), IEOHPNX (4 words),
XENPHOI (2 related answers), and XIHPEON (2 words). Answer and Solvers.
Don Reble: "10^100+37 is 87719765535727771
times a composite number; 10^100+39 is 39640576062095087 *
41313840579541273 * 87719765535727771 *
I haven't found any interesting semiprimes,
yet. With Proth,
I found that 914 5^(5^5) +1 is prime. I wanted to find a case for
k 6^(6^6) + 1 as prime, but so
far all k up to 29812 have
numbers. 26118*2003^2003 + 1 is prime.
Eric Weisstein put together a wonderful
Mathematica notebook on Kimberling
Centers. You can read more about triangle centers at Kimberling's page.
I decided to make a lot of
pictures., which Jan Schneider turned into an animation
(below). Eventually, I hope to extend that notebook to the rest
the triangle centers, and then to include the Edward Brisse centers as well. Even
now, it's quite a constellation. When I have a spare hour, I plan
to look at which of these centers work in a skew tetrahedron -- pick a
center, then connect each vertex to the center on the opposite
face. If the four resulting lines are concurrent, the triangle
center is also a tetrahedral center. All that will produce a nice
galaxy of a thousand or so tetrahedral points. If anyone needs a
project, there's a nice one.
A constellation of triangle centers.
Element 110 has been named --
Darmstadtium. In other emental news, a few months ago, I helped Theo Gray make
cream with liquid nitrogen, as part of a Popular Science column.
(Take a gander through the latest issue at the newsstand, it's just
packed with good stuff.)
At the Infocenter,
one notebook I'm hoping to play with more soon is the one about Canonical
Polyhedra, (which I should link to George
Hart's page). My idea there is to represent a lot of
polyhedra with ultra-simple Schlegel graphs (which I should write up
with diagrams from my heptahedra
page), which could be popped into canonical forms.
added 12 Jun 2003
I did the the National Public Radio
puzzle of the week this week.
has a page of unfolded 4-D polytopes.
I found a nice crossnumber puzzle in the
August 1974 issue of Games and
Puzzles. Your only clue for the numbers going across and
down is the number of divisors that number has. For example, if
an entry was 21, the clue would be "4", since 21 has 4 divisors (1, 3,
7, 21). Answer and Solvers.
added 2 Jun 2003
The wonderfully creative James Stephens of
Puzzlebeast has created a new type of puzzle he calls the Kung Fu Packing
Crate Maze. In it, you can only walk on crates, and crates
can only be toppled once. This simple set of rules produces
surprisingly complex mazes in a small space.
If you liked the easy pieces puzzle at
Google, Patrick Hamlyn offers a similar puzzle with piece set LLVZ --
make 5 2x2 squares with these pieces. You can try out the US Puzzle
Championship test at the wpc.puzzles.com site.
I particulary like the Spellbound puzzle. Numbers in a number
sequence are replaced by their initial letters. For example: T T
F S E T S N T T T T F F F F F or T S E F T O F T O S T S E F T O.
Rudy Rucker had a class based on A New Kind of Science, and the class produced a number of very nice applets relating to turmites, CA-generated sounds, and other simple program goodness. Next month at this time, I'll be out in Boston for the NKS Conference. For an interesting natural simple program, eat some Brocolli Romanesco.
Two of the more interestingly tiled
bathrooms in this room belong to Alex Feldman (Penrose style)
and Bob Jenkins (Hirschhorn style).
The world's biggest cube is apparently the Atomium,
which I suppose would have square tilings in its bathrooms.
I've recently become interested in Semiprimes,
numbers which are the product of two primes. For example,
14029308060317546154181 × 37280713718589679646221 = 38! + 1 is a
semiprime. I know 10^66 +3 is a semiprime. Each of the 30 numbers
13298267 + 1887270 k, k=0..29, is a
semiprime. For actual primes, you can download Proth
and look for big primes. It found 222×2003100+1
is prime. It seems to be much harder to find interesting
semiprimes. For example, are 10100+37 and 10100+39
semiprimes? 211673 could be called a triprime, as could the next
Juha Saukkola wonders about the greatest
number of knight or queens that can be placed on a board so that each
piece attacks exactly 1, 2, 3, or 4 others. Samples for 3 and 4
are below ... can they be expanded? Another interesting piece is the
Pythag, which can move a distance of 5. A similar problem is
being investigated on the Mathpuzzle Board
added 29 May 2003
added 26 May 2003
Take the word PREDICATE. Add the
full name of an actor, and rearrange the letters to get the full title
of a movie that actor starred in. Although he did not play the
title role in the movie, he has since played the title role in a TV
show. Interestingly, the word NOMINATIVE can be a clue for the
movie. Who is the actor, and what is the movie? Answer and Solvers.
Puzzles at clickmazes.com are well worth a look.
Peter Esser explored the posibilities of using
one piece twice with hexatans.
added 21 May 2003
Tony Delgado started me on a much better
exploration of Interactive Fiction. I learns of such things as
the Interactive Fiction Archive,Baf's Guide, Baf's top games, the SPAG newsletter,Brass Lantern, XYZZY news, The Best of IF,Windows
Frotz (for Windows), Frobnitz
Palm OS), Emulators for
other systems, and Inform
(for creating IF). I have a few dozen Z5 games on my Palm, now,
for odd moments. If I still had my old Infocom games, I could
them on my Palm device, now.
The 2003 Colorado Math
Olympiad had a nice problem by Alexander Kovaldzhi called the Map Coloring Game. "The explorer and
the mapmaker are taking turns in a map coloring game. At every turn the
explorer draws a new contiguous country on the plane with no inside
points in common with the previously drawn countries. The mapmaker then
colors the new country so that no two countries of the same color share
a boundary line. (They are allowed to share one or even finitely many
points.) The explorer wins if he forces the mapmaker to use at least 5
colors; otherwise the mapmaker wins. Find a strategy that allows one
player to win regardless of how the other plays. The explorer,
naturally, goes first." My best strategy for winning against 5
colors needs 9 regions. Can anyone beat that? Answer. How many regions are needed
to beat 6 colors?
George Hart has started a limited run of elaborate
polyhedra in his Acrylic
Series. For something much bigger, you can ponder Bathsheba
Grossman's wonderfully titled Large Scale
Model. The latter is a 3D crystal compilation of the 150,000
galazies and stars in the Sloane
Sky Survey. For much smaller images, you can look at Earth Photographs from
Water seems to flow uphill, in this Escher-inspired garden.
The complete works of Albert
Einstein is now available. I was amused by Popular Science's write-up of
the real-life puzzles enshrined within Dollhouses
of Death. From Science News, I learned of the Stone
Circles of Kvadehuksletta.
Material added 12 May 2003
1000 years ago today, the last
Pope-mathematician died. Gerbert of Aurillac was a great scholar
of the time, and was elected to be Pope Sylvester II in 999. In
Reims, he transformed the floor of the cathedral into a giant
abacus. He was the chief person responsible for the adoption of
arabic numerals (1 2 3 4 5 6 7 8 9), and invented the pendulum
clock. He died on May 12, 1003, and was succeeded by a long line
of non-mathematician popes.
In the game of Hex, players struggle to
connect opposites sides of a board made out of hexes. The game
doesn't work well on a square grid, because deadlock is too easy.
game Akron cleverly gets around this problem -- players may stack
their balls atop each other, making pyramid structures. Only the
uppermost paths matter in Akron. The Akron
site include a free windows program that plays the game -- very
beautiful. Cameron also offers the three player game Triad,
and the electrical charge game Py
No-one commented on my Numbered Boxes
puzzle from last week, so I'll leave that open.
The Chicago Area Mensa group has
interesting weekly puzzles, such
as this one.
For even more games, ponder getting 100
Strategic Games for Pen and Paper by Walter Joris. (Amazon
the wrong author.) Just as the title suggests, the book contains 100
new, simple strategy games, and I found all of them interesting.
There is a process with games. 1. Learn the rules.
2. Look for winning strategies 3. Look for easy
defenses. This is a book that deserves a lot of analysis.
Many beautiful pictures can be seen at the
official site for Indra's
Here's a rolling slab maze I created for
the US Puzzle Championship (but
too long). A 1x2x3 slab must be rolled from the SSS squares to
GGG squares (or vice-versa). The O squares are obstacles that the
slab cannot land upon. Answer.
Material added 6 May 2003
One good learning experience for me was
trying to fold a sheet of paper in half 7 times. I managed to do
it with a large sheet of newspaper. Britney Gallivan managed to fold a sheet of paper in
half twelve times. She also computed the folding limits of
arbitrary sheets of paper, and folded a sheet of gold 12 times.
At Math Magic, Erich Friedman is taking up extensions of Serhiy Grabarchuk's Matchstick Snake problem. I already have several new discoveries. Here are two by Dave Langers, who found the optimal solutions for a diameter 3 circle and a 2x3 rectangle.
Weistein's MathWorld has been nominated for a Webby Award in the
Science category. There are two awards given in each category,
with the "People's Choice Award" awarded on the basis of popular
can vote to support a math site.
David Wilson saw
Dots and Boxes variant in a children's magazine. Normally, each
square is worth 1 point, but in the variant, one corner square was
2 points. Other squares could be given other values. I'll
call this variant Numbered Boxes.
Who wins the smaller game? What if the numbered boxes were each worth
-2 points? Answeer.
NJA Sloane has made a page of
sequence puzzles available.
At Puzzlebeast.com, there is a new kind of
sliding block puzzle by M. Oskar van Deventer.
What is the expected outcome of ±1
± 1/2 ± 1/3 ± 1/4 ± 1/5 ± ... ?
Schmuland wrote a
paper about it for the American
Mathematical Monthly 110, 407-416 (May 2003).
is a compilation of information about all the various unusual formats
seen in a variety of different programs. One format I've been
studying recently is the SVG format. An excellent free graphics
program that uses SVG is the Sodipodi
program at Sourceforge.
Puzzles is the leading supplier of logic puzzles to
publishers. They specialize in Paint by Number puzzles.
added 28 April 2003
The National Public Radio
puzzle of the week is partly mine. Take the word
You can rearrange the letters into three 3-letter words: LAD, EGO, TEN.
If you set these one under the other, you'll have what's called a
double-word square, with LET, AGE, DON reading vertically. Now, take
16 letters of THE CONVERSATIONS, the title of a book by Michael
Ondaatje, who also wrote "The English Patient." Rearrange them into a 4
X 4 double-word square, using only common uncapitalized English
Brandon McPhail -- I'm a finishing
undergraduate student working on a math thesis in the area of puzzles,
specifically NP-completeness. After reading Erich
Friedman's proof that "Pearl" puzzles are NP-complete, I was
to create a small java applet to let one play and easily create pearl
puzzles. Erich seemed pleased and gave me your name and email
address, suggesting that I send you a link to the applet, that it may
interest you and be relevant to your website. So here it is.
Guenter Stertenbrink -- A picture of the 3 public Eternity-solutions is attached. I was able to use one of my old programs from 2000 to generate them, but I had to change it a bit. It calculates the center of mass of a piece and tries to put its piece-number there, provided it fits and doesn't intersect with neighbor-pieces. Looking at the picture, I see that in fact the 81 first pieces were common in my two solutions, although I only fixed 71. The next 10 did apparantly fit well and were hard pieces. Piece 191 in the upper right was also reused. Hmm, maybe I should fix it before starting ?! 3solu.gif or also at the MathPuzzle group.
Jens Lund -- Thanks for your interest. .
This is the first time I try to "distribute" a program, and it
took me some time to learn how the installer-programm works. I use
GP-install - a freeware program - (which has nothing to do with
PureBasic). Please observe, that the 500K file, you have just
downloaded, is this size - not because of PureBasic - but because
of the nature of the installer. The PB.exe is only about 30K. (This is
one of the advantages with PB). If I sound like some advertisement for
PureBasic, then please forgive me, but I'm just happy to have found -
finally - a language I think I can learn. I have nothing to do with
the promotion of PB :-) So - the first program you get is
the "Player", where you can move the balls with the arrow-keys, and try
to find solutions. The "Solver" is not quite finished yet. It does
but I have found some problems I would like to repair first. I
will send it as soon as I can. Meanwhile you can try this :
E-E-N-N-W-S-N-N-N-W-S-N-N-E !! (The original
Thunderball description can be seen in the 16
March entry. The player program can be found at the Mathpuzzle
Many lateral thinking puzzles can be found
at the Brain Food site.
But perhaps you'll like the story about the hermaphrodite
Jon K McLean found a unique 22.5-angle
snake of length 32. 04b4b4b4b5cf844009fb4b4b4b418c5c.
He confirmed that there are 3 45-angle snakes of length 15,
002244610636360, 002244670252520, 002346161643200. He also wrote
a solving program.
Material added 16 April 2003
has the interesting property that the product of the horizontal numbers is the same as the product of the vertical numbers: 264*200 = 52800 = 22 * 60 * 40. There are 22 different 2x3 matrices of this type, excluding those with leading 0's. Find the one that uses the digits 2, 3, 4, 5, 6, and 7. The Matrix Puzzle by Erich Friedman. Solved by Juha Saukkola, Ian Lai, Vahid Moradi, Marcis Petersons, Daniel Scher, Peter Exterkate, Zbigniew Zarzycki, James Lewis Melby, Lyman Hurd, Jeff Smith, Josh Snyder, Michael Veve, David Yamanishi, Deniz Sevki Kayabay, Al Stanger, Bryce Herdt, Carlos Gil Nieto, Robert Reid, Ved Dhyani, Lahousse Gustaaf, Loren Garrison, Darrel C Jones, Alastair Cuthbertson, and S Subramaniam.
Material added 8 April 2003
Susan Hoover found a 30-angle
20-snake. Comments Roger Phillips: "That's fantastic. I doubt
there's much more to find." I agree -- this is a stunning solution,
missed by everyone else. Serhiy Grabarchuk's Snake problem
definitely deserves a wider study. Susan: "It was very
liberating to discard the notion that the solution must start or end in
a corner. I started out with a length-19 snake that started in
corner, but then I saw some wasted space and tried a couple of "what if
I erase these two or three segments and bend it here instead?"
scenarios. Along the way, I found two distinctly different
length-19 snakes, plus a third length-19 snake that is identical to one
of the first two except for a section where there were three vertices
an equilateral triangle, where it uses a different two legs of the
Buss wrote a very nice tiny program for
arabesque patterns - just 181 bytes. It's a Window assembly
program. He also provided well-commented
Altay's excellent series of puzzle competitions continues with PQRST 5 on 12 April.
Puzzle Japan offers
pictorial logic puzzles of various sorts, and is free this month.
I've started a Yahoo Group for math
puzzles. Anyone is welcome to join, and to post, but I'll be
moderating the messages for appropriateness and newness. I might
start sending solution submissions for easier puzzles to
Let me know your thoughts.
It is easy to express 2004 as the sum of distinct positive numbers with
the same digits: 2004 = 725 + 752 + 527, 2004 = 617 +
671 + 716, 2004 = 509 + 590 + 905. It is hard
write 2003 as the sum of distinct positive numbers with the same
The answer appears to be unique. Answer.
HSM Coxeter has passed away at age 96. He was directly responsible for many aspects of today's recreational mathematics, including various works by MC Escher and Buckminster Fuller.
I like the sequence of graphs
which represent automorphism groups. It's just numbers, now,
but I think I'll make a picture. If you have Autocad and
Mathematica, you can look at my
methods for doing so. If you have Mathematica, you can look at and
experiment with the hyperbolic
plane -- the same object mentioned by Coxeter to Escher.
1.4a has been released. I absolutely love it. This site
is created in Mozilla composer. The tabbed browsing and pop-up
blocking are wonderful.
Bill Gosper took a look at Masonry tilings (tiling with no straight lines going all the way through) with 3x1 rectangles. He found only one non-symmetrical way to tile a 9x9 square as a Masonry tiling with 1x3 tiles. The smallest rectangle with total symmetry is the 11x15, with 2 solutions. For Mirror symmetry, the 9x11 is smallest, with 2 solutions. (No need to send solutions).
added 2 April 2003
Johnson, author of Fool's
Errand, put the First
Annual April Fool's Day Treasure Hunt Errand on his site yesterday.
Susan Hoover found a degree-30
20-snake! Clinton Weaver and Roger Phillips both found a
material added 1 April 2003
Older Material - 29 Oct 02 to 13 Dec 02
Older Material - 3 Sep 02 to 21 Oct 02
Older Material - 16 Jul 02 to 26 Aug 02
Older Material - 13 May 02 to 9 Jul 02
Older Material - 4 Mar 02 to 6 May 02
Older Material - 5 Dec 01 to 25 Feb 02
Older Material - 19 Aug 01
to 4 Dec 01
Older Material - 17 Feb 01 to 5 Aug 01
Older Material - 2 Jun 00 to 11 Feb 01
Previous Puzzles of the week are here.