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Math Games
Social Golfer Problem Wolfram Demonstrations Dodecahedral Tilings Power Sums Melbourne: City of Math Snakes on a Plane Prime-generating Polynomials The Fano Plane G4G7 Kobon Triangles Beautiful References Times Square Magic From Keyfobs to Ringtones Sudoku Variations Vector vs. Raster Tournament Dice Deadly Rooms of Death Fair Dice Chessboard Tasks A Zillion Connection Games Keen Approximations NUMB3RS TV show Cyclopedia of Puzzles Sliding-block Puzzles Rulers and Gracefulness Hoffman-Singleton Game Ten Trillion Zeta Zeros Evil Numbers Manifolds in Genesis 64K or Less - Demoscene Modern Burr Puzzles Egyptian Fractions Nob Yoshigahara 2D Turing Machines WireWorld Multiplication Crossword Rules The Quantian Distribution Integer Complexity Supermagnetic Polyhedra Gaussian Numbers Number Games Guilloché Patterns Prime Megagap Wolfram Functions Site Cubic Symmetric Graphs Superflu Modeling Domino Graphs Sequence Pictures Square Packing Multi-state Mazes Loculus of Archimedes Matrix Revolutions Möbius Function Paterson's Worms Revisited
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Awards Book Recommendations Chaos Tiles Characteristic Polynomial ChebyshevU Chromatic Number of the Plane Coin Weighing Complex Numbers Connecting Dots Crosswords Digital Sums Dissecting Convex Figures Domino Games Encryption Eternity Puzzle Fair Dice Gambling Odds Graphs Gravity Guestbook Golay Code Group Theory Iamonds Knots Leapers Links Locales Math Software Mini-Go Multistate Mazes Number Fun Parity Partridge Puzzle Path Problems Pentagon Tiling Philosopher Logic Phonetic Alphabet Polyominoes Prize Page Programs Repeating Decimals Retrograde Analysis Silly Puzzle Similar Dissections Superflu Stability of the Atom Riemann Hypothesis Tic Tac Toe Triangles WGE123 Word Play World Puzzle Championship 2000 Zome
Material added 27 June 2020
NPR Puzzle
The NPR Puzzle is mine this week. Think of a 5-letter animal. Remove the middle letter, and two opposites remain. What animal is it?
Heesch 4
Craig S. Kaplan‎: It seems appropriate to spend some of this quarantine computing coronas! Here's the smallest polyiamond with Heesch number 4, a 20-iamond.
iamond heesch
We're all in this together
We are still here tonight.
Puzzle Fun Magazine
A new issue of Puzzle Fun is out.
Proofs From The BOOK
Many titles like Proofs From The BOOK are currently available free from Springer.
PsiQuad 3-colorable
Stan Wagon has shown that my Psiquad tiling is likely 3-colorable.
PsiQuad 3-color
A4 Paper
Here's one of those solutions that's so mathematically perfect that I can't believe that I'm the one that found it. All rectangles are similar to the outer rectangle, which has area 200. Theoretically, 50, 64, 72, 81, 98, 100, 121, 128, 144 can be added to get a similar rectangle with area 1058. I picked this set because there were lots of potential edge-matches, but I got forced to a solution where no edges matched.
A4 Dissection
Didrafters and Taco Chips
George Sicherman: The below shapes are the 13 didrafters. Arrange 8 of the 13 didrafters to make a solid rectangle. Then make nine convex shapes. Then add a few more pieces for the Taco Chips puzzles.
Food Menu
I found this food menu amusing.
Dump Truck
Foster 084 A Graph
While at a meeting for making nice graph embeddings out of graphs that looked lousy, I made what is likely the first nice embedding for the Foster 084A graph.
Foster 084
Venn 7
Venn 7 from Andreas Gammel.
Venn 7
Extraordinary Conics
With five lines/points, find the conic.
Interstellar Blackboards
The Blackboards of Kip Thorne in the background of Interstellar.
1D CA Carpet
This 2D fractal can be made with a 1D Diagonal CA.
Rule 420
Unholey Deciamonds
Patrick Mark Hamlyn‎: The 444 unholey deciamonds in 37 congruent shapes. Took about 24 hours to develop piece tilability data for the BFS for this shape, and 43 minutes to tile the 37 shapes using that data.
Graceful Graphs
The first is the unique minimal heptic graceful graph. The next is a minimal octic graceful grace. The third is a graceful rook grid with all differences 1 to 48.
octic gracefulGraceful Rook
Conway Knot Problem
Lisa Piccirillo has solved the Conway Knot Problem.
Safety First
Things are getting safer here in Illinois ever since we went hardcore on it back in March.
Safety Last
Conway, Berlekamp, Guy
Soon after Conway, Berlekamp, and Guy signed this for me, Conway hunting me down. "Ed, we need to borrow our books!" So I handed them over. They met with A.K.Peters and Tom Rodgers to arrange for the second edition 4-volume set of Winning Ways. So long old friends. Remembering John Conway. Remembering Richard Guy and John Conway.
Winning Ways

Me Berlekamp and Guy

Material added 27 Feb 2020
New Tetrad
Walter Trump has found a new tetrad.
Sparse Rulers Mostly Solved ||||.......................|....|...|...|...|...|...|..|..|
I solved sparse rulers and it's all written up in my blog article Hitting All The Marks. Sparse rulers allow the measuring of all distances up to length n. B. Wichmann almost completely solved the problem in 1963. From Wichmann's sparse ruler recipe, Mn - rnd(√(3 n+9/4)) can be shown to usually be 0 or 1 (the excess). The length 58 ruler at the top is conjectured to be the longest sparse ruler with fewer marks than a Wichmann construction, but no proof is known. Adding a single mark to the end of Wichmann's construction allows the building of excess-01 rulers for all lengths over 257992. My code found solutions for hundreds of difficult smaller lengths such as 1792, 5657 and 16617 to establish that excess-01 rulers always exist.
ruler 50
Orchard Problem
In a previous blog on the orchard problem I mention "If you do not like points at infinity, arrange 3 heptagons of 7 points to make a 4-configuration of 21 lines through 21 points." In the next paragraph I mention the record for 22 points. That record was beaten Dec 2019. This new solution is the 21 point 4-configuration with an extra tree at the center. How could I have missed spotting that?
Submarine Turmite
At github Turmites are many results on 2-D Turing Machines. One of them is what I call the Submarine turmite. Here it is at 7 trillion steps.
{{{1,2,0}, {2,1,0}, {3,4,0}, {1,1,0}}}
101 Enigmatic Puzzles
101 Enigmatic Puzzles by frequent contributor Mark Wolf is now available at Amazon.
Material added 26 Jan 2020
New Heesch Tilings found by Craig S. Kaplan
Some new Heesch tilings. For more, see the Numberphile Heecsh video. Craig also made the Good Fences app.
7 hex10iamond17 Omino
17 omino11 hex13 hex
vZome 7.0
vZome 7 is available for download, and also has a launch video.
Different Rectangles
Can a 3D object be made with different rectangles? Yes.
thirty rectangles
Star Polytopes
Nan Ma has been making excellent visualizations of star polytopes.
1161 Unholey Hendeciamonds
Patrick Hamlyn can divide these into 43 symmetric shapes.
Balanced Magic Square
If weights of the same amount are substituted for the numbers of each square, the balance point will be at the exact center of each square. The first is a magic square, the others are its 2nd, 3rd, and 4th power. 48 order-4 magic squares remain balanced under squaring, but this is the unique square to remain balanced with the 3rd and 4th power.
balanced magic
6-punch Triangles
There are 24 ways to put spots on the corners/edges of an equilateral triangle, with 1-sided rotational symmetry. They fit nicely into a hexagon with matching. With 2-sided rotational symmetry, there are 20 ways -- can they make an icosahedron with edge matching? Bryce Herdt: "First, each vertex must be surrounded by either 0 or 5 corner holes, and there must be six of each vertex. There are four tiles with no corner holes; these must share the six no-hole vertices, and can do so in one of two ways. Note the four tiles have twelve corners total. By the Pigeonhole Principle, we can say that at least one vertex belongs to two or more triangles. But since each vertex has only five neighboring vertices, then whether or not the two triangles just mentioned share an edge, there must be two triangles that actually do share an edge. Repeating this logic shows that three, then all four triangles with solid corners are connected in one tetriamond. This can't be the C shape, as that would make five faces with no corner holes, but the other two shapes are fine. With that fiddly part out of the way, note that in either case, the four solid-corner tiles form a single cluster with six perimeter edges and six internal edges in three pairs. These four tiles have six holes among their edges, and an even number of these must be internal, leaving an even number (two or four) for the perimeter. Now consider the six triangles with one corner hole. Call the edge opposite this hole the "base"; these bases ought to match up to the cluster's six edges. But three of these bases have holes, and three do not. There is no way to match the three holes to the cluster's even number of perimeter holes, QED."
dots in triangle
5-touch hexominoes
Every hexomino touches 5 others, by Erich Friedman. I didn't know that was possible. At the link you can see how George Sicherman did it with pentominoes.
A perfect triangle dissection
Triangles of size 2-10 pack into a 16x18 rectangle perfectly.
triangles 2-10
Infinite Ammann Chairs
Infinite sequentially-sized Ammann Chairs can be packed into a rectangle.
Infinite Ammann
No Shared Edge Dissection
At Triangle dissection, no shared edges, I have seven solutions. Are there others?
No shared edge triangle
Pi from Menger
Squeezing Pi from a Menger Sponge worked out well, and was put into standupmaths.
Puzzle Box
The Puzzle Box series has a few dozen puzzles by me.
Screw Crossword
After making a demo on Ponting's Squares, I wondered how it might look like as a crossword.
warped Crossword
Material added 12 Jan 2020
Plastic Pentagon
Let p=1.32472... be the root of x^3 - x - 1, the plastic constant. The points in the figure are pairs of triples (a,b,c) used in sign(k) sqrt(|k|) where k=a p^0 +b p^1 + c p^2. Numbers on the lines represent integer powers of square roots of the plastic constant. In 3D there is a 19 point powered clique set.
plastic powers
Every positive integer is a sum of three palindromes
Every positive integer is a sum of three palindromes, Javier Cilleruelo, Florian Luca, Lewis Baxter:
Biggest Little Polyhedra
Biggest Little Polyhedra. A talk I gave at the G4G conference based on the Demonstartion Biggest Little Polyhedron.
The 34 canonical heptahedra.
Almost Perfect Tan Solution
This is a complete set of triangles with side 1 to 11, and the same triangles scaled up by sqrt(2). Just a small square hole remaining. Edit: George Sicherman: By the way, your 1..11 + 1q..11q can be done as a 20 x 38 rectangle.
tan1 to 11
Magic Areas
Magic square with areas.
magic square of areas
Mondrian Square 26
A size 26 square divided into rectangles of area 18 to 24, each used twice in different orientations. As a puzzle, do the same with the 10x10 square.
mondrian twice
51 points in 51 lines of 4
A 51 point/line 4-configuration. The equations of the points are the same as the equations of the lines.
4-config 51
tiling a rectangle with the smallest number of squares
Let f(m,n) be the minimal squares to tile a rectangle. Can an integer multiple be tiled with fewer squares? I give a few thousand possible examples.
Heilbronn 13
An improvement to a solution by Peter Karpov. My coordinates: {{0, 0.0992502414}, {0, 1}, {0.0879381177, 0}, {0.6551614146, 1}, {0.7485503739, 0}, {1, 0.4613325715}, {0.96481495015, 0.0876289126}, {0.0879381177, 0.6145067772}, {0.8969384849, 0.90254564724}, {0.3450133066, 0.90150725976}, {0.5001806186, 0.1492347859}, {0.7613458201, 0.4419966218}, {0.3284895496, 0.6333566986}}.
27 lines of 4
25 points in 27 lines of 4.
27 in 25
New Yorker Error
This ad by Architectural Digest in the January 6, 2020 New Yorker magazine has a huge error.
st petersberg
Integer Partitions
Connecting Distinct and Complete Integer Partitions by George Beck has some nice new relations.
Square into 46 45°-60°-75° Triangles
A square can be divided into 46 45°-60°-75° triangles.
Mondrian Cubes
There are 11 distinct cuboids of volume 25 to 36 that can fit into a 7^3 box. Turns out they all fit perfectly in a unique way.
There are 13 distinct cuboids of volume 12 to 24 that can fit into a 6^3 box. Leave out the 1X3x5 and they all fit perfectly in a unique way.
mondrian cubes
Contiguous Partridge Tiling
George Sicherman: The monodom also has a contiguous partridge tiling (not reverse partridge)! These are hard to find.
contiguous partridge
Material added 2 Jan 2020
Colonel's New Year Puzzle
George Sicherman: You can download my 2020 New Year Puzzle. My best regards to you all, and wishes for a very happy new year! (George also has a new page on Pentomino Pair Oddities)
Order-6 Configuration
At 96_6 Configuration, a discovery of L. W. Berman.
Angle Numbers
Each number has that many angles.
Angle numbers
12 Octagon Toroid
It's possible to make a toroid with 12 octagons. By Ivan Neretin. Part of Minimal surfaces for planar octagons and nonagons.
octagon toroid
Error-correcting Codes
I like the free book Error-Correction Coding and Decoding. Many of the SpringerOpen books are free.
Canonical Tetragonal Antiwedge Hexahedron
The weirdest of the seven hexahedra, the tetragonal antiwedge, is a self-dual polyhedron that has a volume of 3.141 in its canonical form.
Four-Chromatic, No 4-cycles
Part of 4-chromatic unit distance graph with no 4-cycles.
4 chromatic
Orchard Problem
As a part of Cultivating New Solutions for the Orchard-Planting Problem, I found a new sporadic solution:
orchard 16
Space-filling Polyhedra
I'd love to see lots of polyhedra based on On Space Groups and Dirichlet-Voronoi Stereohedra.
Fractal Cow-Nautilus
One of the possibilities of the Narayana Cow Triangle Fractal.
cow nautilus
A4 Paper Dissection
I found this dissection of an area 200 A4 rectangle into smaller A4 rectangles. Rectangle dissection into similar rectangles. Related is the Delian Brick. I also have this long list of A4 Dissections from Patrick Hamlyn.
A4 rects
Ghee Beom Kim
I really like the math art of Ghee Beom Kim.
Material added 1 Jan 2020
I'm in Art of Computer Programming
A snapshot from Art of Computer Programming, Volume 4. (Pegg). Here's Knuth's Christmas Lecture.
Knuth book
Shattering the Plane
I've found a lot of new simple substitution tilings. These are also available as demo, Substitution Tilings. And a long image list of tilings. Related are Power Clique polyhedra, Wheels of Powered Triangles, and Mersenne Twister and Friends.
psi quad
Sparse Rulers
I solved the upper bound problem for Sparse Rulers. Soon to be a blog article and proof. I also have a few demos: Sparse Rulers and Wichmann-like Rulers. I wrote Excess01Ruler that can make a sparse ruler for any length. The key to solving the problem is what NJA Sloane called "Dark Satanic Mills on a Cloudy Day". Also, there is my new demo Wichmann Columns.
dark clouds
Venn Illusion
Akiyoshi Kitaoka and Ed Pegg Jr. Based on Venn-5.
The left black circle appears to be smaller than the right one, though they are the same size. Probably won't be an Illusion of the Year.
venn illusion
Unit Distance Graphs
In Dec 2017 I posted Moser Spindles, Golomb Graphs and Root 33. Aubrey de Grey really liked it and had some ideas for improvements. He then managed to solve the Hadwiger–Nelson problem. The latest 5-color graph is down to 510 points.
Tetrahedra and Other Polyhedra
Solid and Dihedral Angles of a Tetrahedron, Canonical Polyhedra, Similarohedrons, and the Tetartoid. The last program was used to make skew dice.
Mondrian Art Problem
Mondrian Art Problem, I solved it with Blanche dissections. My solutions ended up on Numberphile. I found weird bounds in Mondrian Art Problem Upper Bound for defect. I also found Possible Counterexamples to the Minimal Squaring Conjecture. Ponting Square Packing.
Mondrian Art
Sum of Three Cubes
569936821221962380720³ + (-569936821113563493509)³ + (-472715493453327032)³ = 3
Crossing Numbers 10&11
There are no Cubic Graphs on 26 Vertices with Crossing Number 10 or 11.
Incredible Rep-Tile
Dmitry Mekhontsev‎ ( found this order 8 3D rep-tile. He has a whole page of rep-8 tilings.
Graceful Graphs
The Shrikhande Graph is graceful. And so is the toroidial graph next to it. PUZZLE: Fill in the blank squares with 4 integer values from 2 to 21 so that the 22 queen move differences are the values 1 to 22. Hint to make this solvable by logic: Two of the other numbers are 17 and 21. I also made more sextic toroidal graphs.
Shrikgraceful toroid
rook puzzle
Neil Sloane article at Quanta. And entry A326499 is mine. Also A307450.
Rolling Polyhedra Graphs
I've figured out a lot of Rolling Polyhedra Graphs, including all of the deltahedra. I haven't figured out a good embedding for the dodecahedron. I also like various new mirrored polyhedron sculptures. (and more mirrors) I also found some weird-rolling tetrahedra.
Nauru graph
Cut the Knot
Alexander Bogomolny has died. Working with the family, I've been updating Cut-the-Knot and keeping the site alive.
Wang Loops
I like this puzzle by Aaron Wang.
Aaron Wang Loops
New Coverings Found
L's in Circles, Triangles in Circles, and Circles carrying Circles.
Five Ell
I wondered if there was a configuration of barycentric points and lines where the points and lines were the same. I found a lot of 24_3 solutions and a 27_4 solution.
Facebook Puzzle Fun
From the Puzzle Fun page. Patrick Mark Hamlyn‎: The 196 one-sided heptominoes arranged into 28 congruent shapes of 7 pieces each, 3-colored.
hepto 28Difference Set
The {0,3,4,9,11}_21 difference set in circles and lines. All points 0-20 are connected by a circle or line. Can anyone do the {0,1,3,8,12,18}_31 (0-30) difference set?
diff set
I'm doing okay. Here's a picture of me in my office at my latest birthday. Most of my recent work has been at Wolfram Community, Wolfram Demonstrations, Facebook, or StackExchange. Looks like I'm 6 years behind. So much more material to post.
pegg cake
And one of me at the Periodic Table Table
Puzzlium has worked with a lot of people recently. One by me from their Puzzle Box series: Arrange the numbers 1 to 9 in the boxes below so that each line of three boxes sums to 14. Three numbers have already been placed.
one to nine
Highway Interchange
I liked this Highway Interchange.
Robert Abbott
Robert Abbott has passed away, leaving behind some great games and mazes like Where Are the Cows? For Elwyn Berlekamp, I rather like the Triangle game.
Black Holes
Neutron Star Map. Center of the Galaxy. Black Hole Visualization.
Material added 1 Jan 2019
Print 3D Puzzles
Some discussion of Oskar van Deventer and George Miller is in the article The Puzzle Masters.
Ever See A Tree in a Movie?
Trees for movies are being outsourced to SpeedTree now.
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.
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.
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
Puzzle Solving Cockatoos
Cockatoos are rather good at solving parrot-friendly puzzles.
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.
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?
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.