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Material added 28 February 05

I've long found triangle centers fascinating. Eric Solomon, the inventor of Black Box and many other games, has put together a triangle centers applet that is quite fascinating. More Kimberling Centers can be seen at MathWorld.

225964951-1 is the 42nd known Mersenne prime. MathWorld has a write-up.

The polyforms Yahoo group recently posted a nice supertan rectangle:

New results have been found for the Busy Beaver problem. One particularly taunting discovery is the Busy Beaver Surprise in a Box, which deals with a simple 3-3 machine on a tape of length 60.

Some new wooden tetrahedral puzzles are available. A more difficult puzzle, but a highly enjoyable one, is The Great Circle Challenge. I got a copy of it at Enasco. It's thankfully quite forgiving, the designers went to great lengths to make this puzzle as simple as possible.

Nyles Heise has made a Wireworld Multiplier Train. David Wilson sent me an interesting picture based on the Ulam Sequence.

5×5 Go has been solved by computer, in the thesis of Erik van der Werf. Copies of the thesis can be purchased from him for €17.50. His results are very similar to those of Ted Drange, but there were still a few surprises. A Slashdot writeup is available.

George Sicherman: Yesterday I found that the Lobster and Snake hexiamonds are compatible.

Issue 2 of dice2mice is available. is a nice source for strategy gameplay.

If you have some Lego, you can make an MIT-designed Hybrid Burr.

Material added 16 February 05

Puzzle Japan ( is releasing three new logic puzzles every day, which you can subscribe to for $4 a month. It is maintained by noted puzzlemaker Junichi Yananose. The site is sponsored by Nikoli, which has been the leading publisher of puzzle books in Japan since August 1980. has announced their choices for best 64k demos and 4K demos of 2004. They are pretty close to the picks in my demoscene column from August 2004.

Oskar van Deventer: Markus Goetz made two very interesting JAVA applets. -A ternary variation of Thinkfun's Elephans Spin Out -A triangular version of Thinkfun's Rush Hour. [Ed -- Those triangle puzzles get really hard.]

Bryce Herdt: I recently found a triangular number with the property that its digits can be broken down into the sides of a Pythagorean right triangle. I'm wondering if there are more like it. Answer and Solvers.

A comprehensive article on every possible math joke has been made made available at

Bob Kraus: I have just posted a new puzzle "Balls and Jars" on my website. It can actually be solved with only pencil and paper if desired. It uses some simple math as well as T-F logic. Perhaps you might like to mention it on your site.

Col. G. L. Sicherman: There seems to be a vogue for uniform dissection puzzles. Can you dissect this figure into three congruent polyhexes? (Answer and Solvers)

Tony Noe: Ed, we've seen lots of info about Gaussian primes. However, I've never seen anything about Gaussian squares. A search of the Internet found nothing. I call z a Gaussian square if z = u p1^e1 ... pn^en where u is a unit (1,-1,i,-i), pk is a Gaussian prime, and 2 | ek. The attached graph is very interesting. It shows that many Gaussian squares are near families of parabolas, two of which are shown in red. Also note the many near-circles formed by 8 squares.

What is wrong with this picture? (It's a gorgeous real life zinger set in a slightly odd living room).

USA Today wrote a very nice article about Numb3rs. Eagle-eyed viewers may have noticed a Riemann Zeta Function Poster in the background during an office scene where Charlie research to the student's father.

Hu Zhe has written a Mathematica notebook that does "the best card trick." It's great to have this for practicing Fitch's Card Trick.

Eric Solomon's Cow Puzzle suggests a simple task -- maximize the number of cows in a single field. The best known answer has 21 cows in a single field. Can you match or beat that? If so, write to Eric at his site. It's worth noting that the 25 pieces are mathematically complete. Update from Eric Solomon: "Your link to my 'Cow puzzle' generated around 2000 visits. James Willson of Dallas has proved that 23 cows in one field is an upper bound, and previously provided a solution which is now shown below the applet. The puzzle is now effectively 'dead', but I'll leave it up for a while."

Patrick Hamlyn had a dissection of the 6x6 square sans dominoes in two opposite corners that everybody missed. The complexities of these makes me wonder about the 8x8 square sans dominoes in two opposite corners. Patrick provides the answer. I've also put Patrick's investigations together in a big page. Minami Kawasaki sent me a gorgeous .xls rendition of octomino crackers. Warren Phillips managed to find a solution Patrick missed. How many solutions does this have?

Material added 02 February 05

Arrange 33 coins on an 11×11 board, so that no line of 4-in-a-row exists. Leonel Robert has an excellent webpage about this problem. Also, I've located the The No-Three-in-Line Problem page, maintained by Achim Flammenkamp.

The Goldbug Variations, in the latest issue of the Mathematical Intelligencer, is well worth a look. You can also see the paper at Michael Kleber's website. A highlight is 3 million stages of the Rotor-router model - a spectacular image. For more Propp Circles, visit his page. There is also a Rotor-router applet.

Tommy asks for a proof that the following equation goes to neither zero or infinity. (A certain contour integral is very enlightening.) Answer and Solvers.

The latest pentomino competition asks for the longest corridor that can be made with pentominoes and tetrominoes.

Keith Devlin wrote up a great column on Numb3rs, filled with stuff I didn't know. Suresh Venkatasubramanian adds his own comments on the Geomblog.

Grout is an edge-matching tile program. Aaron Kellner has some very symmetrical scuptures. The Complexity Zoo has added a wall poster of complexity.

Stuart Anderson: I have recently added a new line of tilings to my website. Simple Perfect Right-Angled Isosceles Triangled Squares or SPRAITS's as the abbreviation is known. The theory of SPRAITS's is related to that of Squared Rectangles and Squared Squares and was the subject of a paper recently published by two of the original authors of the 1940 'The dissection of rectangles into squares' paper, in collaboration with JD Skinner. Skinner's discovery in the 1990's of a perfect RAIT square dissection inspired the research paper ; J.D. Skinner II, C.A.B. Smith, and W.T. Tutte, On the Dissection of Rectangles into Right-Angled Isosceles Triangles, Journal of Combinatorial Theory Series B 80 (2000), 277-319. Unfortunately CAB Smith and Bill Tutte have both passed away since that paper was published in 2000. However Skinner and Geoffrey H. Morley have been working on SPRAIT dissections for over a decade and have compiled a large catalog of SPRAITS's tilings. Geoffrey H. Morley has discovered the largest number of SPRAIT tilings. He has made his catalog available to me. It contains over 14600 SPRAITS's. I have created javascript code to display all known SPRAITS's on my website and they can be selected and displayed in manner similar to the Squared Squares and Rectangles on my site at

Material added 23 January 05

PQRST 12 has started with another great set of 10 puzzles. The puzzle blog has started. Crocopuzzle is a great source for anyone that needs custom logic puzzles. The Annual MIT Mystery Hunt has started. has a collection of World Puzzle Championship puzzles.

Russell Towle has done some very interesting work with zonotiles. At the Colored Zonotiles page, you can download "Fifty Decagons".

A Lego Rubik's cube has been built. Various Slashdot postings: Fixing the uncool image of Physics, Breakthrough in JPEG compression, and Newsy Numbers. I devoted an column to NUMB3RS.

Hew Wolff has published an answer to GLAT Problem 19.

Luke Pebody has solved my Smoothness problem. 1, 4, 675, 2600, 83520, 114376, 18264064. Largest integer k and k+1 that are both Gaussian-n smooth, starting at n=2. 7+24i, 119+120i, 110+750i, 42470+29835i, Largest complex k and k+1 that are both Gaussian-n smooth, starting at n=2. 42470+29835i = i(1+2i)(2+i)6(5+2i)2(5+4i). 42471+29835i = i(1+i)(2+3i)(2+5i)2(3)3(3+2i)2. 18264064 = -i(1+i)24(2+3i)(3+2i)2(7)3. 18264065 = -i(1+2i)(2+i)(2+7i)(4+5i)3(5+4i)3(7+2i).

Titus Piezas III has written a fascinating article on e^(pi sqrt(n)). Among the most incredible things there was one I didn't know. Let y be the real root of x3-6x2+4x-2=0. Then y24-24 and eπ sqrt(163) match for the first 32 decimal places. The reason for this is related to the Monster Group, and the perfect packings of spheres in the 24th dimension. I had no idea that the simple polynomial x3-6x2+4x-2 was so important.

Alexey Tarasov has found a Edge Heesch Tiling of 6.

Peter Esser: I have just found some constructions using the tetrahexes and the pentahexes cut from a tricoloured grid.

Dr. Frank Harary has passed away at the age of 84. Dr. Harary was widely recognized as one of the pioneers of modern graph theory, a discipline of mathematics he helped found, popularize and revitalize. An author of numerous books and articles, his book ``Graph Theory'' published in 1969, is a modern classic that helped define, develop, direct and shape the field of modern graph theory. The Frank Harary webpage has many more details. He was very active in Recreational Mathematics.

David Millar (with account drunkoffdietpepsitwist): "In my spare time this week I've been coming up with a lot of 6x6 rolling block puzzles (with a 3x1 block). A lot of them are simple but when I come up with a valid solution path, I try to lengthen it without adding extra obstacles. I designed some layouts for puzzles and printed them to Avery 3x5 index cards and cut out 1 1/8" x 3/8" x 3/8" pieces of wood for pieces, and so far my computer programming teacher and classmates really like playing the puzzles." Here are four of his "Bury the Treasure" puzzles: (1, 2, 3, 4).

Material added 11 January 05

I've seen the first episode of NUMB3RS, a new TV show by CBS, and I thought it was quite good, with a decent amount of math, and a believable mathematician. It will premeire Sunday, January 23, 10PM ET. The below is a screenshot of a problem in the first episode -- you have ten random points that all came from some central point, like ten waterdrops from a sprinkler. Find the point of origin.

Arrange 9 queens and 1 pawn on a chessboard so that no two queens attack each other. A solution is here.

Wei Ji Ma has a great page on Generalized Tic-Tac-Toe. Some findings: 3×3+one square is a first player win in 3-in-a-row. 5×5+one square is a first player win in 4-in-a-row. 4×9 is a first player win in 4-in-a-row. What 12, 11, 10 and 9-ominoes make interesting games in 3-in-a-row?

28593+6426i = (1+2i)5(3)3(3+2i)(5+2i). 28594+6426i = (1+i)3(1+4i)(2+3i)(4+i)2(5+4i)2. The are two large adjacent G5-smooth numbers. Is there a larger pair? I thought I'd settled G5-smooth pairs weeks ago, but these numbers keep surprising me.

Congratulations to all at Seventeen or Bust for finding that 28433×2^7830457+1 is prime. This has 2,357,207 digits and is the fourth largest known prime. There are now only ten k values remaining to prove Sierpinski's conjecture.

Many games have recently been updated at Zillions of Games. Akron is the current game in the spotlight. Speaking of Akron, Connection Games: Variations on a Theme has now released by AK Peters. Cameron Browne did a great job covering the whole genre of connection games.

A powerful CAD system, BRL-CAD, has been released for free.

My site's getting hammered hard by Metafilter (10 Jan 2005) and an army of related bloggers. (welcome, Metafilter)! If I was still on Earthlink, I'd need to pay about $7000 in overage charges. Fortunately, I'm with Yahoo now, so if I go over my 200GB bandwidth limit this month, each extra GB will only cost $1, instead of the $100 that Earthlink charges.

Material added 02 January 05

I've put Sam Loyd's Cyclopedia of 5000 Puzzles, Tricks, and Conundrums, sometimes called the Cyclopedia of Puzzles, entirely online. You can view individual pages, or download a zipped file of the entire book. Page 138 is corrupted in the zip, and I needed to rescan page 80. I'll try to fix those. Loyd's works are in the public domain-- I won't lay any claim to them for having scanned them. Feel free to use Loyd's pages files however you wish. I did an MAA column on some of the highlights.

Emrehan Halici has launched -- a weekly problem and games website.

David Millar greatly enjoyed getting River Crossing 1&2 as a Christmas present, so he made a new challenge for it. Serhiy Grabarchuk sent a different challenge, involving 7 coins.


Claudio Baiocchi sent me foffofx.TeX for the foffofx page, which winedt readily turns into foffofx.pdf.

What were the top 10 movies of the year? I rather like the huge chart at

Erich Friedman notes that each of the following polyforms has the center of gravity centered on one of the tiles. What larger polyforms without rotational symmetry possess this property? Answers and Solvers. (11-ominoes, 12-ominoes)

M. Oskar van Deventer: I would like to let you and your readers in on a secret. The secret is that plastic Rokenbok blocks match perfectly with the Thinkfun TipOver puzzle. By gluing Rokenbok blocks together, one can easily build and play e.g. Adrian Fisher and Erich Friedman's Rolling-Slab Mazes and Richard Tucker’s Rolling-Megalith Maze (see Robert Abbott's page). Obstacles are easily made by cutting a Rokenbok blocks in half. The fact that TipOver matches so well with Rokenbok blocks is not a coincidence. Working on the TipOver mechanism, I realised that the board would be suitable for rolling-block puzzles. Knowing that rolling-block puzzles are a bit too abstract for Thinkfun, I decided to match the dimensions of the TipOver prototype with Rokenbok blocks. So now, all components for a rolling-block puzzles are commercially available, albeit from a combination of vendors. The only thing that is still missing is a set of 40 challenges for all rolling-block enthusiasts ...

Richard Bean released a short paper on the 69352859712417 positions that are possible after black's fifth move in chess. His complete lists of fast mates are also quite interesting.

Material added 23 December 04

For the new year, you might like to try out a 12-sided calendar -- available in pentagonal or rhombic!

Erich Friedman: I made many generalized full house puzzles. Yellow squares must be passed through twice, once vertically and once horizontally, in either order. When you hit a green square, you must turn. As before, you have to start somewhere and fill every non-black square. (More Full house puzzles are available at and Erich's Puzzle Palace.) (Answer and Solvers)

Harry Nelson sent me a lovely Alphametic puzzle. Answer and Solvers.

                E D
P E G G ) M A T H P
          U Z . .  
            Z . . .
            . E . .
            . 1 . .

A number is 11-smooth if it has no prime factors > 11. For example, 11859210 = 2×34×5×114 is 11-smooth. It could also be called 19-smooth, since it has no prime factors over 19. 11859211=7×13×194 is also 19-smooth. This is the last pair of consecutive 19-smooth numbers (see A002072 for higher consecutive k-smooth numbers). I thought 2×34×5×114 +1= 7×13×194 was pretty neat, so I found 7949+9650ι = ι×(2+5ι)4×(3+2ι)×(4+ι) = -ι×(1+ι)3×(1+2ι)3×(2+ι)2×(2+3ι)2×(6+ι) - 1. Both 7949+9650ι and 7950+9650ι can be expressed as the product of small Gaussian integers, which suggests a Gaussian extension to the concept of smoothness. I'll say that 7950+9650ι is G6-smooth, since all of its prime Gaussian factors have real and imaginary parts ≤6. A larger adjacent G6-smooth pair is the following: 10259+8570ι = ι×(1+6ι)×(2+3ι)2×(4+ι)×(4+5ι)2 = -ι×(1+ι)2×(1+2ι)4×(1+4ι)×(2+ι)×(5+2ι)2 - 1.

On page 199 of The Phantom Tollbooth, the Mathemagician gives Milo a letter that he sent to King Azaz. I tried to figure out if anyone has ever translated this, but was unsuccessful. I suspect that the opening is "King Azaz," but wasn't able to discern more. Can you decipher the Mathemagician's letter? Comments. This has been discussed at rec.puzzles a few times, without a solution being posted.

   "He's much too unreasonable," interrupted the Mathemagician again. "Why, just last month I sent him a very friendly letter, which he never had the courtesy to answer. See for yourself."
   He handed Milo a copy of the letter, which read:
           4738 1919,
              667 394017 5841 62589
           85371 14 39588 7190434 203
           27689 57131 481206.
                                 5864 98053,

   "But maybe he doesn't understand numbers," said Milo, who found it a little difficult to read himself.

Martin Gardner's Mathematical Games: The Entire Collection of His Scientific American Columns is now available for pre-order at Barnes&Noble. I'm still waiting on Amazon and to list it (ISBN 0883855453). I haven't gotten my hands on a copy yet. It will be released by on CD rom in February 2005. Catalog Code: TDG/FW04. ISBN 0-88385-545-3. List Price: $54.95. This is all 15 of Martin's Mathematical Games books on a CD-Rom. These are the best books ever written. I'd like to do a "behind the scenes" look at Mathematical Games for an article. Martin did lots, but he avidly used a lot of help from hundreds of people. I'd like to write an article about the background help. If you ever assisted in one of Martin's columns, I'd like to hear from you -- a note about what you did. I'll use these notes in my article. Marv Schaefer: Bad news. Yet another technical publication snag according to Don Albers at MAA, and he says it's going to take "more than a few minutes to get around this one". The new anticipated publication date is February.

Various programs I use a lot have all gotten updates recently. OpenOffice 1.1.4 is a great word processor, spreadsheet, and drawing program, which I use for opening all Microsoft documents. (I like it more than Microsoft Office.) WinEdt is an inexpensive TeX editor. FireFox 1.0 has been out for a few weeks -- it makes webbrowsing much nicer. 7Zip is a free compression program. ZoneAlarm is a free firewall that has blocked 487959 attempts to access my computer. If you'd like to clean and speed up a Windows system for 2005, download FireFox, AdAware, ZoneAlarm, and Grisoft. (use the free versions, they're fine). Disconnect from the internet, install all four programs. Run AdAware and Grisoft, and eliminate any bugs and spyware. Use Firefox to browse the web with ZoneAlarm as the firewall. Finish up by researching mysterious start-up programs at Pac's Portal.

Many new chess logic puzzles are available at the Retrograde Analysis Corner. I rather like the presentation of geometry facts at Numericana. The Complete Sherlock Holmes is handy.

Did you know that the specific fraction 450359962737049600/450283905890997363 is named a monzisma? A List of Intervals is available for all of the fractions which have names.

The state of mathematics education in the world has recently been discussed on SlashDot: Math Skills Survey Shows U.S. Lags Behind. A shorter article is at Lessons in Perspective.

Nyles Heise and Karl Scherer have successfully implemented a WireWorld multiplier in 3-tick logic. This scheme allows for two 32-bit binary numbers to be multiplied in 6000 ticks.

Any group looking for a great party game over the holidays should try out Mafia. The new single-page PDF is quite nice. An online version of the game is at

Material added 13 December 04

Over at, I have a new Math Games column on Sliding-block Puzzles. I decided to rebut the recent claims about a "hardest puzzle" made in The Economist. While preparing the article, I accidently chanced upon a 68-move 4-piece puzzle. Nick Baxter added it to his new 4x5 puzzle page. Jared Weinberger's has some hard constructions by Bob Henderson. I should also note the 2-piece, 6x6 record. Incidently, Ivars Peterson's coin packing column is also nice.

Erich Friedman: Divide a square into L tetrominoes of various sizes so that each L touches exactly 4 others. Answer and Solvers.

NetLogo 2.1, just released, is a very visual implementation of the Logo language. kseg allows an easy exploration of ruler&compass geometry. Group Explorer provides 3D pictures of various small group theory objects.

Material added 06 December 04

Mathematica 5.1 has been released. You can try a free trial, if you'd like. Last month, Springer published volumes 1&2 of Michael Trott's Mathematica Guidebooks, and it is doing so well that they are immediately publishing volumes 3&4. Michael has also made 380 pages of additional material available, with short pieces of code for making objects like the below image.

Tribute to a MatheMagician is available, the third book in the Gathering for Gardner series. At, you can get the first book from the series, The Mathmagician and the Pied Puzzler, for free. Another great book you'll soon be able to get from AKPeters is Connection Games, by Cameron Browne.

Patrick Hamlyn has done a number of polyomino dissection problems lately. Divide each holey square into the given number of identical pieces. For example, in the first one, divide the 4x4 square with the upper left corner missing into 3 identical pieces. The 4 piece puzzle has 3 solutions. Answer and Solvers. For perfect packings of this type, see the Rectifiable polyomino page, and Michael Reid's box collection. Also, see the Similar Divisions page.

Speaking of squares, I recently added a favicon, . While I was at it, I also made one for the site, .

The Economist recently wrote an article about the Quzzle, by Jim Lewis ( It then appeared on Slashdot. Jim states that this is the hardest 4x5 sliding block puzzle where one piece must get from one corner to another and all the pieces are 1x1, 1x2, or 2x2. Quzzle was one of the entries at the 2004 IPP Design Competition. Some of the best puzzles are at Nick Baxter's Sliding Block Puzzle Page. Taniguchi's solver and both have sliding block puzzle analyzers, in case you'd like to try making one.

An excellent primer on putting math into webpages is at In addition, the physics lab side of the site is worth a look.

The Poppy Seed Bagel theorem was recently in the news at Vanderbilt.

What can you make if you only have Uranium and Hydrogen? Anything, it turns out. An experiment launched uranium into liquid hydrogen, and pretty much got every element out of it.

It's my birthday soon, on Dec 7th. That's an astrological clue for a puzzle: how can you find the points that make a D10 -- a ten-sided die? Here's the answer, in ROT13 format: Fxrj cragntbany cevfz nobhg bevtva. Guvpxarff, clguntbehf, naq uggc://zngujbeyq.jbysenz.pbz/Fntvggn.ugzy.

The Torito puzzle shop has some nice boxed gifts, if you can pay in Yen. Gridworks is a very interesting logical game (and deserves its own short URL). George Olshevsky has made some 3-D nets of 4-D objects.

There are 30 ways to cut a tetrahex from a 3-colored hexagonal grid. Peter Esser shows them below. A smaller puzzle is to make a hexagon out of the 10 similarly constructed tri-hexes.

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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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