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Math Games
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Material added 23 December 2010
Vi Hart Vi Deos
Vi Hart's videos have been the most popular math item in months, even showing up on cable news shows. My favorites are Doodling Snakes and Binary Trees. Her father is the polyhedreally-famous George Hart, who is currently the chief advisor for the Museum of Mathematics (which recently started selling the Tetraxis).
Odd Dice
Speaking of polyhedra, various odd dice are available at Shapeways. Click of a picture for the item page.
Wolfram|Alpha Spikey
A couple years back I designed a cutout for W|A Spikey, and a lot of people have now built and photographed Spikeys.
City Mazes
Robert Abbott: I just added something to my web site about City Mazes, and I thought you might want to say something about it. Mostly I talk about the use of walk-through Logic Mazes in cities, and I discuss three festivals that used them.  But something more important is going on here.  These festivals all try to show kids the fun side of mathematics and logic. One of the festivals was in Lima, Peru, and the organizers sent me some fantastic pictures, which are now on my site.  The pictures show mazes by me and by Andrea, and they also show tables where the kids are playing various small mathematical puzzles. I recognized a Tower of Hanoi puzzle, but I couldn’t recognize any of the other puzzles. [Ed- Gorgeous Pictures. And Robert is right, there is a lot of learning involved in solving one of these mazes.]
Remove the letters HKRBI from "I should have known better" to get an item in the news. Answer.
Erich Friedman's Holiday Puzzles
Erich has put together a new set of holiday puzzles.
Decorating the tree
Sudoku Variations
Since I wrote an article on Sudoku Variations, many have done a better job. The books Mutant Sudoku and Sudoku Masterpieces are the best books of the genre. The best sites for variants are Passion for Puzzles, sachsentext, WPC-Style Puzzles, Puzzlemix, Puzzle Picnic, and The Griddle. The most prolific composer of pencil&paper puzzle variants is Inaba Naoki (more english-friendly version)(I would love to get a copy of his book). Below is the first sudoku, by Howard Garns in 1979. Sudoku is popular with all life forms, since bacteria can solve sudoku.
Howard Garns First Sudoku
Programs I Use
Some of the free programs I use are Irfanview (most picture formats), LibreOffice (instead of OpenOffice), Golly (cellular automata), Blender (3D graphics), Burr Tools (puzzle analysis), Notepad++ (text editing), VideoLan (most video formats), TeXnicCenter (TeX), and Simon Tatham's Puzzles (lots of puzzle types). My favorite non-free programs are Mathematica, Adobe CS5, Geometry Expressions, and Crossword Compiler. On my non-free program wish list are Rhino 3D, Solidworks, Autocad 2011, and Maya.
Theo Gray in Japan
Theo Gray and Denjiro competed in Japan for Mad Scientist of the Year, and tied.
Morpion Solitaire
In August 2010, a new record was set in Morpion Solitaire.
Loyd's King Ptolemy Puzzle
Will Shortz recently used the King Ptolemy puzzle on NPR. How many triangles are in the below figure? John Kiltinen sent me a King Ptolemy Analysis.
King Ptolemy
Paperfolding morphisms, planefilling curves, and fractal tiles
Michel Dekking ( An interesting class of automatic sequences emerges from iterated paperfolding. The sequences generate curves in the plane with an almost periodic structure. We generalize the results obtained by Davis and Knuth on the self-avoiding and planefilling properties of these curves, giving simple geometric criteria for a complete classification. Finally, we show how the automatic structure of the sequences leads to self-similarity of the curves, which turns the planefilling curves in a scaling limit into fractal tiles. For some of these tiles we give a particularly simple formula for the Hausdorff dimension of their boundary.
Nimbers are inevitable
Julien Lemoine, Simon Viennot ( This article concerns the resolution of impartial combinatorial games, and in particular games that can be split in sums of independent positions. We prove that in order to compute the outcome of a sum of independent positions, it is always more efficient to compute separately the nimbers of each independent position than to develop directly the game tree of the sum. The concept of nimber is therefore inevitable to solve impartial games, even when we only try to determinate the winning or losing outcome of a starting position. We also describe algorithms to use nimbers efficiently and finally, we give a review of the results obtained on two impartial games: Sprouts and Cram.
Tron: Legacy Easter Egg
I wrote a Tron: Legacy blog, and listed the scientist names I recognized in the Disk game scene. I haven't been able to figure out some of the other names yet. I did like the movie, but I think it would have been better with more mathematics.
Tim Berners-Lee developed the World Wide Web.
Thomas Eugene Kurtz developed the BASIC programming language.
J. Presper Eckert developed ENIAC.
Peter J. Landin developed ALGOL.
Ed Logg developed Centipede and Gauntlet.
William Wulf developed the BLISS programming language.
John Backus developed FORTAN.
Alan Turing developed many of the initial concepts of computers.
Seymour Cray developed supercomputers.
How to Win at Coin Flipping
I also wrote a blog on the mathematics of Coin Flipping.
Richard Pendered and Frank Lewis, Codebreakers, pass away at 89 and 98.
Telegraph, via John Gowland: Richard Pendered, who has died aged 89, was one of the small team of Bletchley Park codebreakers who broke the Shark Enigma cipher used by German U-boats during the Second World War. Frank Lewis, also at Bletchley Park, died a few days earlier.
Material added 15 Dec 2010
Fusible Numbers
You are given several fuses, each of which burn for exactly one minute. The fuses burn don't burn uniformly, so you can't predict how much fuse will be left after (say) 15 seconds. What is the smallest interval of time over 0, 1, 2, 3, and 4 minutes that you can measure? Turns out that it is 0+2^(-1), 1+2^(-3), 2+2^(-10), 3+2^(-1541023937), and the number for 4 is really, really huge, probably between Skewes number and Graham's number. This is all given by the wonderful presentation Fusible Numbers by Jeff Erikson, given at the Gardner event at UIUC.
Fuse Solutions
Back Online
The weekend of Dec 4, Comcast was offline due to server issues. The weekend of Dec 11, I was offline again. I thought Comcast was at fault again, but it turned out to be that the old cable to my house had decayed to unusability, and that has been replaced by a helpful Comcast technician.
Material added 9 Dec 2010
Stained Glass Magic Square
John Hardisty: Love your website. I make is a stained glass window you might like. [Ed - Indeed, very lovely.]
Golly 2.2
The wonderful cellular automata program Golly has gotten an update to version 2.2. Bounded grids, turmites, and much more has been added.
Material added 7 December 2010
A slew of puzzle columns in todays NYTIMES=SCIENCE+SECTION. I haven't found a best way to link to all of them, but there are monuments, egyptian scrolls, construction, crossword moments, jigsaws, and more.
Material added 17 November 2010 now operational
Neil Sloane: there is a new version of the On-Line Encyclopedia of Integer Sequences (the OEIS). There are two parts: the main OEIS page,, which has all the sequence data, and a wiki,, which has discussion pages. This means that after 45 years, I no longer have to process every single contribution to the database. [Congrats, Neil. FWIW, I suggested to Neil that he buy years ago, and he did.]
Mathematica 8 Launched
Mathematica 8 has launched. I work at Wolfram Research, so my glowing praise is probably biased, but I love it. A Home edition for hobbyists is available for $295. You can peruse New in 8 to see some of the new features. Stephen Wolfram has written up multiple blog entries, including the Mathematica 8 launch, freeform linguistics, a sneak peek, and natural language programming.
Wolfram Mathematica 8
Global Math and Science Town Hall
Today, there is a global Math and Science Town Hall meeting, hosted by Al Gore and Mythbusters.
Material added 1 November 2010
A new Science-rich Website
Al Seckel - I have been working on, which covers many areas (quantum computing, perception, cognitive science, artificial intelligence, evolutionary biology, theoretical physics, theoretical mathematics, etc). Jeffrey Epstein is one of the largest private benefactors of science. We are also simultaneously working on his philanthropy site ( which will be taking applications for large grants in support of major in fundamental areas of scientific research and humanitarian causes. [Ed - A very worthy cause]
World Puzzle Championships
From Team USA topped the leader board for most of the first two days, but slipped to second place in the last round, giving up a small time advantage to a strong Japanese team. But on the final day, the Americans crushed the team relay round, and survived a no-notes physical Skyscraper Sudoku puzzle to win the team championship. Japan placed second, and Germany third. Americans Wei-Hwa Huang and Thomas Snyder, both qualified for the 8-person individual playoffs, but were eliminated in the first round. In the final round, Taro Arimatsu of Japan won first place, followed by Ulrich Voigt of Germany, Hideaki Jo of Japan, and Ko Okamoto of Japan. [Ed - A strong showing from Japan. The Nikoli books must be helping.]
World Puzzle Championship Style Puzzles
For lots of puzzles not based on language, trivia, or tricks, do check out
Material added 20 October 2010
Holeless Hexomino Compatibility
George Sicherman: Here I show minimal known hexomino compatibility figures without holes. If you find a smaller solution or solve an unsolved case, please let me know.
Benoit Mandlebrot
The trailerblazer for fractals, Benoit Mandelbrot, has died. My own first interaction with him was back when I was doing research for Piers Anthony, when he was writing the Mode series. Fractals were relatively new back then. As a part of the research, we asked Benoit for permission to use the Mandlebrot set. He replied "You probably don't need my perrmission, but I'm happy to grant it to you." Much later, Benoit borrowed a cylinder of uranium for a speech. A blog post I wrote should go live soon. I've looked at a lot of fractals -- I think my favorite is Trip to E214.
Worldwide Gatherings for Gardner - Thursday October 21
Martin Gardner's birthday, October 21, will have gatherings in over 50 locations worldwide. I've written a blog item about this Celebration of Mind. One of these events, the one I'll be at, is the Champaign-Urbana event - Four Colors Suffice and Other Math Wonderments. That Facebook event link gives our latest speaker schedule, I think we have 18 speakers at the moment. (Yes, Ed Pegg Jr is on Facebook.) If you'd like to attend the Champaign event, please write me.
Four Colors Suffice
Burr Tools .6
Version .6 of Burr Tools was released on 10/10/10. It's a fantastic free program for 3D puzzle analysis. For more detailed 3D work, I've been experimenting with Blender 2.54.
Penney's Game
In Penney's game,'s_game  , I have HHT, and you have HHH.  We start flipping a coin, and the winner is the first to get their sequence of heads and tails. In HHH beats HHT, the number of wins on turn 3, 4, 5, 6 ... form the Fibonacci sequence.  All of the below famous sequences are linked by Penney's game, which doesn't seem to be well known. I've also compiled all the 4 flip Penney sequences. Only one sequence in the 3 flip games was not in, so now that one, A171861, seems to be the Penney's game sequence.
1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597 HHH beats HHT A000045 - Fibonacci
1,1,1,2,4,6,9,15,25,40,64,104,169,273,441,714,1156 HHH beats HTH A006498
1,1,2,3,4,6,8,11,15,20,27,36,48,64,85,113,150 HHH beats HTT A023434 - Dying rabbits
1,1,2,3,4,6,9,13,19,28,41,60,88,129,189,277,406 HHH beats THT A000930
1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65 HHH beats TTH A000931 - Padovan
1,2,3,5,8,12,18,27,40,59,87,128,188,276,405,594,871 HHT beats HTH A077868
1,2,4,6,9,12,16,20,25,30,36,42,49,56,64,72,81 HHT beats HTT A002620 - Quarter squares
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 HHT beats THH A000012 - All 1's
1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32 HHT beats THT A004277
1,2,4,6,9,13,18,25,34,46,62,83,111,148,197,262,348 HHT beats TTT A171861 - Penney's Game
1,2,2,3,6,10,15,24,40,65,104,168,273,442,714,1155,1870 HTH beats HHH A070550
1,1,1,2,3,4,6,9,13,19,28,41,60,88,129,189,277 HTH beats HHT A000930
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17 HTH beats HTT A000027 - Natural numbers
1,2,2,3,5,7,10,15,22,32,47,69,101,148,217,318,466 HTH beats THH A097333
1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2 HTH beats TTH A040000
1,2,3,4,6,9,13,19,28,41,60,88,129,189,277,406,595 HTH beats TTT A068921 - Tatami mats
1,2,3,5,7,10,14,19,26,35,47,63,84,112,149,198,263 HTT beats HHH A054405
1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9 HTT beats HHT A008619
1,1,2,4,6,9,14,21,31,46,68,100,147,216,317,465,682 HTT beats THT A038718
1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584 HTT beats TTH A000045
1,2,4,6,10,16,26,42,68,110,178,288,466,754,1220,1974,3194 HTT beats TTT A128588
Material added 26 September 2010
Gardner Remembrance - Thursday, October 21
About 50 different locations will hold events in remembrance of Martin Gardner's 96's birthday. I'll be talking at a small event here in Champaign IL. Other people attending events at their localities are Jim Gardner, James Randi, Stan Isaacs/Don Knuth, Magic Castle, Card Colm, Bill Ritchie, Ivars Petersen, Les Barton, Hans Schepker, Jerry Ferrell, Doug McKenna, Al Seckel, Jordini, and many others.
Morpion Solitaire Record
A new world record in Morpion Solitaire has been set, beating the record set in 1976. Chris Rosin set the record.
Geometry Expressions 2.2
One program I find incredibly useful for the highly accurate mathematical diagrams I like to draw is Geometry Expressions. A new version just came out. This is the best available algebraic Geometry drawing program I know of at the moment.
Single tile non-periodicity
Joshua Socolar has updated his paper on single tile non-periodicity.
Paley 49
Here's a puzzle to try out. Each square has seven rows, columns, left broken diagonals, and right broken diagonals (broken diagonals wrap around). The eight lines containing the "1" have been given. Notice that the other numbers from 1-49 are also on these lines. In the finished diagram, every pair of numbers shares exactly one line. Finish filling in the second square. Send answer.
Paley 49
Jumping Maze on iPod
Todd Neller has put together a paper on jumping mazes (also called rook mazes). He has also released a Jump Maze Ap. Robert Abbott has a web version at Logic Mazes.
Sidenote on the iPod
I finally have an iPad Touch, and I am profoundly unimpressed. There are very few good aps available (Move it!, Vexed, Subway Shuffle, The Elements, iCut Deluxe), mixed among thousand of self-replicating and similar lousy aps. Heaven help you if you have two computers, and you try to add an ap from the "wrong" computer. I just lost hours of work. Also, iTunes has no tabs, so evaluating or even looking at lots of ap pages is many times slower than normal web evaluation. Jobs mentioned in his last address that the iPod is taking over the mobile games market. After looking over about fifty iPad games, their quality is for the most part terrible compared to Nintendo DS and Sony PSP games, both of which I've tried quite a few of. Nintendo and Sony games both have dozens of sites critiquing their every offering. There is nothing similarly comprehensive for iPod that I can find. Also, Apple seems to have no quality control at all on aps. Great device, but terrible DRM.
Amazon Kindle
Since I just had a rant on the iPod Touch, I might as well mention my experiences with Amazon Kindle. I like it a lot. The reading surface is very crisp. For one, it's fine with me having more than one computer. I also like the free WhisperNet a lot. During a town power failure last week, I had no trouble staying connected. It's fun to read outdoors. I like the text to speech feature - I've been listening to the works of Balzac, Baum, Chesterton, Conrad, Cooper, Crane, Dickens, Doyle, Dunsany, Dumas, Futrelle, Garrett, Haggard, Howard, Hugo, Kipling, Lang, London, Melville, Piper,Scott, Shaw, Shakespeare, Sinclair, Stevensen, Tolstoy, Twain, Verne, Wells, and Wilde. Most of these works are available as free books. One of the many free books I've liked is Shackleton's South. I like the Wikipedia-based Encyclopedia. It's all very relaxing for me. The browser is kinda lousy, but free. Pdfs look quite good on it, so it's worth looking at such PDFs as Euclid's Elements, Triangle Geometry, Lecture Notes on Graph Theory, Math Model for Mafia, or Lombardi Drawings of Graphs.
The Vegan Werewolf
Amazon gave me a very odd recommendation a few days ago.
Excellent Penrose Animations
Andreas Röver (burr tools): just found this animation explaining the penrose tiling. Quite nice...made only with povray.
Linux Wallpaper
I find this Linux Wallpaper quite useful on the Linux side of my system.
Was it a Cat I Saw?
At, Walter Tross programs a classic puzzle by Sam Loyd. I put a copy of his Cyclopedia of Puzzles into the public domain a few years ago.
Material added 6 September 2010
Ruler and Compass
Getting a few items at WalMart recently, I noticed a shopping cart full of pretty decent drafting kits, with a high quality compass, 30-60-90 and 45-45-90 triangles, protractors, straightedge, and a pretty good quality mechanical pencil -- marked down from $15 to $1. I found that kind of depressing, that basic drafting-by-hand seems to be an obsolete art now.
Sweet Clyde's Inversion Theory
Futurama writer Ken Keeler, who is also a PhD mathematician, developed Sweet Clyde's Inversion Theory for the episode Prisoner of Benda. They also put up a full board of the proof.
Tetrahedra Packing
There have been many results on tetrahedra packing lately, such as the .8547 packing density. I wrote up a blog column that includes the new best known result, a .8563 packing density, by Chen, Engel, and Glotzer.
Enumeration of Polyhedra
I've been using the table of polyhedra enumerations quite a bit recently.
Cyber Command Logo
The US Cyber Command Logo, part of the DOD, has a code within its design. The code has been cracked. There are many logos with hidden messages.
Shapeways and Spoonflower
Shapeways computer prints a variety of user-designed 3D objects. For 2D, Spoonflower prints user-designed patterns in fabrics. There is also LaserExact, which laser cuts various puzzles.
Visual Word Puzzle
NPR put together a Visual Word Puzzle.
H-Bombs in Space
In 1962, a hydrogen bomb was exploded in outer space.
Game of Life in HTML 5
Pure HTML 5 can run Conway's Game of Life.
Panel cracking and other fractals
The cracking of a painted panel greatly resembles a close-up of a leaf. There is also a 3D Fractal animation by Hömpörgő of incredible detail based on the Mandelbulb 3D set.
Material 19 August 2010
Fields Medals
NPR does a segment on the 2010 Fields Medals, with Julie Rehmeyer from Wired/Science News.
US Qualifier for the World Puzzle Championship
On Saturday, August 21, the US Qualifier for the World Puzzle Championship will be held. Good luck solving my puzzle, if you try it. The Practice test is available right now.
Penrose Staircase
Also on Wired, puzzlemaster1 Mike Selinker and puzzlemaster2 Eric Harshbarger put together the article Neverending Stories, refering to the movie Inception. Mike asked Warner Brothers for permission to use an image from the film, so they asked Chris Nolan for an image, and he went frame by frame through the scene to pick one for the article. Famous director assisting with a recreational math article -- wow.
Math Puzzle on Futurama
Tonight's episode of Futurama (Mind-Switcher) involved mind-swapping. After the first experiment, though, they found that once two bodies had swapped minds, they could never swap minds again. Then other swaps occurred. Puzzle: how many new bodies are required to get all the correct minds back into the correct bodies? A full solution is given right in the episode (the Futurama writers all have math degrees).
Material 9 August 2010 (an update to the update)
God's Number is 20
It takes 20 or fewer moves to solve any Rubik's cube position. With significant help from the GoogleNet, Morley Davidson, John Dethridge, Herbert Kociemba, and Tomas Rokicki prove that God's Number for the Cube is exactly 20. More details at
An article on OEIS
Julie Rehmeyer wrote the article The Pattern Collector, which talks about Neil Sloane and
Two from Arxiv
Counting Links and Knots in Complete Graphs by Loren Abrams, Blake Mellor
We investigate the minimal number of links and knots in complete partite graphs. We provide exact values or bounds on the minimal number of links for all complete partite graphs with all but 4 vertices in one partition, or with 9 vertices in total. In particular, we find that the minimal number of links for $K_{4,4,1}$ is 74. We also provide exact values or bounds on the minimal number of knots for all complete partite graphs with 8 vertices.
Circle Packing for Origami Design Is Hard by Erik D. Demaine, Sandor P. Fekete, Robert J. Lang
We show that deciding whether a given set of circles can be packed into a rectangle, an equilateral triangle, or a unit square are NP-hard problems, settling the complexity of these natural packing problems. On the positive side, we show that any set of circles of total area 1 can be packed into a square of size 8/pi=2.546... These results are motivated by problems arising in the context of origami design.
Material added 9 August 2010
Computer Update
This is my first try at using Adobe CS5 on a brand new computer. It's fairly easy to get a bona fide supercomputer for 1K-2K these days, so I've been doing a number of big runs on square packing. Part of the new system is two hard drives, one Windows 7 64-bit, the other Ubuntu Linux. For analyzing billions of planar graphs (with help from plantri), and general fun, I prefer the Linux system. It's definitely easier to find and install a lot of good free software on the Ubuntu system, just by selecting it. Windows (compared to Linux) definitely has the edge on running commercial software (like Adobe, Autodesk, and Crossword Compiler). On the other hand, I had to download a scad of software to make sure my Windows system stays clean (CCleaner, AVG, Spybot, Ad-Aware). FWIW, Mathematica runs just fine on either operating system.
"You have two minutes to make a maze I need a minute to solve."
That quote is from Inception, which I consider a must see for any visitors of this site. Here's another Christopher Nolan quote: "I always find myself gravitating to the analogy of a maze. Think of film noir and if you picture the story as a maze, you don't want to be hanging above the maze watching the characters make the wrong choices because it's frustrating. You actually want to be in the maze with them, making the turns at their side, that keeps it more exciting...I quite like to be in that maze." His logo for Syncopy Films is a maze. The Penrose stairs are explicitly mentioned.

Hofstadter, Godel, Escher Bach (discussing Escher's Concave and Convex, p. 106): "Achilles: What happens if you then find a picture within a picture which you have already entered, and you take another swig from the pushing-potion? Tortoise: Just what you would expect - you wind up inside that picture-in-a-picture."

Hofstadter (p. 338): Suppose a friend who has borrowed your car telephones you to say that your car skidded off a wet mountain road, careened against a bank, and overturned, and she narrowly escaped death. You conjure up a series of images in your mind, which get progressively more vivid as she adds details, and in the end you 'see it all in your mind's eye'. Then she tells you it's all been a joke! In many ways, that is irrelevant. The story and images lose nothing of their vividness, and the memory will stay with you for a long, long time.

Hofstadter (p. ix): Little Harmonic Labyrinth. This is based on the Bach organ piece of the same name. It is a playful introduction to the notion of recursive - i.e. nested - structures. It contains stories within stories. The frame story, instead of finishing as expected, is left open, so the reader is left dangling without resolution.

I was reminded of a fractal maze while watching this. I made the new one below -- copies of the brown frame are copied inside itself. Following the fractal path, get from S to F, on the correct level. It's okay to spend more than a minute on it.
Some Recent Purchases
Here are some of my recent purchases that I've liked: , Oskar&Bram's Gear Cube, Gems of Geometry, Dig-it, Blokus Trigon, Thinkfun Z-Knot,, Prof Layton and the Unwound Future, Sudoku Masterpieces (a fantastic exploration of variants by Wei-Hwa Huang and Thomas Snyder), and Anti-virus (by Oskar and PuzzleBeast, superb puzzles, finally available in US). I've also put together a list on Amazon, Games for Math Fun.
Gear CubeGems of GeometryDig-itBlokus TrigonZ-KnotProf layton and Unwound FutureSudoku MasterpiecesAnti-virus
The Most Beautiful Geometry Book from the 1800's
Byrne's version of Euclid's Elements replaced most of the words and verbiage with images in red, yellow, and blue. It was an exacting work, nearly impossible to print, and ultimately bankrupted the publisher. Surviving copies sell for thousands. Gorgeous book, now available as a reprint.
Byrne's Euclid's Elements
Five Trillion Digits of Pi
Alexander J Yee and Shigeru Kondo have calculated 5 trillion digits of Pi.
Vinay Deolalikar claims to have a proof of P!=NP, with copies of his paper at or at scribd. "I am pleased to announce a proof that P is not equal to NP. The proof required the piecing together of principles from multiple areas within mathematics. The major effort in constructing this proof was uncovering a chain of conceptual links between various fields and viewing them through a common lens. Second to this were the technical hurdles faced at each stage in the proof. This work builds upon fundamental contributions many esteemed researchers have made to their fields. In the presentation of this paper, it was my intention to provide the reader with an understanding of the global framework for this proof. Technical and computational details within chapters were minimized as much as possible."
A Book made from Books
Eric Angelini tells me about an unbooking of the book Napoleon.
Squares made of Squares
Rectangles made entirely of squares can be wonderful problems.

Bill Gosper: "Search the Web for 'squared rectangles'. If I were king, one of these diagrams (undimensioned) would appear daily in the newspaper puzzle pages." rectarith12.pdf or Googebra.htm  (with Solving techniques)

I made a few thousand puzzles involving Squared Rectangles.  In each case, the dissection is "nowhere neat", in that no two squares share a full edge.  Also, all of the squares in each dissection have a size less than 100.  I give two different types of puzzle for each dissection. MondrianPuzzles Here's my best nowhere-neat dissection of a size 35 square.
Mondrian Squares
Puzzle Party
Brian Pelcher has written up some details of the Hakone Puzzle Party. Also, the results of the Nob Yoshigahara Puzzle Design Competition are available.
Sawing a Man in Half Trick
Here's a new take on a classic trick -- saw a man in half -- I have no idea how it's done.
A Proposal to Use Arabic Digits
Fibonacci (or Leonardo of Pisa) writes me about a new method for representing numbers -- with Arabic numerals instead of Roman numerals. The first edition came out in 1202, and none are known to survive. The second edition was written 1228, and there are 13 partial copies, and 2 complete copies. All this comes via David Singmaster, who took some great photographs.
Arabic Numbers
Also, he sent me something called the Rabbit's problem, with a curious sequence of numbers. Click on either picture for a larger version.
Fibonacci Numbers
Nine Cool Points
To promote some demonstrations, I wrote a blog entry, Nine Cool Points on the Complex Plane.
Modernist Cuisine
The most advanced cookbook series ever prepared, Modernist Cuisine, by Nathan Myhrvold, delves deeply into the science of cooking, and what is possible. The photographs at his site look amazing.
Huge Ice Sheet
Greenland is now 260 sq km (100 sq miles) smaller, due to a large ice island breaking away.
Material added 25 June 2010
New Glider in Game of Life
Andrew J. Wade has constructed a spaceship with period (5120,1024)c/33699586. Announced at Game of Life News, the ConwayLife forum, the Game of Life Status Page, New Scientist, BoingBoing, and Slashdot. Golly runs it.
Two MG Puzzles
Carlos Penedo: Here are two easy puzzles for remembering Martin Gardner:
Martin Gardner Issue of College Mathemetics Journal
Submissions are sought for the January 2012 issue of The College Mathematics Journal which will be devoted to the mathematics of Martin Gardner. In order to avoid duplication of effort, authors are strongly encouraged to write the editor of CMJ (cmj at describing their proposed paper. Final submissions must be received by February 1, 2011 to be assured of consideration for this issue.
Point Covering Problem
Naoki Inaba, who runs the fantastic Naoki Project (with hundreds of new puzzle types), gave me a puzzle in person at G4G9, and I'm finally getting around to transcribing it. You have a hundred identical coins. What is the minimal set of points that the coins cannot cover? Some thoughts on this -- you could spread out 101 points. You could also pick 99 points randomly in a small rectangle, and at least one of the points would be in the triangle between 3 coins. Send Answer.
Point Problem #2
From Joshua Socolar: There are 6 points. Any triangle of 3 points has an area ≤ 1. What is the maximal area for the hexagon defined by the 6 points? Send Answer.
Flash Anzan
20 four digit numbers flash in front of you in 15 seconds. Can you instantly give the sum? In Japan, this has become a popular sport. Originally, it came out of clubs that were very, very good at using an abacus. After awhile, the abacus became a mental object, in a technique called flash anzan. There are also various youtube videos of flash anzan. Here is more anzan, and still more. [Update - Joseph Cooper: I saw the recent entry on your site about anzan software. I have just released a highly functional anzan practice app on Sourceforge: Little Soroban. under the GPL 3.0. It allows the user to: 1. Select the time interval between summands (to a decimal value), 2. Select the number of digits. 3. Select the number of summands.]
Polycube Symmetries
What are all the different symmetries of a polycube? George Sicherman has identified 33 distinct polycube symmetries.
Hirsch Conjecture Counterexample
From the abstract (arXiv:1006.2814) by Francisco Santos: The Hirsch Conjecture (1957) stated that the graph of a d-dimensional polytope with n facets cannot have (combinatorial) diameter greater than n-d. That is, that any two vertices of the polytope can be connected to each other by a path of at most n-d edges. This paper presents the rst counterexample to the conjecture.
Rhombus Puzzles
From 8 August 2002: Oyler has sent me a zip file with 20 number logic puzzles by Rhombus, from The Listener. Here is list of known errors. "They all appeared between 1960 and 1980. I've typed them in rather than scanning them as some of the photocopies are not very good. I was out with a magnifying glass for some of the puzzles in the Rhombus3 folder to try and discriminate between c and e and also 3 and 8 in some puzzles so be warned. I think they are all correct. I haven't myself got round to trying them as yet!! The numbers after the title refers to the number of the puzzle in The Listener Crossword series. Rhombus's last puzzle was All Square ( 2547 ) and appeared on 8th May 1980." If you have trouble with DOC files, has a good substitute program. [New -- Scott Marley sent a correction.]
Cast H and Other 3D Puzzles
Oskar has a new commercial puzzle out; Cast H. It's a very nice 2-piece maze, but not nearly as hard as it's rated. Other recent Cast puzzles are Cast Square and Cast Marble. 378% More Oskar is at the YouTube OskarPuzzle Channel, and also at the Shapeways Oskar shop. Shapeways recently featured Oskar in their column Puzzles, puzzles, and more puzzles. In addition, there is the Shapeways 3D Puzzle page. One of Oskar's puzzles was recently used in a wedding proposal.
cast h
Square Packing
I've made more progress with the Mrs. Perkins's Quilt problem. I've found lots of interesting papers... for example, a paper by Ian Gambini with square packings on cylinders. Erich Friedman adds a different one -- pack 33 unit squares into the smallest circle. "An easy problem solved for the first time." (by Maurizio Morandi). More square packing is in the June 2010 Math Magic.
mrs. perkins quilt
100 Strategic Games
I found a blog by Walter Joris, who wrote one on my favorite books.
Outer Billiards, Arithmetic Graphs, and the Octagon
By Richard Evan Schwartz (arXiv:1006.2782): Outer Billiards is a geometrically inspired dynamical system based on a convex shape in the plane. In the case of the regular octagon, the case we study, the arithmetic graphs associated to periodic orbits are polygonal paths in R^8. We are interested in the asymptotic shapes of these polygonal paths, as the period tends to infinity. We show that the rescaled limit of essentially any sequence of these graphs converges to a fractal curve that simultaneously projects one way onto a variant of the Koch snowflake and another way onto a variant of the Sierpinski carpet. In a sense, this gives a complete description of the asymptotic behavior of the symbolic dynamics of the first return map.
Blue Ball Machine
How many balls in the machine?
Material added May 23, 2010
Martin Gardner: October 21, 1914 – May 22, 2010
Martin Gardner died yesterday in Norman, Oklahoma, at a retirement home (near S. Canadian Trails Dr and Chautauqua Ave). At age 95, he still had a fairly active life, working on a number of books, accepting many visitors, and corresponding with many people. For example, David Blaine recently visited him, then gave a private magic show for the other residents. So far as I know, his neighbors never found out why high-level mathematicians and magicians kept stopping by. Various remembrances of him are already online: Scientific American, Discover Magazine, Boing Boing, Wired, SlashDot, Huffington Post, Associated Press, You-tube, and the Richard Dawkins Foundation.

On Friday, I was showing David Talbot of Technology Review around the Wolfram office. His son was with him, so we focused on the periodic table we have there. Near the end, he asked me of what mathematical books I would recommend for his son, and I told him to get Martin Gardner's Mathematical Games. I had no idea that it would be the last time I gave him a strong recommendation while he was still alive.
Don Knuth Metapuzzle
Don Knuth - The Wikipedia page for MASYU has an example 10×10 puzzle with 23 clues. My problem for you is to show that only 20 of those clues are needed. (Solution - Alan O'Donnell: Numbering as a large chessboard, the white circles at E9, C5 and J3 are not required, so as to leave a uniquely solvable Masyu.)
New Erich Friedman Puzzles
Erich Friedman: Some of the new puzzle types since your last update are Aztec Math, Gap puzzles, One of Each, Chess Pack, Triple Letter, Chess Loop, Weight Equations, Precise Pointer, Birdwatching Puzzles, and Color Strip. [Two of his latest Math Magics: Packing Two Shapes and Strata and Ringed Tilings are also well worth a look.]
Mrs. Perkins's Quilt
I've been busy lately studying the Mrs. Perkins's Quilt problem, with Richard Guy and Stuart Anderson ( At Wolfram Demonstrations, I've put together a demo with the first thousand solutions, many never seen before. In the weeks since I've posted it, though, we've made about 20 more improvements.
mrs perkins's quilt
I Pentomino Exclusion
Naoki Inaba: For a given polyomino P, find a polyomino [without holes] Q such that P can't be covered with Q under the following conditions. You have enough Q and you can rotate or turn Q, but Q must not overlap with other Q. For example, If P is "I" pentomino then the attached picture is a solution of Q. I don't know solutions of the case that P is "U", "V", and "Z" pentomino.
Noaki Inabi
Oskar Interview
Oskar van Deventer was recently interviewed by Shapeways.
Material added April 10, 2010
8000 Euros in Prizes for Magic Squares Results
Christian Boyer has announced some new Magic Square Prize Problems, in a recent press release. He's picked some interesting hard problems here. His main site is For more on this topic, my favorite site is the Harvey Heinz Magic Squares site.
The new location of the Online Encyclopedia of Integer Sequences is now available, along with various goodies. For example, sounds of the Recaman sequence, or the OEIS Movie. NJA Sloane has turned over all rights to OEIS to the new OEIS Foundation, which is looking for help and donations.
Some Packing Problems
Bram Cohen asks if 16 4×5×6×7 cuboids will pack into a 11×11×11×11 box. A nice problem of his with interesting solutions is to pack 8 7×9 and 8 8×11 rectangles in a 35×35 square.
DNA Fractal Globule
The DNA molecule is about 2 meters long, but fits in a very tiny space. There is now strong evidence that it packs into a fractal globule, similar to the hilbert curve.
Hyperbolic Mappings
Conformal mappings of hyperbolic geometry is a fascinating talk by Vladimir Bulatov.
Bulatov Hyperbolic
Shuffling with ordered cards
I liked the paper Shuffling with ordered cards by Steve Butler and Ron Graham.
Single Tile Aperiodicity found
For a long time, whether a single tile exists that tiles only aperiodically has been an open question. The paper Aperiodic hexagonal tile by Joshua E. S. Socolar and Joan M. Taylor gives a solution.
Fermat Number Factoring
When Richard K. Guy turned 80, John Conway bet him $20 that there would be no new factorization of a Fermat number within the next 20 years. The last complete factoring was of F11 in 1988. Now 93, RKG was quite happy with recent breakthroughs in factoring: F12 has 17353230210429594579133099699123162989482444520899 215 + 1 as a factor. March 27, 2010. Michael Vang.
F14 has 1784180997819127957596374417642156545110881094717 216 + 1 as a factor. February 3, 2010. Tapio Rajala.
F19 has 8962167624028624126082526703 222 + 1 as a factor. July 18, 2009. D. Bessell & Woltman.
F22 has 3853959202444067657533632211 224 + 1 as a factor. March 26, 2010. David Bessell.
F52 has 81909357657279 254 + 1 as a factor. March 15, 2010. Cedric Vonck.
The color-changing card trick
The color-changing card trick at is amazing, definitely a must-see. If you are able to see the trick, you can follow up by solving the perceptual whodunnit.
color changing card trick
Trick Art Museum
Even more visual tricks can be seen in japan, at the Takao Trick Art Museum.
Math Movies
Burkard Polster: As I mentioned, I have been collecting math-related movies. In particular, here is a list of mathematical movies that we've collected. Hope you find something of interest to you there.
Nine Point Cubic, a Cubic Curve Gallery, and Triangle Cubics
My demonstration Nine-Point Cubic lets you move around 9 points to find the cubic equation that goes through those points. There are 45 basic types of cubic curve, which are compiled at the Gallery of Cubic Plane Curves by Steven J. Wilson. These are explored further in the gorgeous site Cubics in the Triangle Plane.
triangle cubics
Approximation Feed
Kenneth Hammond: I have anapproximation Twitter feed where I post, once per day, an approximation that I find using Mathematica the night before. Two good ones that have come out of this experiment so far are K*716^(1/5) = 9.9999999723... (where K is Khinchin's constant), and the pandigital one shown here. I'd like to encourage others to join in by posting their own expressions on Twitter and hashtagging it with "#approx" so that it can be found on the search page. I believe there are some wonderful minds out there that will surely best--if not outright eclipse--all of these.
Mixed Polyhex Compatibility
George Sicherman: My Mixed Polyhex Compatibility now includes tetrahex-hexahex pairs. Some of them look highly improbable!
Mixed Polyhex
Random (blog)
Some of the events of G4G9 are detailed by Neil Bickford at his Random (blog).
Just in time for a South Park episode that attacks people getting too sucked-in to Facebook, I've gone and joined Facebook.
Material added 22 March 2010
Brain Busters
I've been doing the Brain Buster in the Japan Airlines in-flight magazine Skyward for a few years now. I put together a Brainbusters PDF that has most of these puzzles. Do feel free to mail me comments.
Brain Busters
Grigori Perelman Wins Millennium Prize
From Slashdot: The Clay Mathematics Institute has announced its acceptance of Dr. Grigori Perelman's proof of the Poincaré conjecture and awarded the first Millennium Prize. Poincaré questioned whether there exists a method for determining whether a three-dimensional manifold is a spherical: is there a 3-manifold not homologous to the 3-sphere in which any loop can be gradually shrunk to a single point?
Gray Matter
Theo Gray's science experiments are now all online at Popular Science, under Gray Matter. The magazine has also put the entire 137-year Popular Science online.
Beaded Polyhedra
John Gowland: I do rugs and flat beadwork, and think of it as three dimensional graph paper. However, I am impressed with this 3D polyhedra beading; she got third prize at the TOHO (a type of bead) competition she entered.
Record 73-digit Factor of 2^1181-1 Found
From Number Theory List: We are pleased to announce that on the occasion of its 25th anniversary, ECM smashed through the 70-digit barrier by finding the 73-digit prime factor 1808422353177349564546512035512530001279481259854248860454348989451026887 of 2^1181-1 (thereby completing the factorization of 2^1181-1). It used a Playstation cluster.
Nob Yoshigahara Puzzle Design Competition
Nick Baxter: This year is the 10th annual Nob Yoshigahara Puzzle Design Competition. The entry deadline is May 31. The official web site ( has all the particulars, and a history of the past entries.
Pi Day at Google
Google used the following logo for Pi day:
Pi Day Google
Material added 26 February 2010
A Fresh Look at Zome
A long time ago, I had a Zome System challenge to build a Petersen Graph with 15 blue struts of the same length. David MacMahon claimed the prize with an impossibility proof. Today, there is now a Zome Puzzle Kit available. There are also new lengths -- supershorts, extra longs, and half-lengths. There is also the excellent vZome software by Scott Vorthmann. In vZome, one can play with hypotheticals like the oranges and purple struts, which have not been manufactured. Brian Hall found a way to make the complete graph K9 in Zome, using these hypothetical struts. With the current system, K7 is possible in two different ways. Zomepad software is also available, currently in a reader form. Various papers and sites such as Mathematics of Zome (PDF by Tom Davis), and A Look at Zome (site by Chris Henrich). There's also the nice Metazome site.
Zome K7Zome K9
4×4 Kakuro
Johan de Ruiter (of Puzzle Picnic): Bram de Laat (reigning Dutch sudoku champion) wondered about the existence of 4x4 kakuro puzzles. I found there are 5 essentially different ones (although not too much different) and they are particularly evil. Kakuro rules: Write a digit from 1 through 9 in every box. The numbers above the diagonal line represent the sum of the horizontal digit combinations, the numbers below the diagonal line give the sum of those vertically. Within a combination, no digit occurs more than once. [Ed - Nice find. Interactive versions are available at Puzzle Picnic.)
4x4 Kakuro
Tony Madison II: I wanted to turn you on to a brand new online puzzle which I have created. The game is called Diceagons and it is a very challenging math puzzle which involves polygons and dice. This is the first of a series of online games that I will be developing for my website Findagame.
Not a Wake (3.14)
Mike Keith: Not A Wake is, quite simply, the longest Pilish text (in which the number of letters in successive words is required to follow the number of digits in the number π) ever constructed, and the first book-length work written entirely in Pilish.  Divided into ten sections of 1000 digits, each written in a different style, its words "spell out" the first 10,000 digits of the number π = 3.14159265358979323846... [Ed - Certainly a record-setter.]
New Orleans Saints = Winners, last season
The Football Pool Problem, which gets its name from a lottery-type game where participants predict the outcome of soccer matches, is to determine the smallest covering code of radius one of ternary words of length v. For v = 6, the optimal solution is not known. [Ed - So, six games, with win, lose, draw. Find a set of 65-70 tickets that will always match at least 5 of the games. It's been unsolved for 30 years.]
A New Polyomino Oddity
George Sicherman: A pentomino compatibility is a shape that can be covered with two different types of pentomino. An odd pair is a set where the number of pentominoes used is odd. So, what's the smallest figure coverable with an odd number of X or T pentominoes? Here's a 137-piece solution.
Odd Pair
Omnificent English Dictionary In Limerick Form
The OEDILF (Omnificent English Dictionary In Limerick Form) is a project to write limerick definitions of every word.
Material added 16 February 2010
Stacked Decks in Texas Hold'em Poker
Ben Joffe: I've been running simulations to try to solve ideal stacks for Texas Hold'em poker. The decks have the property that no matter which way that are cut, the same player will win the hand. I've solved it for 2, 3 and 4 players, and now trying to solve it for 5, see the results: [Ed-Neat! For example, the below is his solution for four players. The dealer will always win, for any cut.]
Texas Holdem Stacked Deck
Puzzle Fun Online
One of the best defunct puzzle magazines was Puzzle Fun, which was run by Rodolfo Kurchan from 1994-2000. Rodolfo has now put the entire series of Puzzle Fun Online, and has started making new issues, all available for free.
Puzzle Fun Hexominoes
Morpion Solitaire Updates
Christian Boyer: A new update of is now online: look at the February 2010 Morpion news! With a very new record from Japan. And with a scan of Bruneau’s “historic” letter of 1976 including his handwritten grid of 170 moves, which is still today the world record.
Morpion Bruneau record
A New Keen Approximation
Gerson W. Barbosa: Is the following interesting enough to be a Keen Approximation: e - atan(e) = 1.49999892344176761360? [Ed - I think so. A nice find in the realm of almost integers.]
Triangle Dissections
From An enumeration of equilateral triangle dissections. We enumerate all dissections of an equilateral triangle into smaller equilateral triangles. We con rm W. T. Tutte's claim that the smallest perfect dissection has size 15 and we nd all perfect dissections up to size 19.
Lots of digits of Pi
A new Pi computation record has been set by Fabrice Bellard - 2699999990000 decimal digits. The BBC did a news report. Of note, this was the first pi record set by a single personal computer.
The Christmas Star Rises
Don Knuth: Sudoku for Christmas. Here's a type of "jigsaw sudoku" that has a holiday message. [Click the link for Don's explanation.]
Christmas Star Rises
Factor of F_14 = 2^(2^14)+1 Found
The first factor of Fermat Number 14 has been found by Tapio Rajala, as part of GIMPS. The factor is 1784180997819127957596374417642156545110881094717 · 2^16 + 1.
Recursive Tilings
Herman Haverkor: This paper defines the Arrwwid number of a recursive tiling (or space- lling curve) as the smallest number a such that any ball Q can be covered by a tiles (or curve sections) with total volume O(volume(Q)).
Losing as Little as Possible
Vittorio Addona, Stan Wagon, and Herb Wilf (from Suppose Alice has a coin with heads probability q and Bob has one with heads probability p > q. Now each of them will toss their coin n times, and Alice will win iff she gets more heads than Bob does. Evidently the game favors Bob, but for the given p, q, what is the choice of n that maximizes Alice’s chances of winning?
Thirteen Spheres
Oleg R. Musin, Alexey S. Tarasov (from The thirteen spheres problem is asking if 13 equal size nonoverlapping spheres in three dimensions can touch another sphere of the same size. This problem was the subject of the famous discussion between Isaac Newton and David Gregory in 1694. The problem was solved by Schutte and van der Waerden only in 1953. A natural extension of this problem is the strong thirteen spheres problem (or the Tammes problem for 13 points) which asks to find an arrangement and the maximum radius of 13 equal size nonoverlapping spheres touching the unit sphere. In the paper we give a solution of this long-standing open problem in geometry.
Material added 9 February 2010
Two Decs, No Recs, 365 Solutions
I asked George Sicherman to extend his 9×9 results shown on 5 February in Two Shape Irregular Sudoku Squares. The rules: make a 10×10 square with two different decomino shapes, such that there are no internal rectangles of two or more pieces. It turns out there are exactly 365 solutions, an entire years worth. A twodec-norec Zipfile of pngs is available, as is a twodec PDF file. There is also a large twodec picture available, showing all 365 solutions at once (click on smaller image below). George wonders "I wonder how sparse a sudoku you can make with these." I'm wondering the same thing. Is there a good program available for working on irregular sudoku? Solutions by Bruce Harold.
Twodec Norec small
Handy Programs
Looking for a handy math program? My current list of regularly used free programs includes Burr Tools, Golly, IrfanView, Jenn, SeifertView, Simon Tatham's Puzzles, Sudoku Susser, VLC Media. Some of my favorite free programming languages are Cygwin, GAP, Lua, PARI, Perl, Python, MiKTeX. My list of favorite purchased programs includes Mathematica, Crossword Compiler, Textpad, Geometry Expressions, Adobe CS4, E Editor, Stella 4D. Are there any good programs I should know about? Let me know.
Material added 8 February 2010
Orthogonal Game of Hip Update 2
Combining various entries here. The Game of Hip, by Martin Gardner, is played on a 6×6 board. Red and Blue alternate placing a stone of their color on the board. The first person to complete a square of their color, in any orientation, loses. Ignoring rotations and reflections, there is a unique tie game. Josh Geifer wondered what the largest solution would be if only orthogonal squares were considered. Both he and I managed to find several 10×10 solutions with no orthogonal squares, but none of them seemed extendable to 11×11. As a sidenote, Josh mentioned he's a son of TV writer Lewis Greifer, who wrote The Prisoner episode "The General" under pseudonym Joshua Adam, from his and his brother’s first names. We showed an 11x11 solution. William Rex Marshall computer-found a 12×12 drawn position with 72 counters of each colour, with rows in binary (71, 2716, 2505, 3186, 2846, 1461, 1699, 3129, 1356, 870, 1745, 3215). WRM followed up with a computer-found 13×13 drawn position, with rows in binary (427, 5346, 4663, 1925, 5420, 7321, 2548, 1119, 3890, 2665, 6467, 5080, 2702). I noticed a couple of repeating shapes in that solution, so I experimented with that shape, and found the below 14×14 position, with rows in binary (10724, 9551, 3882, 8825, 14611, 5064, 2718, 7764, 1266, 12839, 10129, 5436, 15529, 2533). This particular design cannot be extended to 15×15. Is there a drawn 15×15 position? Let me know.
Hip draw order 14
New Orleans Saints = Winner, last season.
I've had a few accolades for anagrams, recently, at I also set a record for the two longest well-mixed pair of single source anagrams, with omnidirectional antennas, Daniel Constantine Marino. Some of the recent anagramming winners were:
Scarecrow, Tin Man, Lion, Dorothy = Mind, rosy heart, crown, location. (Ed Pegg Jr)
McDonalds Restaurant chain = Standard lunch to Americans. (Andrew Brehaut)
Skeletons in the cupboard = Bones unlocked their past. (Tony Crafter)
Director James Cameron = Set major cinema record. (Meyran Kraus)
Triple chocolate square = Atherosclerotic plaque. (Ed Pegg Jr)
Golden Anniversary = Dear Granny's in love. (Dharam Khalsa)
Righteous indignation = Idiot shouting in anger. (Paul Lusch)
A crisis on Wall Street ~ will start a recession. (Ellie Dent)
If love isn't here ~ then life is over. (Meyran Kraus)
Economist = :( :( :( emoticons. (Ed Pegg Jr)
"Astro Boy" = Say "Robot." (Adie Pena)
Material added 5 February 2010
Two Shape Irregular Sudoku Squares
I asked George Sicherman if he could find all ways to make a 9×9 square using only 2 enneominoes such that the 9×9 squares had no subrectangles of 2 or more pieces. He cranked up his programs and quickly sent me all possible solutions. The next question would be which of these have valid Sudoku solutions.
Two Shape Sudoku
Eternity II Remains Unsolved
The two million dollar Eternity II puzzle remains unsolved. Eternity II is a very, very hard edgematching puzzle, similar to MacMahon squares. There is an Eternity II Yahoo group with much discussion. (And a Yahoo MathPuzzle Group, of course).
Math Dice
Eric Harshbarger: Ed, thought you might get a kick out of these math dice I created. I'd probably sell a set of 3 for $3.00 (discounts for large orders). [Ed - One puzzle from Eric's Logolog blog: make a 4×4 word square with 16 different letters. The link gives a list of all the solutions, all of them using obscure words.]
math dice
183-digit Prime-period Oscillator in Game of Life
Adam P. Goucher: The oscillator repeats every 2^607-1 generations. It is accomplished by a p8192 base loop (2^13) followed by 506 period doublers (2^506) and 44 period quadruplers (2^88) and a 1 generation glider advancer (-1). This corresponds to the Mersenne prime M607, which has 183 digits. This method can be extended to yield any Mersenne prime oscillator, including the recently discovered 12-million-digit megaprime. To avoid timing issues, the glider advancer is asynchronous. The input glider is allowed to arrive at any time (apart from a small 'window' where it collides with the clock glider). The oscillator (requires a cellular automata program like Golly to run).
Seven Staggering Sequences
Neil Sloane's paper Seven Staggering Sequences [PDF] is available on
Material added 4 February 2010
Magic Tile
Roice Nelson: MagicTile is a new Rubik's Cube analogue which extends the original to regular polygonal tilings spanning all three constant curvature 2D geometries (spherical, flat, and hyperbolic).  In this abstraction, Rubik's Cube follows from the special case of a tiling of squares on the sphere. A heptagonal tiling of the hyperbolic plane leads to a 24-colored puzzle based on Klein's Quartic! [Ed - This is an amazingly versatile and beautiful program.]
Magic Tile
Math Magic - Spaced Out Polyominoes
The latest Math Magic has a very interesting challenge based on spaced-out polyforms. George Sicherman has already found some amazing solutions. Recent previous episodes have gotten beautiful results in sparse tilings, unique packings, and 32 + 42 = 52 tilings. George also recently found some vastly improved Odd Pairs.
N pent 345       Odd Pair
More Erich Puzzles
Erich Friedman has also made several new puzzles for for his Puzzle Palace. These include Knight Tour, Latin Square, Line Segment, Chess Avoidance, and Chess Loop.
knight loop
Polyhedral Calendar
Jean-Charles Meyrignac: I don't know if this is a new idea, but I found a free hexahedral calendar to do yourself. [Ed - Very nice. Two 6-pentagon stars, put together flat and with a rubber band weaving between the corners, will pop up into a dodecahedron, a fact mentioned by Martin Gardner.]
Multimagic Update
Christian Boyer: Various news, new enigmas, and recently developments on additive-multiplicative magic squares is available at
Oskar's Cube on iPhone
M. Oskar van Deventer: Oskar's Cube (mechanical version) can also be played on iPhone now. The design and programming was done by Albert Leung and Mikheal Kruk, presented as Amazing Cube Maze at Whifflebird. It can be bought at iTunes as Amazing Cube Maze (link launches iTunes). [Ed - Object to App.]
Oskar Cube Maze
Subway Shuffle as Puzzle
Bob Hearn: Just thought I'd drop you a line to let you know that my Subway Shuffle game has now been physically produced, by Popular Playthings. It was re-themed, and is now called "Athena" [Ed - App to Object.]
Subway Shuffle
Material added 31 January 2010
Prime Curios
The book Prime Curios by Chris K. Caldwell and G. L. Honaker Jr. is well worth a look over at Amazon. Might be worth a look at Barnes and Nobles (Prime Curios), too, due to Amazon de-listing publishers, recently (Update: Amazon has relisted the publisher). Another source is the publisher, CreateSpace (Prime Curios). You can also visit the author's site - Prime Curios. Opening the book at random, 14593 is listed as the largest prime factor of 12345678, and 16033 is the first prime both followed and preceeded by 20 composite numbers. All sorts of interesting prime facts, as you might suspect.
Sky Williams: I only have like 100 hits, ever, compared to your 4 million. How's Google #1? Anyways, at, I have a number of my puzzles that have been in Games Magazine. [Ed - A very interesting collection of puzzles.]
Crossword Sudoku
George Sicherman: A recent issue of Games Magazine included four Crossword Sudoku puzzles by Ken Futamura.  This is the cleverest twist on sudoku I've seen yet!  You use the clues to fill in the words, then fill in the rest of the matrix as a sudoku with letters instead of digits. They're almost as much fun to construct as to solve. Here's a Crossword Sudoku that I made in Ken's honor.
ACROSS.  2. Buck Rogers enemy Killer ______.  4. Suffix with mino-.  5. 2nd Amendment org. (abbr.)  8. Sound of hesitation. 9. Counterfeiter.  11. A giant word.
DOWN.  1. N.Y. baseball player.  2. A character in Street Fighter.  3. A kind of line. 6. A kind of sleep (abbr.).  7. What Crossword Sudoku puzzles may find.  8. Educ. inst. near DFW airport (abbr.).  9. What one may have with Crossword Sudoku puzzles.  10. In the morning (abbr.).
crossword sudoku
Four Million
Chris Lusby Taylor: I've just noticed that, at the bottom of your home page you claim "Yes, over 3 million" visitors. Actually, you are now over 4 million! Congratulations and many thanks for maintaining this wonderful site which never ceases to stimulate and amuse. Best wishes for 2010. [Ed - many thanks to you and all.]
Polynomial Plot
John Baez: "A while back, my friend Dan Christensen drew a picture of all the roots of all the polynomials of degree at most 5 with integer coefficients ranging from -4 to 4." Ed - That's the start of a fascinating column on Polynomials Roots, which has a fantastically detailed set of pictures.
Five Configurations
Arrange points and lines so that exactly 5 lines go through every point, and 5 points are on every line. Here's a new answer.
five configuration
Material added 18 January 2010
Planetary Art
Each of the below is a close-up picture of some planet. Each picture is linked to the original source.
2010 MIT Mystery Hunt
The latest MIT Mystery Hunt just finished -- "The coin was found by Metaphysical Plant at 5:50 AM Sunday, January 17."
Fractal Pinwheel
Natalie Priebe Frank And Michael F. Whittaker wrote a very interesting paper called "A Fractal Version Of The Pinwheel Tiling."
pinwheel fractal
RSA Composite 768 factored
The 232-digit number known as RSA-768 has been factored by a large team.
I sent a 2010 puzzle to Will Shortz for National Public Radio, and it got a lot of response from people like Joe Becker, Emrehan Halıcı, Robert Wainwright, Michael Reid, Alain Zalmanski, Jordi Domènech i Arnau, Juha Saukkola, and John Grobben. Some commentary is a Mazurland.
2345*6/7 = 2010
2/3*45*67 = 2010
36 * 57 - 42 = 2010
Σ (n^3 - n/3),  n = 0...9   = 2010
8*12*67*95 / 304 = 2010
0^7 + 1^9 + 2^8 + 3^6 + 4^5 = 2010
2*3*5*(7+11+13+17+19) = 2010
9+8*7+6*54*3*2+1 = 2010
2700 Billion Digits of Pi
Fabrice Bellard has managed to compute a lot of digits of Pi on a normal PC. All previous records were set on arrays in supercomputers.
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