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Math Games
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Material added 17 December 2006
When Chesspieces Attack
Bernd Rennhak reminded me of Erich Friedman's Attacking Chess Pieces Math Magic of the Month. Erich has had many great investigations lately, including Polyline compatibility, building trees, polyform addition, and triangulations. Bernd also did his own analysis.
W and H oddity Improvements
George Sicherman: Tonight I shrank the H pentahex oddity from 59 to 29 tiles and the W pentahex oddity from 149 to 41. I cheated a little--I used a computer!
Hofstadter Butterflies of Bilayer Graphene
A sheet of graphene can have fascinating patterns.
Expandible Circular Tables
Fletcher Capstan Tables have the amazing ability to double in size with a 3 second spin. The videos are well worth a look.
Christmas Letters
Jed Martinez: The ten four-lettered words below contain four different things associated with Christmas. The first three of these things can be found by taking one letter from each four-lettered word, and reading it from left to right. Circle one letter from each word, to spell out the first thing - this way, you won't make the mistake of reusing said letters, in order to determine what the other two things are. (NOTE: The first three things associated with Christmas each consists of two or three words. It is up to you to figure out how many letters there are in each word.)

When you have accomplished this task, there will be ten letters left (one from each word) without a circle. Rearrange these ten letters to spell out the fourth Christmas-related thing (which is a one-word answer). Good luck... and "Merry Christmas"!
Loop the Loop
David Millar: I have some sudoku-esque puzzles called 'Loop de Loop' that I am making for my site. I have a few posted at The Griddle and I also have them available through Zoho Sheet, which is a rather cool online spreadsheet application. I'm also branching out and trying other online spreadsheet utilities like Google Docs and Spreadsheets and EditGrid. It's quite interesting to see all of the options there are out there. [Ed - Quite fascinating.]
Material added 9 December 2006
Rectilinear Crossing Number Project
The Rectilinear Crossing Number Project has made a number of discoveries since 2000. They've resolved the K10 problem, then K11, and then all the way up to K17. K18 is still unresolved. They've also solved K19 and K21, shown below. Those red dots are pts = {{35, 67}, {18, 28}, {63, 50}, {27, 69}, {17, 23}, {87, 66}, {8, 79}, {18, 16}, {69, 53}, {5, 86}, {0, 0}, {94, 66}, {4, 88}, {5, 6}, {115, 69}, {9, 74}, {14, 13}, {108, 67}, {6, 85}, {2, 2}, {111, 68}}.
Google books vs Amazon books
For looking up info on obscure topics, I have often gone beyond Google to Amazon, which can find quotes inside books. Now, Google Books is available. Both are excellent sources for researching a topic. For "hexecontahedron", for example, Amazon gave 17 books, and Google Books 12 books.
Polycube solutions
Bob Harris: I've also uploaded a nice rendering of my favorite polycube puzzle-- the 10x10x10 filled with the 166 hexacubes (and a 2x2x1 void in the center of one face).
World's Smartest Person Contest
Matt Sheppeck: Thanks to you (and Ken Duisenberg) for posting that link to the 2006 World's Smartest Person Challenge. I took first place along with seven others, earning me $62.50 and a T-shirt. Maybe the WSP title gets me a discount on a cup of coffee somewhere? Eleven people were given a set of tie-breaking questions. When I saw that the tie-breaking questions were relatively easy, I considered submitting one wrong answer in the hopes of being the sole winner of the $200.00 second prize, rather than a small share of the $500 first prize. As it turned out, three people split the second prize, earning them $66.67 each. So who's smarter? Thanks again for your site, always a delight.
Ken Nordine Fibonacci
The video fibonacci numbers by Ken Nordine was nicely done. I've always been a big Ken Nordine fan, so I'm glad to see he's making videos.
Overview of Math
A nice overview of mathematics is given on page 2 of "Is the theory of everything merely the ultimate ensemble theory?" by Max Tegmark.
After mentioning spidrons last week, I was reminded by no-one less than Daniel Erdély that I should link to the main Spidron page,
Material added 13 November 2006
New Largest Probable Prime
Norm((1+I)^1127239+ 1)/5, or (2^1127239+2^563620+1)/5, has 339333 digits, which puts it at the top of the Probable Primes list. Norm((1+I)^n+1)/5 is proven prime for n = 5, 6, 7, 9, 11, 13, 17, 29, 43, 53, 89, 283, 557, 563, 613, 691, 1223, 2731, 5147, 5323, 5479, 9533, 10771, 11257, 11519, and 12583. For n=23081 , 36479, 52567, 52919, 125929, 221891, 235099, 305867, 311027, 333227, 365689, and 1127239, all the probable prime tests work, but no method is known for proving primality. Many congratulations to Borys Jaworski for setting this new record.
Andrew Clarke: Here is as a Sudoku type problem. In each 6x6 square all rows and columns contain 1-6 as does each hexomino. See also Numbered Polyiamonds and Numbered Polyominoes.
The Popular Science Periodic Table
Theo Grey's Periodic Table is now a gorgeous interactive screen at If you get the latest copy of Popular Science, the Dow ad features a beautiful fold-out poster of all the elements. One of these items is recently in the news -- polonium-210, the active ingredient in a camera brush.
Wordplay DVD
My longtime friend Will Shortz now has a DVD out: Wordplay. This film about intelligent puzzle people wound up being one of the best reviewed movies of the year.
Powerful vector support
Geometry Expressions, the most math-intensive vector drawing program, now interfaces directly with Mathematica.
Snakes on a Plane
The records for the matchstick snake contest are in. Vadim Trofimov is the big winner. I hope to show pictures of most of the solutions soon.
Fractal Maze
Martin Windischer: On I´ve read about the fractal mazes, but they don´t looked very attractive for me. If you insert the original maze in each of the boxes it will be too small in very few steps. So I tried to make another fractal maze with only one limit point. Here is the result. Send Answer.
Irfanview 3.99
My favorite free fast art editor, Irfanview, has just come out in an improved version.
The Lobster-Snake puzzle
George Sicherman: Here are two of the 12 hexiamonds, the Lobster and the Snake. Make two congruent shapes by adjoining the same hexiamond to both. This arose from Erich's latest Math Magic. Erich Friedman: Find a polyomino that can be joined with each of the two pictured hexominoes to get two congruent shapes. Send Answers. Colonel Sicherman also sent me a note that the T and X pentahexes are compatible.
Non-unique Town Placements
Given six road lengths between 4 towns, can the town placements be discerned? Turns out it's ambiguous. Can 5 points be done ambiguously? Can 4 points be set up so that all distances are integers? Send Answer.
Bimagic Square of Primes
Christian Boyer has made the first Bimagic square of primes. Under 701, only the primes 2, 3, 523, 641, and 677 are missing. It's a magic square that remains magic if all the terms are squared.
137 131 317  47   5 457 541 359 467 353 683
401 277 239 647  23 421 229 181   7 419 653
463 269 701  59 157 257 563 557 179 191 101
593 311 379 503 197  83  53 521 149 619  89
307 617 397 241 571 661 109 107  79 127 281
373 443  29 587 383  61  19 409 631 389 173
 73  11 607 433 613 577 263  97 227 313 283
 43 599 151 199 509 487 223 163 293 691 139
673  37 113 271 193  31 601 431 331 337 479
 67 233 103 439 499 251 547 659 491  41 167
367 569 461  71 347 211 349  13 643  17 449
Polyspidrons at Kadon
Jacques Ferroul's Polyspidrons are now available as a handsome set from Kadon Enterprises. Polyspidrons uses the famous Spidrons of Dániel Erdély. Ivars Petersen wrote an article about the Spidrons, Swirling Seas. I've also made Ivars a permanent link, in the upper right.
120-cell Sculpture
Ivars also wrote about a recently dedicated 120-cell scupture. Here's a picture from that event, showing the 120-cell trapping John Conway's arm in the 4th dimension. It was part of a dedication for Coxeter, whose biography, King of Infinite Space, recently came out.
Jenn 3D
Poor John Conway... when he tried to blow some bubbles after the dedication, they looked like the below. Actually, this figure is from Jenn3d, an excellent free program for playing around with higher polytopes - beautiful stuff.
Lorenz and Modular Flows
A very tubular and very beautiful set of animations is available at Lorenz and Modular Flows at, in an article by Etienne Ghys.
Material added 29 October 2006
Interview with Martin Gardner
Colm Mulcahy's latest Card Colm column is lengthy interview with Martin Gardner, in honor of Martin's 92nd birthday.
Paul Halmos passes
Paul Halmos, author of many books, including I Want to be a Mathematician, passed away on October 2 at age 90. Kirk Bresniker: He was one of the fastest walkers I'd ever seen, and could always be seen zipping around the campus at SCU. Several years ago he was walking during the winter and slipped and detached a tendon in both legs. Since he could not walk as quickly during his recuperation, he calculated the number of calories he was no longer burning and deducted that amount from his daily intake. That's discipline!
More highlights from Games magazine
Robert Abbott: I got my Starry Night program working on the GAMES web site, and I just put a notice about it on my home page.

Serhiy Grabarchuk: Just to add to your note on the latest GAMES Magazine, I though you'd be interested to read a big article of mine among the issue's features. It's named Playing with Puzzle Classics, and contains a lot of interesting puzzle materials I've never published before. [Indeed, great article.]
Making Salt the Hard Way
My friend Theo Gray wanted to salt some popcorn, so he filled a net with popcorn, put it over a bowl of molten sodium, then started bubbling chlorine gas through it. For awhile, everything was working fine, but then a piece of popcorn fell out from the net, starting a water-sodium reaction and explosion. As more heat was generated, the net holding the popcorn started to melt, and then chaos ensued. Theo had the wherewithal to turn off the chlorine gas in the midst of it all. The video and pictures are spectacular.
Material added 21 October 2006
Games 100
The latest Games Magazine is out, with the Games 100 list of top games. Robert Abbott has a devious logic maze on the cover. I was very pleased to see that the Thinkfun Gordian's Knot was the puzzle of the year - congrats to Frans de Vreugd and George Miller for designing it. Gordian's Knot recently showed up at the end of Numb3rs episode The Mole.

Even more amazing -- the Hoffman-Singleton game was selected as one of the 100 top games of the year. I thought this elegant item from mathematics would make a great card game, and many playtesters agreed with me. Column about the game.
Hexed Chess Robots
Erich Friedman: I've added a number of new puzzle types to Puzzle Palace: Hex turn, Chess attack, Robot mazes, Color strip puzzles, Chess mazes, Unequal length puzzles, Color mazes, and Left right mazes. (wowsers.)
Math Pick-up Lines, and N E W S
Alan O'Donnell: I thought you might like some of these 'Math pick-up lines' - no warranty provided or implied! eg: My love for you is a monotonically increasing unbounded function, Let's take each other to the limit to see if we converge, and I wish I were a derivative so I could lie tangent to your curves.

I've always noticed that American urban architects like their roads to travel N/S and E/W, but did you know they sometimes they get a bit anal about this? (An amazing road design) That image made me shake my head slowly in disbelieving astonishment - thought it might amuse you too...
Peter Hugo McClure - Math Artist
Peter McClure: Greetings... I am independent artist and maths/geometry/puzzles have been my greatest inspiration see my web-site at: [Ed - 129 fascinating thumbnails of math art, right off the bat]
Fractal Forum
There are forums on many topics... one I just learned about is the Fractal Forum. What other good forums are out there?
Pseudoku and Sudorku
Timothy Y. Chow: I've made several new sudoku variants based on objects other than Latin Squares. The first variant is the closest to ordinary Sudoku, and is the one I like best, because it is closely related to an unproved mathematical conjecture of mine, as mentioned in the endnotes. If you have any suggestions about how I could spread the word about this variant, or encourage other puzzle creators to make other instances of it, I would be interested to hear them. [Also very cute is a recent Foxtrot comic. Another puzzle is Hard Corners, by Joseph White]
World Puzzle Championship Blog
The final results are in for the World Puzzle Championship. The WPF site has more info. Wei-Hwa Huang (link to his excellent puzzle gadgets) came in second. Thomas Snyder (4th place) wrote a WPC blog item about the event.
Repeated squares, and the Smartest Person Challenge
Ken Duisenberg's latest Puzzle of the Week, Unique 2x2 Squares, is quite nice. He has a link to the 2006 World's Smartest Person Challenge, put on by the High IQ Society. A nice set of puzzles.
Material added 11 October 2006
Solving Techniques
If some of the puzzles at the WPC seem difficult, Cihan Altay offers an essay of various solving techniques (PDF).
Magnetic Tetris
If you'd like to cover your refrigerator with Tetris pieces, has them. Just 278 rubles per set.
Patterns in Ramanujan Tau
Simon Plouffe's homepage has hundreds of fascinating patterns. The lastest patterns explore the Ramanujan Tau function.
W-pentahex Oddity
Col. George Sicherman: I finally found a full-symmetry oddity for the W pentahex!
Emma Lehmer's 100th birthday
Emma Lehmer, a famous mathematician, will soon be celebrating her 100th birthday. People are invited to send best wishes to her mailbox, 1180 Miller Avenue, Berkeley CA 94708.
Magic Square of Cubes
Christian Boyer: Nobody knows a 4x4 magic square of cubes, using distinct positive integers. But very interesting step with the first known 4x4 semi-magic squares of cubes (semi-magic means non-magic diagonals), by Lee Morgenstern, USA. Who will construct a 4x4 magic square of cubes, with 2 magic diagonals? [Much more news is given at]
  16^3  20^3  18^3 192^3
 180^3  81^3  90^3  15^3
 108^3 135^3 150^3   9^3
   2^3 160^3 144^3  24^3
Halloween Themed Squiggly Sudoku
Bob Harris: I've added a new batch 43 puzzles with halloween related clues. Some are pretty easy (the first five might be nice for a children's halloween party). The ones at the bottom of the page are probably hard.
Color Mazes
Erich Friedman: A new type of puzzle for you: Color mazes. Find a path from Start to Finish (moving only horizontally and vertically and never passing through a square more than once) so that you pass through each (non-white) color an equal number of times.
The Tangent Circle Problem
Dick Hess: I ran across this one recently in the Pi Mu Epsilon Journal and wondered if you'd seen it before. Four planar circles are pairwise externally tangent. Three of them are also tangent to a line, L. If the fourth circle has unit radius what is the distance of its's center from the line? It's amazing to me the distance in question doesn't depend on the relative sizes of the other three circles. Send Answer.
Me and Numb3rs
The local Gazette-Telegraph wrote an article about Wolfram Research and Numb3rs. The Numb3rs season 2 DVD ($38) just came out, and it's loaded with great stuff. Near the start of episode Dark Matter, Andy Black and Cheryl Heuton graciously mention a story about me on the commentary track. You can still get Numb3rs Season 1 ($40), which curiously has fewer episodes at a higher price. If you prefer digital versions, Numb3rs Amazon Digital offers individual episodes for $2 each.
Material added 30 September 2006
Asymmetrical Looney Gear
M. Oskar van Deventer: A few months ago I challenged people to design four gears fitting inside a fifth gear. Andreas Röver took up the challenge. The sketch below and this gear animation shows some of Andreas' results. Andreas realised quickly that it would be hard or even impossible to find a solution that would result in an exact fit. In an exact solutions, the gears would fit perfectly and mesh perfectly. Instead, Andreas developped an algorithm that searches for approximate solution. Andreas found several criteria which a solution would satisfy, like a maximum mis-mesh of x%, gears not having common denominators and gears fitting "nicely" together. Finally, Andreas performed a quasi-random search for good-enough solutions. The resulting gear arrangements look quite nice. The Looney Gear animation demonstrates how they can rotate as a set of planetary gears, albeit quite asymmetrical. Two questions remain. 1) Do exact solutions exist, or only appromiate ones? 2) Do these Looney Gears have any useful applications? [Ed - The animation is fascinating. Other gears can be found at the Kinematic Models for Design library]
Pentagon Rule broken
A buckyball molecule has 12 pentagons and some number of hexagons. This can be proven with Euler's formula, V+F-E = 2 (vertices, faces, edges). Another buckyball rule has been that the pentagons don't touch, but a new buckyball built around triterbium nitride violates this rule.
Sudoku Variations
Uwe Wiedemann: Because it was your page where I found at first a collection of sudoku variants I will point you to my own sudoku variant page: There are well known variants and less known, and also some which I've created. Most of the puzzles aren't easy. [An excellent collection of variants]
Numb3rs Puzzle
Texas Instruments is continuing their excellent activities for the Numb3rs TV show, with season 2 and season 3 now available. One of them is mine, the missing city exercise. From the last update, I mentioned Colby, Kansas and Milford, Utah -- most of the cities in the route have huge crop circles visible in their overhead maps, a feature spotted only by Walter Hoppe.
Math Factor on iTunes
Chaim Goodman-Strauss: I am writing to advertise the Math Factor, a short weekly segment that airs on our local NPR affiliate, KUAF 91.3, Fayetteville Ark., and then is podcast around the world. We've been having a great time-- last week we had Jeff Weeks explain the Poincaré conjecture! We've discussed cardinality, encryption, paradoxes, puzzles, rates of change and much more. For a show about math, it is surprisingly well received by our general audience. To find it, simply search Math Factor on iTunes, or the podcast directory of your choice, or enter the Math Factor url into your podcast browser.
Perfect Sequential Rectangles
Brian Trial: Your prizes page lists a $100 bounty for the first perfect rectangle consisting of sequentially sized squares 1..n. For now the money is quite safe, but how close can one get? For a given n, what is the highest percentage of the squares 1..n one can use to make a perfect rectangle? The smallest perfect rectangle, 33x32, uses 9 squares in the 1..18 range, or 50%. lists a 60x84 rectangle that uses 17 out of 1..33, or 51.5%. It appears that rectangles that do better than 50% are uncommon, and I've only found 5 that do better than 60%, for n=30,32,33,36, and 41. My best so far, 66.6% and 65.9%, are attached. Pascal Huybers has independently found the 80x89 66.6% solution. For larger values of n, would the percentages increase, or do they plateau at some value of n? [Ed - Is the rectangle below the best possible?]
Material added 17 September 2006
Ultimate Periodic Table
A long-term local project is the ultimate periodic table by Theodore Gray, which I've mentioned here many times. At considerable expense, Theo has now made a spectacular illustrated periodic table poster, along with photographs of every element. Many sizes are available at reasonable prices. The large scale views are amazing.
New Mersenne Prime
A new biggest prime exists. 232,582,657 - 1. See the story on MathWorld.
Coxeter: The Man Who Saved Geometry
A delightful book has just been released by Siobhan Roberts: The King of Infinite Space: Donald Coxeter: the Man Who Saved Geometry. In my latest column, I mention the love of geometry that Washington and Jefferson had. At the turn of the century, geometry was on it's way out. Largely by himself, Coxeter revived the entire field. A great book, with lots of wonderful math along the way.
Lightforce Games
Jean-Charles Meyrignac: I just discovered Lightforce. There are a lot of nice puzzles on this site.
Numb3rs Season Premiere
The first episode of the third season starts Friday, 22 September. Among other things, I was given the task of picking out data points across the United States as a part of the plot. Here are two of them: Colby, Kansas and Milford, Utah. Can you figure out what trait many of the data points have in common? Lots of cool stuff is coming up in the next few episodes.
Replacement Puzzles
Erich Friedman: I've made a number of replacement puzzles. For example, Rule #1: {2,1,2}-->{1,1}, Rule #2: {2}-->{1,2,2}, Rule #3: {1,1}-->{2} QUESTION Using only these three replacement rules, one at a time, on some consecutive substring, get from {2,2,2} to {1,2,1,1,1} in 13 moves. You never need a string longer than 6 digits.
Mathematical Equivoque
Ken Suman: I have seen on various sites that you are interested in crosswords and wordplay as well as mathematics. So am I. I have been building a collection of mathematical wordplays that I hope you might find interesting.
Coordinates of the Harborth Graph
Eberhard Gerbracht: I have prepared a much deeper analysis of the Harborth Graph. I put some more thought to the problem and finally (with the help of MATHEMATICA) was able to produce minimal polynomials of degree 22 which completely describe the coordinates of the vertices of the Harborth graph. Thus now the problem is not horrifying anymore, but has become quite nice (at least in my opinion). Thus I found it worthy of a more thorough exposition.
Non-crossing knight tour
Alexander Fischer: Hello dear friends of mathematical recreation. Here is another improvement concerning the longest noncrossing leaper tour on a 14*14 chess board, it's of 135 steps length. By board extension some other records can possibly be improved. [Alexander also found 163 moves on the 16x16.]
Melbourne, City of Math
My latest column is about the spectacular math architecture in Melbourne. Christian Boyer pointed out some other unusual buildings. Also, A-R-M has more math buildings in the works.
Rolling Block Ramp
Cihan Altay: A red block, with a length of 39 units, is standing next to another block which supports a ramp that doesn't move. Roll the red block around the board, without exceeding the rectangular border, so that it ends up standing upright on the supporting block. A move is to roll the red block using one of its edges (touching the ground) as the axis of rotation. There is a chasm on the board on which the red block can not stand upright, but it may lie down. What is the minimum number of moves to achieve the goal, following these rules? [A simple solution is flop the block S, then to roll W a lot, N at the proper spot.... but that is not the shortest path.]
Material added 27 August 2006
New Mersenne Prime
Robert Bosch: This summer, I finally had the time to put together a page on my "TSP Art". It has samples of my recent pieces, a link to a second page describing how I do what I do, and links to related pages by Craig Kaplan and Mo. [The below is a simple loop, not a knot.]
Anisohedral Tiling Database
Joseph S. Myers has compiled a tiling database of interesting tilings possible with polyominoes, polyiamonds, and polyhexes.
Erich Friedman: I've made a page of Polyomino Addition Problems. The first two are surprising! They might make good puzzles for your readers.
Sudoku Variations
A large variety of new sudoku variations is on display at
New Hanayama Puzzles
Various new metal puzzles have been minted by Hanayama, and are available at Puzzle Master. All of them are exquisitely made, and feature only 2 to 4 pieces. Some of the newer puzzles are Oskar's Cuby puzzle, and Serhiy's Coaster puzzle.
Survo Puzzles
Seppo Mustonen: During the past 45 years I have made statistical software, one of them being the SURVO MM (Survo) package, which is a general environment for statistical computing and related areas. As an application of some special computational features (especially those related to combinatorics) of Survo I got an idea of cross sum puzzles having something of the flavour of Kakuro but with unique and mathematically challenging features of their own. I call them Survo cross sum puzzles or Survo puzzles (link to PDF). As an example, arrange the numbers 1 to 12 in the grid below so that the rows and columns sum to the given numbers (problem 10 on page 24). [The survo website has been further updated.]
  21 10 18 29
Poincare Conjecture Proved
Back in 2003, Perelman proved the Poincare conjecture. After several years of study, his proof has been accepted. An excellent report on the full story is given in The New Yorker story, Manifold Destiny. Another Fields medal winner is Terence Tao.
Clickmazes Update
There are new logical mazes available at Andrea Gilbert's Oskar: When writing the challenge generator for the Active Maze, I discovered that I could not use a genetic algortithm, as even changinging the color of only one square would change the entire maze. I wonder whether there would be any intelligent approach to such generator, other than brute-force search.
Combinatorial Object Server
Many combinatorial objects are explained at the Combinatorial Object Server.
Davis Megamaze
The Davis Megamaze (in Sterling, Mass.) is a huge cornfield maze designed by Adrian Fisher. A feature of the maze, in addition to the many bridges, is a series of gates which allow for 24 different mazes. The honor system prevents a person from passing through various gates. Oskar sent a photo. Aerial shot.
Site Goals
Martin Gardner celebrates math puzzles and Mathematical Recreations. This site aims to do the same. If you've made a good, new math puzzle, send it to My mail address is Ed Pegg Jr, 1607 Park Haven, Champaign, IL 61820. You can join my moderated recreational mathematics email list at