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Material added 13 Sep 04

I did a column for about the orbit for the Genesis mission. I set things up for Wolfram Research to watch the mission right over our Periodic Table, but far too early, it was all over. Still -- the mathematics behind the mission was interesting and flawless. has the latest news.

Nick Hobson runs the very nice Nick's Mathematical Puzzles: "A collection of puzzles ranging over geometry, probability, number theory, algebra, calculus, and logic. Hints are provided, along with answers, fully worked solutions, and links to related mathematical topics. Many of the puzzles are elementary in their statement, yet challenging. New puzzles are added on a regular basis."

85359^5 == 85282^5 + 28969^5 + 3183^5 + 55^5 was just found by Jim Frye. The last solution of this type was found by Lander and Parkin in 1967.

A little oddity I found is Sqrt[3109]+Pi (try it!). Some very large pictures have been drawn via GPS. A new version of Quantian is available. A book for the SIAM 100-Digit Challenge is now available.

Brian Trial: "I've written a crude program to fit consecutive cubes inside a larger cube. Nothing earth shattering here, but the pictures are interesting, and it's interesting to observe the packing behavior as the number of cubes becomes large. Enclosed is POV source for cubes 1-412 in a 1988 cube for a 92%+ packing density, and cubes 1-1009 in a 6496 cube for a 94%+ packing density."

Christian Blatter: "1. Print the attached file MacMahon.pdf on a color printer. 2. Cut out the six squares. 3. Choose an arbitrary one of these squares as model. Now comes the task: 4. Select four of the remaining squares and use them to compose a 2x2 square that is an enlarged copy of the chosen model. Maybe one of your readers has a computer with the power to create a really "fractal" version of it."

Peter Esser has made a very nice page about Polyspheres. He has many interesting polyform pages. Just recently, he put all the decominoes together: all decominoes: 4460 pieces without holes, 182 pieces with a single hole and 13 pieces with a double hole plus 5 additional single holes and 1 double hole = 46765 squares = 235x199.

Material added 26 Aug 04

Andrea Gilbert has put some new mazes on "Erich Friedman's full-house puzzles are superb, now you can try them clickmazes-style. James Hutton of Melbourne, Australia has produced a version of marble mazes for the Nokia phone, try the free download here. Last but not least, a few additions to Oskar's collection, including the haunted vending machine puzzle." Another set of Oskar mazes (Pathfinder) is at Andrea also sent me an advanced copy of River Crossing 2, which should be available in stores very soon.

The Online Encyclopedia of Integer Sequences is having a party.

AK, publisher of many recreational math books (Winning Ways, for example), has updated their website.

My latest column is about the Demoscene -- 64K or Less. I include my favorite 256B, 4K, and 64K programs, along with pictures, history, many great links, and explanation. Many excellent, tiny assembly programs are available, for your perusal.

For more mathematical graphics, Tim Rowley's page on SIGGRAPH 2004 papers has everything.

From the hard-to-move-pieces dept: "It is a truly colossal project, a $2.6 billion gamble on the game of chess. The sprawling desert emirate, which has already vowed to build the world's largest mall, theme park, archipelago of man-made islands and tallest building on earth, now says it aims to conquer the age-old world of chess by creating the 64,000 square meter International Chess City, which will feature 32 buildings designed to mirror the image of a traditional black and white game board, Dubai hopes to corner yet another niche industry by winning over millions of chess enthusiasts worldwide."

Wolfram Research (my employer) has released GUIKit (a free toolkit for building and running graphical user interfaces) and Publicon (a $150 product for making technical documents).

An interesting article on steganography (hiding messages within pictures) is at

Don Knuth: NIIHAU ± KAUAI ± OAHU ± MOLOKAI ± LANAI ± MAUI ± HAWAII = 0. That can be 0 or O at the end -- either alphanumeric has a unique solution. This is featured in the pre-release of The Art of Computer Programming, Volume 4.

Brian Trial mentions the Persistence of Vision Raytracer below, which just came out with a new version. From ( A new POV raytracing competition will run from now until mid-October. The first prize is a raytracer's dream machine worth more than US$10,000.

RSA-150 was recently factored, to show off a new factoring technique.

Would anyone like a email address? 2 Gigabytes storage, free, through the recently upgrade Yahoo Business Mail. Write to me if you would like one.

Material added 17 Aug 04

Richard Sabey recently found A = (1+9^-4^(7*6))^(3^(2^85)), an approximation to Euler's constant e=2.718281828... How many decimals of accuracy does this approximation have? a) 10 digits of accuracy, b) 86 digits of accuracy, c) 18457734525360901453873570 digits of accuracy, or d) none of the above? This was found as a part of Erich Friedman's monthly Math Magic.

Brian Trial (ourmail (at) I've enjoyed the entries on your web site concerning fitting consecutive squares inside the smallest possible square and I wondered if the idea could be extended to three dimensions. Enclosed I have a picture of consecutive cubes 1x1x1 to 69x69x69, orthogonally fitting inside a cube 186x186x186, for a packing density of 90.6%. The picture was created by using the ray trace program Persistence of Vision (, which has a nice tutorial, is pretty simple to program, and nicest of all, it's free. Enclosed is my POV source code as well, which also includes the exact coordinates of each cube. I suspect an orthogonal solution with a higher packing density is out there, though so far I have not managed to get 1x1x1 to 70x70x70 to fit inside 189x189x189. If someone submits a solution to you, you might want to sent it to me first, I have a program that checks for overlaps, and it kept me honest on many occasions that my 2 dimensional brain was deceived.

Brian Trial's packing of cubes 1-69 in a 186 cube. Full, expanded picture.

George Miller's website features many new 3D-printed puzzles, designed by Oskar van Deventer -- "Oskar's Exotics". Many different 3D printing technologies were investigated to produce these puzzles. All technologies build physical models layer by layer.

Inkjet printer technology that cures starch or plaster (3D color prints!). Glue printers that bind steel powder which is then sintered.
Lasers heaters that fuse nylon powder or liquid particles together. Laser cutters that build layered paper models.
Nozzles that deposit wax droplets for lost-wax casting. Extruders that deposit thin wires of ABS.

George Miller is using the last technique, the ABS extruder, which he describes as a "giant glue gun". All technologies are mainly used for rapid prototyping, as it only takes several hours to create accurate fully functioning prototypes. Each technology has its own specifications for resolution/accuracy, strength and (im)possibilities. All technologies are still rather expensive, both in terms of equipment and materials. Despite that, George predicts that in ten years many people will have a 3D printer next to their desk.  For now, this printer technology allows construction of puzzles that could not otherwise be built.

Tube Maze has deep undercuts, and a complex mesh of tubes. Leaky Membrane is a 3-dimensional periodic minimal surface (see takahashi-5).
Moby Maze is a completely "uncastable" maze based on a Mobius strip.
Turn Apart has circular dovetail connections in a Rubik-like form.
Snake Ball has helical grooves on a torus surface.
French Fries is a six-piece burr "with a twist".
Tanacube Too is a 7-piece tangrammic cube dissection.
Magno Maze is a 3-D rolling block maze.

And there are many more "Exotics" on display, in total 13 new 3D printed designs now. George Miller offers a limited edition of Oskar's Exotics for sale. There are two ways of looking at the price of the 3D printed puzzles. On the one hand, 3D-printing is the least expensive way to produce these puzzles. On the other hand, it is also the only way. Some of Oskar's less complicated designs have been made by Hanayama Toys and Bits&Pieces.


Material added 05 Aug 04

Results of the Nob Yoshigahara Puzzle Design Competition have been posted on Puzzle World. Congratulations to all of the winners, and to all that entered. That looks like a fine collection of new inventions.

I received a very old puzzle in the mail. It's the 5 rooms problem. Draw a path through the walls of all 5 room, without going through any wall twice, and withouth crossing your path. I spent a long time during the third grade on this problem. There is no need to send answers to it. I decided I would make an update to the problem -- two rooms have been added to the original house. Now, you must pass through all but two doors of the house, without crossing your path. In addition, you must alternate going through red and green doors. The first house has already been solved for you. Answers.

Kadon Enterprises is having a contest, Narrow Passage, to see who can make the longest snake with the hexominoes. Below is a snake of length 31 with the pentominoes, by Guenter Stertenbrink. Can you find a longer one? If you can find a good answer for the contest, send it on to Kadon.

My latest Math Games column is on Modern Burr Puzzles. Get that Boxed Burr from Bits&Pieces while it lasts. While putting the column together, I came across some puzzles by GarE Maxton, which features elaborate dissections of a cube into interlocking pieces made from many different types of metal. For different dissections, Rinus Roelofs has spiral dissections of a cube and a sphere into 2 equal pieces.

Material added 25 Jul 2004

Markus Frind, Paul Jobling and Paul Underwood discovered the worlds first 23 primes in arithmetic progression.
56,211,383,760,397 +K*44,546,738,095,860 for K =0 to 22.

Erich Friedman. For each puzzle, find the starting hexagon and draw a path that passes through each open hexagon exactly once.  For each segment in the path, you must go as far as possible, changing direction only when you are blocked by the grid's edge, a black hexagon, or a hexagon already visited.

I learned of the 96K Windows game .kkrieger recently, and tried it out from the .theprodukkt website. It's a first-person shooter type game, much like Doom2, with full 3D rendering, monsters, weapons, and soundtrack. I'd show a screenshot, but it's 377K, more than 3 times larger than the entire game. After checking out the game, I learned about the "demoscene" -- a program under 64K in size that produces a full 3-D movie, complete with a sountract. For example, I really liked Project Genesis and Beyond, along with everything else made by Conspiracy. I also loved the Kings of the Playground invitation to Evoke 2004. More info on demoscenes can be found through Minimalistic mathematical primitives create realistic effects, and top-notch data compression does the rest.

Speaking of data compression, Stephen Hawking has changed his mind about Black Holes. Data compressed by a singularity eventually escapes, but in unrecognizable form (lossy). Also, astronomers have determined that some black holes have torii. The Hubble Telescope has found a runaway black hole that moves 4 times as fast as any other star.

I've switched to the Firefox browser. Get Firefox Below is a screenshot of it in action. I use extensions Adblock, Basics, Compact Menu, FoxAmp, MiniT, QuickNote, Statusbar Clock, Tabbrowser Preferences, and Web Developer.

Quat is software for making three-dimensional fractals (Julia sets in quaternion space). The five part course What is Chaos? is available free.

Material added 13 Jul 2004

Google has launched a series of puzzles (spoilers).  Stage Two can be found at {first 10-digit prime found in consecutive digits of e}.com .

Cihan Altay: PQRST 10 -- Puzzle Competition starts on July 17th Saturday at 20:00 (GMT+02). You'll have one week to solve 10 puzzles.

Kiki the Nanobot is a nice 3D version of Sokoban, available for free at Sourceforge. 

John Gowland:  Herman Goldstine died, a colleague of von Neumann and one of the creators of the ENIAC computer.

William Waite: I thought I'd let you know about the new wooden puzzles at the Nemmelgeb Murr Import Shop: the DoubleCross (engrossing), the StellarBurr (sculptural), the HexTrivet (practical), and the CoasterToaster (practical).

In the news recently, a student asked Jeb Bush to name the angles of a 3-4-5 triangle.  He didn't have the answer on the top of his head.  Then the student proceeded to give the wrong answer: "The angles are 30-60-90." Many major news organizations then published the exchange, without posting the correct answer.  Didn't anyone at the New York Times pass basic trigonometry? I can excuse a person for not knowing 36.8698976458440212968556125591º and 53.1301023541559787031443874409º off the top of their head (I didn't know them), but everyone involved should have known the method (sohcahtoa).  So, what are the angles of a 3465 - 49552 - 49673 triangle?  As a slightly different question, can you find a triangle where each side has a length equal to the opposite angle in degrees? Solved by Earl Gose, Cpt. Lettuce, John Abreu, Alan Lemm, Chris Lomont, Juha Hyvönen, Joseph DeVincentis, Chris Lomont, Jon McLean, Carlos Gil, Jirka Kosina, Kirk Bresniker, David Stigant, Sandy Grace, James Mahaffey, Kimberly Pinkham, Bryce Herdt, Eugene Salamin, Shyam Santhanam, Alec Mihailovs, and Luke Pebody. Several gave a uniqueness proof using the Law of Sines. For a more difficult problem, find an exact solution for a right triangle where two of the sides match their opposite angles. Send Answer.

I would love to see some good progress made in the Progressive Packing Problem, now featured at Math Magic. The suprise solution of Shigeyoshi Kamakura still amazes me.

More by Erich Friedman.  Full House (as seen at the World Puzzle Championship Qualifier).  For each puzzle, find the starting square and draw a path moving horizontally and vertically that passes through each open square exactly once.  For each segment in the path, you must go as far as possible, changing direction only when you are blocked by the grid's edge, a black square, or a square already visited. Answers and Solvers. The analysis by Kirk Bresniker is quite interesting.

Full House ExampleFull House 7  Full House 8 Full House 9

Patrick Hamlyn solved a challenge problem using the 108 planar heptominoes.

108 planar heptominoes

Documents on Weaving, Textiles, and Related Topics isn't kidding.  There are hundreds of documents there discussing the mathematics behind weaving.

For Battleship Logic Problems, an excellent site is

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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 Other math mailing lists can be found here.

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