Fair Dice, Iamonds,Eternity, Mazes, Atom Stability, Dissections, Kites&Bricks,Triangles, Retrograde, Pentagons, Prize Problems, BOOKS, and Partridges. Math Resources, Puzzle Resources, News, Search Engines, References

Older material below.  One ongoing note ... you can always subscribe to my moderated mailing list at http://groups.yahoo.com/group/mathpuzzle/

This week, I'm at the NKS2003 conference.  You can see the Wolfram Atlas of Simple Programs, or obtain a free trial 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)

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 Solvers.

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 murdEr On tHe orIeNt eXPress.):  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 * 69609259115904932249803147029051381732497164971097 (Hans Havermann).  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 given composite 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 of 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 some ice 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.

I did the the National Public Radio puzzle of the week this week.

Weimholt.com 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.

Jean-Charles Meyrignac did a marvellous analysis on Knights attacking other knights.  He includes his searching code.  Many of these patterns I've never seen before.  The below is a summary of his findings.

You can place
20 knights, each attacks 3 others, on a 7x7 board.
32 knights, each attacks 0 others, on a 8x8 board.
32 knights, each attacks 1 others, on a 8x8 board.
32 knights, each attacks 2 others, on a 8x8 board. (single path possible)
40 knights,
each attacks 2 others, on a 9x9 board.
48 knights, each attacks 2 others, on a 10x10 board.
32 knights, each attacks 3 others, on a 8x8 board. (the solution is unique).
36 knights, each attacks 3 others, on a 10x10 board.
48 knights, each attacks 3 others, on a 9x9 board.
16 knights, each attacks 4 others, on a 7x7 board.
36 knights, each attacks 4 others, on a 11x11 board
68 knights, each attacks 4 others, on a 12x12 board
80 knights, each attacks 4 others, on a 13x13 board
(10x10 appears to have also the same solution).

Elvis lives.  I loved an analysis of Calculus in a recent College Mathematics Journal, and Ivars Peterson did as well.  Woof.

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 6 numbers.

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 at yahoogroups.

The 2003 Google U.S. Puzzle Championship will be held online on Saturday, May 31, starting at 1pm EDT (GMT-4). The deadline for registering is today, 29 May 2003, 9PM EDT.  The top U.S. solver will be crowned the U.S. Puzzle Champion; and the top two will qualify for the U.S. Puzzle Team and participate in the World Puzzle Championship in the Netherlands in October. You can also see the Google page for this.  Solve Bob Wainwright's marvelous little puzzle there. Amazingly, Google has a link from their Front Page

There is an extensive review of The Mathematical Explorer posted on the MAA website.  Reviewer Marv Schaefer: "There is a tremendous variety of good mathematics in the Mathematical Explorer, and users are certain to find exciting mathematical concepts, insights and challenges aplenty."  I'd love to see some TME notebooks.  If you've made some good ones, send them to me.

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.

The Wriggle Puzzles at clickmazes.com are well worth a look.

Peter Esser explored the posibilities of using one piece twice with hexatans.

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 (for 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 play 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 Digital Sky Survey. For much smaller images, you can look at Earth Photographs from Space.

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.

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.  The new 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 (potential Y).

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 has 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 Pearls.

Here's a rolling slab maze I created for the US Puzzle Championship (but was too long).  A 1x2x3 slab must be rolled from the SSS squares to the GGG squares (or vice-versa).  The O squares are obstacles that the slab cannot land upon.  Answer.

`XXXXXXXXXXXXXSSSXXXOXGGGXXXXXXXXXXXOXXXXXOXXXXXXXXXXXX`

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.

Eric 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 vote.  You can vote to support a math site.

David Wilson saw a Dots and Boxes variant in a children's magazine.  Normally, each square is worth 1 point, but in the variant, one corner square was worth 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 ± ... ? Byron Schmuland wrote a paper about it for the American Mathematical Monthly 110, 407-416 (May 2003).

Wotsit.org 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.

Conceptis Puzzles is the leading supplier of logic puzzles to publishers.  They specialize in Paint by Number puzzles.

The National Public Radio puzzle of the week is partly mine.  Take the word ELONGATED. 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 the 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 words.  Answer.

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 inspired 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 work, 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 Group.)

Many lateral thinking puzzles can be found at the Brain Food site.  But perhaps you'll like the story about the hermaphrodite stowaway.

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.

264
200

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.

Oyvind Tafjord has found a second snake of length-20.  There are 64 length-19 snakes.  List and snaky comments.

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 one 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 in an equilateral triangle, where it uses a different two legs of the triangle."

Frank Buss wrote a very nice tiny program for arabesque patterns - just 181 bytes.  It's a Window assembly program.  He also provided well-commented source code.

Cihan 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 mathpuzzle@yahoogroups.com.  Let me know your thoughts.

Erich Friedman:  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 to write 2003 as the sum of distinct positive numbers with the same digits. 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.

Mozilla 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).

Cliff 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 clock-19-snakes.

Up in the International Space Station, astronaut Don Pettit wanted to experiment with soap bubbles, but tried the experiment with normal water first.  Here, you can read what happened. This surprising result involving tension reminded me of my own recent experiments with Tensegrity.  For example, you can make a structure with 3 pens, 6 paperclips, and 9 rubber bands. Or 15 straws, 30 rubber bands, and 60 paper clips. You can even make a structure with 6 struts, and elastic shockcord.  My favorite structure involving tension is Nova Plexus by Geoff Wyvill. Only 23 were made -- each machined from 12 steel rods, with micron-accurate hyperbolic divots. George Hart replicated the feat with pencils,and made the fragile structure below (Nova Plexus is very sturdy).  I made one myself, but used 12 small rubber bands instead of making divots.  That turns out to be very stable.  If you have 12 pencils and 12 rubber bands, I urge you to try making one.

If you'd like to join an online event of word puzzles, take a look at Intercoastal Altercations

Shade in triangles in the figure below so that every vertex touches the corner of exactly one shaded triangle.  It is related to the 3-bones problem.  Toby Gottfried wrote the problem as a very nice applet, Tri-shade. Solved by Porter TwoThreeFive, Clinton Weaver, Darrel C Jones, Toby Gottfried,  Marcis Petersons, David Molnar, Matt Elder, Scott Purdy,

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