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Math Games
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Material added Novemeber 14 2013
Print 3D Puzzles
Some discussion of Oskar van Deventer and George Miller is in the article The Puzzle Masters.
Seven Touching Cylinders
Seven mutually touching infinite cylinders has new solutions. Can 7 indentical toruses touch?
7 cylinders
Ever See A Tree in a Movie?
Trees for movies are being outsourced to SpeedTree now.
Not Photoshopped
The below is a single picture by Béla Borsodi. For how it was done, see his VLP video.
VLP Illusion
Martin Gardner -- The Best Friend Mathematics Ever Had
Colm Mulcahy wrote a blog article on Martin Gardner.
Snake Polyominoes
Shade 16 squares in a 5x10 rectangle so that the remaining square make an obvious path. The resulting snake polyomino is unique.
New Morpion bound
Morpion Solitaire 5D: a new upper bound of 121 on the maximum score.
n-body Orbits
I love the 5 pointed star orbit in this Scientific American blog.
five pointed star
Ruben's Tube
Sound, fire, and sine waves are shown in a Ruben's Tube (youtube).
Loss of a Mail Service
For fifteen years, I had a system for doing updates involving mail sorting and sending messages to tabs. Sometimes dozens of tabs as I boiled down an update. Unfortunately, Yahoo decided to kill the mail service I was used to, and it's taken me a long time to recover. Part of the process was keeping up with the mail, which I find harder to do in the "new and improved" systems. I'll agree that they look better.
Party Like it's 19999999...
2 10^1059002 -1, or 19999999999999...9999, is prime. It's the first near-repdigit prime found with over a million digits. Found by Serge Batalov.
Impossible Fractal Sphere
I liked this impossible fractal sphere.
Attacking a Triangle
For an order 20 triangle, 9 vertices can be selected that attack all other vertices.
All the colors
At allRGB, interesting math produces 4096x4096 images where every pixel is a different color.
Doyle Spiral
The Doyle spiral circle packing is quite beautiful.
Lines through the United States
Connect two points of the same US state with a straight line. What's the most other states the line can pass through? Lines through States has the answer.
Prisoners Dilemma
Prisoners do better with Prisoner's Dilemma than normal students.
New Puzzles
The 2013 Nob Yoshigahara Puzzle Design Competition has the 2013 entries and winners.
Balloon pyramid
I also liked this enormous Sierpinski tetrahedron made of balloons.
balloon pyramid
Some Anagrams
Climate change deniers ~ maligned earth science. 
Wimbledon Tennis Tournament = Mountbatten-Windsor Linemen.
Lives of the Noble Grecians and Romans = Conan O'Brien, Marvels' Agents of SHIELD
Animated TV series ~ is a advertisement. 
Incarnadines ~ in sardine can.
Malnourishment = The norm in a slum. (Anagram by Christopher Sturdy)
A cheap costume = Cape, moustache. (anagram by nedesto)
Professor Stephen Hawking = He gets known for his papers. (anagram by Larry Brash)
"I'm so ready for a new chapter" -> "Fancy a road trip somewhere?" (Twitter's Anagramatron)
Being sober = No big beers. (anagram by Ivan Andonov )
Eleanor Rigby = Yer ignorable. (( anagram by Mike Mesterton-Gibbons ))
Secret Trig Functions
From the Onion -- everyone neglects the Hacovercosine ... including me.
Seven Consecutive Consonants
A "backlit LCD screen" has 7 consonants in a row. Find some other items with this property.
1. A well-known phrase.
2. A well-known movie.
3. A 2nd grade math topic.
4. A beverage.
5. A piano tune.
Material added 7-11
Harvey Heinz passes
The maker of the foremost magic squares site, (mirrored at, died on July 6, 2013. In 1958, he got 5 tons of obsolete equipment from a telephone company and start a computer club with high school students. In 1973, his efforts to bring electronic games to market were unsuccessful, so he turned to bookbindery until 1991 and retired. A biography is at
Harvey Heinz
No Three In A Line Paper
I was looking for Flammenkamp's page on the No-three-in-line problem, and found the paper "Martin Gardner's minimum no-3-in-a-line problem". It deals with the opposite problem -- put a minimum number of counters on a board so that it's impossible to add another counter without creating three in a line.
no three in line
Puzzle Solving Cockatoos
Cockatoos are rather good at solving parrot-friendly puzzles.
Wolfram Technology Conference, October 21-23
I'll be at the Wolfram Technology Conference.
Poolazoid Puzzle
From George Sicherman: Here is a puzzle to cool off with. It's called the Poolazoid Puzzle. Just put the pieces together to form a right triangle. You may flip them over. (You can also take a look at the solution.)
Roll Three 8-sided Dice
The IBM Ponder This problem for July 2013 throws an 8 sided die 3 times, and can get 120 possible different positive integer sums. If all the faces have positive integer sides, what is the lowest possible value for the highest face? For an  n sided die, the maximal number of different sums is atetrahedral number. Here are my best results for d3-d7.
d3: 1, 2, 5 -- 10 sums
d4: 1, 2, 8, 12 -- 20 sums
d5: 1, 2, 16, 19, 24 -- 35 sums
d6: 1, 3, 12, 27, 43, 46 -- 56 sums
d7: 1, 2, 8, 51, 60, 79, 83 -- 84 sums

The d8 is the contest problem, so don't post that answer. I'm curious about d9, d10, and so on. I used a sieve method, but it doesn't scale up well. (I posted this at There is also a reddit discussion.)
Number Maze
joedev made a great series of number mazes.
Advance in Elliptic Curve Theory
There are advances on the Minimalist Conjecture.
Ancient Greek Geometry Challenges
These geometry challenges are quite well done, all with an HTML5 interface.
Bruce Oberg pointed me to Anagramatron, which scrapes Twitter and looks for anagrams.
I should not be awake right now = I wonder who she is talking to.
"Holy crap I'm late" ~ "metaphorically"
Absolutely apathetic = Especially about that.
21061-1 Factored Last Year
The Cunningham Project factored 21061-1 last year.
Crawling Metal Polyhedron
I liked this slow-moving walking metal polyhedron.
Rolling Shutter Photos
Based on the below image, determine the number of propellers and how fast they were moving. Or see the rolling shutter photo analysis.
rolling shutter
Ghost Diagrams
At Ghost Diagrams, shapes with rules try to fit together.
Paul Nylander's Math Artwork
I'm not sure if I've linked to Paul Nylander's Math Artwork site, yet.
Proofs with Problems
I liked the wikipedia Proofs with Problems, and also the mathoverflow discussion.
Covering the Alphabet
Queen Alexandra of Yugoslavia and Herbert Norman Schwarzkopf, Jr. are two names in Wikipedia that cover the alphabet. Is there a shorter pair of names with the same property?
Material added 28 May 2013
Prime Gap
The ternary Goldbach conjecture is proven. Every odd number over 5 is the sum of 3 primes. The proof by H. A. Helfgott is just 133 pages long.
Prime Gap
Zhang Yitang has proved that for some number N less than 70 million, there are infinitely many prime pairs that differ by N. His paper was accepted by Annals of Mathematics in early May.
Naoki Inaba Puzzles
Naoki Inaba has been updating his awe-inspiring puzzle site with pictures. If you have Chrome, the page can be translated to the language of your choice.
Naoki Inaba
Siggraph Papers
video of Siggraph technical papers. I want to effortlessly do all these sorts of graphics NOW, dammit.
Reel Physics
Movies frequently contain a lot of bad physics, and some of these instances are explored in Reel Physics.
Laser Maze
In Laser Maze, a 25-square grid has targets, mirrors, beam-splitters, gates, and a laser. With a certain amount given within 60 challenges, you'll need to light up all the targets. Another great puzzle from Wei-Hwa Huang.
Least Efficient Packing Shapes
Least efficient packing shapes: We address the question of which convex shapes, when packed as densely as possible under certain restrictions, fill the least space, leaving the most space empty.
Martin Gardner in the Twenty-First Century
Martin Gardner in the Twenty-First Century has columns on Geometry, Number Theory, Graph Theory, Flexagons, Packing, Playing Cards, and more. Of the 41 articles, 8 were written by Martin Gardner. Many articles update material from Martin's columns.
Material added 5 5 2013
The State of the Unit: The Kilogram (Successful!)
I've been supporting various Kickstarter projects, and one of them is by my Numb3rs colleagues, Michael Trott and Amy Young. The State of the Unit: The Kilogram collect up the huge amount of interesting facts about the kilogram, and attempts to move away from the platinum cylinder in France. Please consider supporting this documentary about the current state of the art in physics, and all the amusing things that can go wrong when going after .99999999999 accuracy.
kilogram movie
Mrs. Perkins's Quilts
Lorenz Milla and Stuart Anderson recently extended the known results of Mrs. Perkins's Quilts.
Period 43 Oscillators in Life
AP Goucher reports that stable reflector in Conway's Gaome of Life has been found by Mike Playle. The reflector has a recovery period of 43, which allows all oscillators of periods 43 and above to be made. Goucher also put together known spaceships into Ford circles, including a tiny new tiny c/7 spaceship by Josh Ball. Dave Greene: "I see you've updated your Prize Page for a small 90-degree stable reflector in Conway's Life. I called this object a Snark back when it was just a mythological hypothetical creature. I've sent the $100 to Mike Playle, so I'm finally free from prize promises after a drawn-out dodecade. Mike has decided to continue the tradition, now in its third round, of recycling part of the prize money into a new prize."
Life Ships
After 100 Years, Ramanujan Problem Solved
My colleague Oleg Marichev noticed the line "We do not record the value here, because it is not particularly elegant" in one of Berndt's books on Ramanujan, but Oleg went ahead and tried his hand at it, and found a short, elegant solution. It's very likely a solution Ramanujan meant to write down. I helped write up the whole story at After 100 Years, Ramanujan Gap Filled.
Sicherman Updates
George Sicherman has sent in updates on Didrafter Compatibility, Polyking Exclusion, Triangular Enneiamond Compatibility, Knight's Move Cell Shifts, Holey Heptomino Compatibilities, Pento-Tetro-Tetrominoes, Ying/Yang Pentomionos, Polycube Symmetries, Ways to make an irregular Sudoku grid with 3 shapes (1, 2, 3, 4, 5). He also sends a puzzle. "These are the six didrafters. Join two of them, then join two others to make the same shape. There are four solutions. (sol 1, sol 2, sol 3, sol 4)". Below that, he shows how the H and triangle 7/9-iamonds are compatible.
Highcastle Chess
The Zillions version of Highcastle chess at the Chess Variants site has had an error, which Karl Scherer fixed for me. In this variant, any piece may move two squares towards a target piece, and the target is moved to the square moved over. Highcastle Chess Zillions.
Temple Trap
Raf Peeters: I noticed on Amazon that you liked my Temple Trap puzzle game I designed for SmartGames. A digital version of Temple Trap has been releasedin HTML5. All challenges of Temple Trap on the digital version are different from the ones in the physical version.
Tiled Rectangle
What is the fewest squares an n×m rectangle can be cut into? Bertram Felgenhauer has found solutions up to 300300 rectangles, which I've also turned into a Minimally Squared Rectangles demonstration.
Nontrivial Games
Jean-Charles Meyrignac: I just discovered the Nontirivial Games blog. It's about logic puzzles on the iPhone and iPad. What is interesting is that the author is named Nicola Salmoria, and he is the original author of Mame.
Math Lab
There has been a Math Lab Bust.
3-4-5 Pentagon Dissection
Serhiy Grabarchuk: In your material added on April 20, 2011, I've spotted George Sicherman's solution for the 3^2 + 4^2 = 5^2 dissection problem (at Math Magic) for the pentagon. I've improved it, getting two slightly different solutions with less parts (-1). Solution 1 is especially interesting, since it contains in the final big pentagon three smaller unbroken pentagons.
345 Pentagon
Rectangle coloring
Can you four-color a 17×17 rectangle so that there are no monochromatic rectangles? Bernd Steinbach and Christian Posthoff solved it, and then bumped it up to a 18×18 coloring.
3 Notch Tetrominos
Peter Esser notes that there are 64 T-tetrominoes with 3 notches, and they will fit into a 16×16 square with all the notches lined up.
T-tetromino 3 notches
Uncrossed Leaper Tours
Alexander Fischer has put together a page on Uncrossed Leaper Tours. Related to this is Zigzag Paths results. Jean-Charles Meyrignac: "This reminded me about the impressive Russian site on uncrossed kinight's tours."
Polyhedra site
A nice polyhedra site is Java Applets for Visualizing Polyhedra.
Attractive Attractors
At Behance, a nice set of Strange Attractors was put together.
Life is a Game
Emrehan Halici's Life is a Game TED talk is up.
Advanced Roshambo
The higher levels of Roshambo are quite impressive.
Scooping the Loop Snooper
It's possible to prove the undecidability of the halting problem, as Seuss.
A Boy and his Atom
IBM has successfully replaced actors with carbon monoxide molecules in A Boy and his Atom.
No Repeated Distances
The Al Zimmermann Programming contests have had many interesting results. I recently did a demo for No Repeated Distances.
Glass Crack Count gives Projectile Velocity
It would have been nice on Numb3rs: Counting cracks in glass gives speed of projectile.
GED Prep
Aspen High School. One of the older online high school and GED prep websites, offers an online high school diploma class which covers basic subjects including Math.
Material added March 5th
A Square in 50 Similar Acute Triangles
Lew Baxter has divided a square into 50 similar triangles with angles 45-60-75. With b=sqrt(3), the points are {{0,0}, {3492-210b,0}, {3890-140b,0}, {4288-70b,0}, {4686,0}, {3000-116b,492-94b}, {3398-46b,492-94b}, {3597-11b,492-94b}, {3796+24b,492-94b}, {4194+94b,492-94b}, {2262+25b,1230-235b}, {2859+130b,1230-235b}, {3456+235b,1230-235b}, {1260+45b,1260+45b}, {1746-105b,1260+45b}, {2232-255b,1260+45b}, {1428+51b,1428+51b}, {1278+213b,2214-423b}, {1278+213b,1278+213b}, {1980+517b,2706-517b}, {0,1491+639b}, {1278+213b,3408-213b}, {0,4686}, {1305+465b,0}, {2736-237b,756+27b}, {1278+213b,756+27b}}, and the triangles use points {{6,7,2}, {3,2,7}, {7,9,3}, {4,3,9}, {9,10,4}, {5,4,10}, {11,12,6}, {8,6,12}, {12,13,8}, {10,8,13}, {14,17,15}, {15,17,18}, {16,15,18}, {19,17,18}, {1,21,19}, {21,19,22}, {23,22,21}, {13,11,20}, {20,18,11}, {22,20,18}, {1,14,24}, {2,25,24}, {24,26,25}, {25,16,26}, {26,14,16}}.
Lew Baxter Square
I wrote a column on Turmites 9 years ago, and these have shown up on Wikipedia's Turmite / Langton Ant pages. They can be run in Golly, with the help of a 32-bit Python installation. More recently I've been interested in the Busy Beaver problem, which I reformulated as "When does a machine become predictable?" Certainly a Halt starts a perion of predictability, but also Traps, Highways, Spirals, Wedges and Sequences are predictable behavior. All sorts of activity is now documented at (I didn't pick the name) Ed Pegg Jr's Busy Beaver Turmite Challenge. Here are the results so far -- for single state machines with cyclic coloring, Right Uturn Left Straight (RULS) can describe the rule.
9,977 steps
Langton's Ant
67,620,060 +10 steps
9 unresolved
~6,650,200,000 steps
Georgi Gochev
94 unresolved]
~217,782,000,000 steps
Ed Pegg Jr
615 unresolved
~3,511,330,000,000 steps
Georgi Gochev
570 unresolved
1.9*10^61 steps
Georgi Gochev
I am currently looking at some five color rules: RULUU, RLRLU, RLURL, URULU, URLRL, UURUL. Here they are at step 5 trillion. If you'd like to run some of the 615 unresolved rules, head to the page.
Turmites step 5 trillion
Largest Prime
The largest known prime is 257,885,161 − 1, with 17,425,170 digits. The top 100 Prime list, and the top Probable Prime list might also be of interest.
Tetrahedron Centers
I've always liked triangle centers (ETC, mathworld, wikipedia, cubics), so I've often wondered about tetrahedron centers. I've got a method for calculating lots of them, but I haven't found elegant form of Trilinear Coordinates. Any insights would be appreciated.
tetrahedron centers
Button Men
I've recently gotten obsessed with the old game Button Men (wikipedia, boardgamegeek, cheapass, randomdice, blog, iOS version).
Payback Percentages
Payback percentages are the amount a slot machine must return, and they are defined by law. According to a site explaining how to win at slot machines, the state with the lowest minimum payback is Nevada with 75%, and the state with the highest is Maine with 89%. I never would have guessed that. Much of optimizing life as a gambler now involves researching the law. For example, for poker, the site covers New Jersey Law as it relates to poker.
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 My facebook page is at Ed Pegg Jr.