MathPuzzle.com

Material added 27 Oct 05

Michael Trott's 4 volume Mathematica Guidebooks now in print: With help from Eric Weisstein, I wrote up an article about Michael Trott's Mathematica Guidebooks. You can also see the Guidebooks site, or the SIAM Review (direct link). Michael has added a special gallery of images from the 4 volume set. If you've ever wanted to make beautiful mathematical graphics, this is the tome that will teach you everything.

Matchstick Cubes: Vladimir Zahoransky: I am sending you photos album, where you can see cubes made from matches without glue. It's a well known activity realized by seminaries. The creator here is Ivan Masaryk.

New Seventeen or Bust Prime: Congratulations to Richard Hassler and Seventeen Or Bust for finding another one - in a double check: http://www.seventeenorbust.com. That's 9 of 17. Halfway there? The last 8 will probably be much harder.
New Chess Record: For those interested in the numeric aspects of chess ... Marc Bourzutschky and Yakov Konoval have broken Lewis Stiller's record for the 'deepest' known chess win, which was a KRNKNN position with depth = 243. This is a KRRNKRR win with depth = 290. KRRNKRR was recommended by Noam Elkies as worth investigating early. As Noam recommended several other endgames, it is possible that 290 and even 300 will be beaten in 7-man space, and we may not have long to wait. See http://www.xs4all.nl/~timkr/chess2/diary.htm, item 298 to play through the moves.
Hexomino Oddities: George Sicherman: I've put up the Hexomino Oddities (mostly by Mike): Do you remember the Polyomino Oddities that Mike Reid and I worked on? More recently we've done some Polyhex Oddities, and some of them are elaborate.
Binary Dice, Sliding Puzzles: Ivars peterson has two excellent Recreational Math Columns at maa.org, Sliding Around and Binary Dice. All of his MathTrek Columns are good.
FoxTrot Mathematics: Lyman Hurd let me know about a FoxTrot cartoon with some math content. Bill Amend was kind enough to allow Eric Weisstein to add it to MathWorld.

Material added 12 Oct 05

Magic Multiplication Squares: About one century ago, sometime between 1905 and 1916, H.A. Sayles wrote about the 3x3 and 4x4 multiplicative magic squares with the smallest possible magic product (216 and 5040), identical on rows, columns, and main diagonals. Christian Boyer is apparently the first to find the smallest 5x5 multiplicative magic square with magic product 302400 (below).
   12 35   1 40 18
   36   2 24   7 25
   14 45 15   4   8
    5 16 42 30   3
   10   6 20   9 28
The 4x4 case is now the National Public Radio Puzzle of the Week: In a standard 4 by 4 magic square you arrange the digits from one to sixteen so each row, column and corner digital totals 34. This is a multiplication magic square: Arrange sixteen numbers in a four by four square so that the product of each row, column and corner to corner diagonal is 5,040. You can use any numbers you want. But they have to be whole numbers and you can’t repeat a number in the square. (And as a hint I’ll tell you the number in the upper left corner is 42.) If you can solve this by Thursday, October 13, send the answer to NPR. If you have commentary, especially on earlier history, email me. The history is hard to pin down on multiplicative magic squares -- Arnauld wrote about them in 1667, and the were published by Boris Kordiemski. Christian Boyer uncovered that Borkovitz & Hwang found a 6x6 square with constant 2000376000. Is a smaller 6x6 square possible? For 3x3, 4x4, and 5x5 semimagic squares, the minima are 120, 4320, 277200, found by Boyer, Pegg, and Boyer (but possibly found much earlier).
Multilingual Polyform Pages: Andrew Clarke: I've taken the opportunity of having the extra space to upgrade the site at recmath to become the original multilingual one. (See The Poly Pages.) It's mostly there now but it will take some time to get it all right. Thanks again for the opportunity of using the site. [Also at recmath.org, Al Zimmermann's Programming Contests have now been fully archived. If you have a math site that you'd like to have considered for permanent storage, email me.]
New Prime records: Cletus Emmanuel: Here is a new Kynea Prime.... (2^281621+1)^2-2. There's a new Fibonacci Prime as well, U(50833), using the same techniques John Renze used successfully to prove the primality of U(37511). The CHG technique has now been used to prove over 20 large primes.
Numb3rs Lecture and Hidden Code: Read about the Gary Lorden Lecture on Numb3rs.
Chess Sudoku: Peter Taylor: I thought your suggested Chess Sudoku sounded fun, so I've made some. It's actually not a very good design space, because there are only 32 distinct solutions. I'm quite happy for you to include one in your page. Answer. [In a Chess Sudoku, numbers 1-8 and a chess piece are in every row, column, and region. Each chess piece attacks all 8 numbers.]

Kobon Triangle breakthrough: Toshitaka Suzuki has found the 15-line Kobon Triangles. That is, with fifteen lines, what's the most triangles you can have that don't share any area? An upper bound proven by Tamura is that no more than 65 triangles could be made with 15 lines.

World Puzzle Championships: The 14th World Puzzle Championships, run by the World Puzzle Federation (WPF), will be held in Eger, Hungary, from 8th to 13th October 2005. 29 nations are taking part. (Blog) More news is on puzzleratings.org.
PuzzleHack: The 2005 ACM PuzzleHack has started. http://www.acm.uiuc.edu/puzzle/
Diamond Solitaire: George Bell: Please check out my recent peg solitaire paper: "Diamond Solitaire" in the Games and Puzzles Journal. [[Wow -- that's a great article! G&PJ has long been one of the two best recreational math journals]]

Material added 29 Sep 05

tones.wolfram.com: As a part of the Wolfram Science Group, I'm happy to announce WolframTones. There should be a MathWorld News Story with some of the tunes I found. If you compile a nice collection of tunes, please mail me the address.

Is Yawning Contagious?: Justin Smith: I watched MythBusters' episode 28 that examined the question of 'Is yawning contagious?' I was amazed when they reached a conclusion that their data did not support.
Introducing Chou Romanesco!: As one of the math advisors to Numb3rs (along with Eric Weisstein and Michael Trott), I get to give math advice on the scripts. As a result of one of my suggestions, Friday night will mark the network television premeire of vegetable Chou Romanesco! As Chou's agent, I hoped to get a percentage of the appearance fees, until I learned Chou got chopped up, boiled and eaten by David Krumholtz, who rather likes brocolli.

Test Taking Strategy: Ryohei Miyadera -- A teacher is going to give students a test. The test consists of 11 questions and along with each question there is an answer. The answer may be correct or false. The students are expected to write "correct" if they think the answer is correct and "false" if they think it false. The teacher has two habits. One is that he never present three correct answers in a row nor three false answers in a row. Another one is that he presents more correct ones than false ones. What is the best strategy for students if they do not know anything about the subject? Answer and Solvers. Ryohei's write-up.
Du-Sum-Oh -- Sparse Geometric Sudoku Arrays: Bob Harris: Well, it took a few weeks, but I have found several uniquely solvable 9x9s. I've posted one, at a page I've added at my web site with a bunch of the 7x7s and about a dozen 8x8s, and a nice 7x7 solving tutorial. I am calling them Du-Sum-Oh puzzles which is a play on words intended to remind someone of Sudoku. I don't feel that the 9x9s I have found make interesting puzzles. For the one shown, my program had to search a tree of depth 18. All all the other puzzles I have posted have depth 0-- fully forced using simple logic (yet they still seem to challenge me). [I called these Geometric Sudoku. That he found a 9x9 geometric with 9 givens astounds me. UPDATE-- He's made some with only 8 givens.]
Orthogonal Sudoku: Paul Vaderlind: After reading your note about variations of sudoku I send you my variant: orthogonal sudoku, in the sense of orthogonal latin squares. Each cell will have a two-digits number (no zero digits) the first digits form one sudoku, the second digits form another sudoku, and no two-digit number occurs twice in the net. (larger pdf). Answer and Solvers.

NetLogo 3.0 Released: Uri Wilensky -- The NetLogo development team at the Center for Connected Learning and Computer-Based Modeling at Northwestern University is pleased to announce NetLogo 3.0. PARI and Crossword Compiler have also gotten updates.
World Puzzle Championships: The 14th World Puzzle Championships, run by the World Puzzle Federation (WPF), will be held in Eger, Hungary, from 8th to 13th October 2005. 29 nations are taking part. (Blog)More news is on puzzleratings.org.
Bathsheba sculptures in Numb3rs: Bathsheba Grossman, a mathematical sculptor I mention frequently frequently here, had one of her sculptures featured prominently within the Numb3rs TV show. Here are two screenshots (click to enlarge). Coincidently, she is working on a Romanesco project.


Puzzle Site Round-up: Other good sites for weekly puzzles include TurkZeka, Macalester College Problem of the Week, Puzzlemonster, PuzzleUp, or Ken Duisenberg's site. Breinbrekers is a Dutch puzzle magazine, published by Puzzelsport (www.puzzelsport.nl). I liked a recent MacPOW problem with palindromic Fibonacci-like sequences (..., -7, -4, 3, -1, 2, 1, 3, 4, 7, ... ). A recent site I've been buying things from is Puzzle Japan. In particular, I like their list of puzzle types, their Puzzle Communication Nikoli magazines (just buy one to start, to be sure you like it), and their Puzzle Cyclopedia. All of these books are in japanese, but heavily illustrated.

Material added 11 Sep 05

Sudoku Variations

The Magic Sudoku puzzle by Alexandre Owen Muniz a few weeks back was very popular. I followed it up with a Math Games column about Sudoku Variations. An unsung hero in this is Wei-Hwa Huang, who sent me an enormously helpful list of Sudoku types. Alan O'Donnell sent solutions (and corrections) for every puzzle in the column -- "I think I can say I've truly conquered this genre of puzzle now." Bob Harris mentioned that the Geometric Sudoku are also known as Latin Squares, with only one given per region. Bob Harris: It's amazing to me how much more difficult generating an 8x8 puzzle seems to be than a 7x7. Here's an 8x8 (below). Probably pretty hard, but at least 3 cells can be labeled quickly and a couple other matches (this cell is the same as this other cell) can also be identified.

Guenter Stertenbrink applied the rules to a 12x12 grid of queens, using one of the solutions to the 12 queens problem. (Can you solve his queens puzzle?) I wondered if the groups of queens could be divided into distinct regions of 12 -- the one solution I looked at had the above symmetry (warning: likely unsolvable). How well do the other solutions divide into regions? Is there a really good Sudoku puzzle with queens?

Vegard Hanssen added several of the variants to his own Sudoku page.

More oddly, I found that the London Times story was partially concocted, according to the interviewee. Tetsuya Nishio: It is true I first found Number Place puzzles in a Dell magazine, and I brought an old magazine I had at home to the interview to show the reporter. It was "Dell Crossword Puzzles Vol. 13, No. 304, May, 1985" and had one Number Place on page 69, with no mention of the creator. Actually, I have the majority my books and magazines stuffed in a rental storage space (you know how tiny the average Japanese house is) and I think there may be some Number Place puzzles in earlier issues in my collection.

Oskar Sculpture -- Connectivity

M. Oskar van Deventer: Attached is a photo of the puzzle sculpture "Connectivity", designed by me. It was made by selective laser sintering of nylon powder ("SLS"), a rapid prototype technique that produces accurate, rugged and affordable models. Its size is 30x30x45 cm, the largest that could be built by the SLS machine. After cooling down, it was metal coated. The object of the puzzle is a parity problem: to move the five rings to the top in the right orientation. Yesterday 26 August, the puzzle sculpture was installed on its pedestal and revealed to the public. The puzzle sculpture can be visited and played with at the gardens of TNO at Schoemakerstraat 97 in Delft, Netherlands. The gardens are accessible from the street by foot.

tones.wolfram.com

As a part of the Wolfram Science Group, I'm happy to announce WolframTones. There should be a MathWorld News Story with some of the tunes I found. If you compile a nice collection of tunes, please mail me the address.

Metal Pyramid

GarE Maxton: I'm releasing my newest puzzle sculpture, Pyramid I. This sculpture consists of a 4.5" tall, four sided stepped pyramid with a hidden recess holding a 1.5" standard brass central cube. It weighs a pleasing 10.5 pounds.

Prime Arrays -- Results of the first Recmath.Org contest

Prime Arrays, also called the Gordon Lee puzzle, were the subject of latest Al Zimmermann Programming Contest at recmath.org. Over a hundred topnotch programmers competed, setting many records. Here are best solutions for Part 1 and for Part 2. Scads of interesting comments have been posted at the Zimmermann Group.

Noam Elkies Chess Combinatorics

Erich Friedman: If you haven't seen it already, go to http://www.combinatorics.org/ and download paper A4 of volume 11 (2). Noam Elkies has some great enumerative chess problems.

New Twin Prime Record

Paul Underwood: A special congratulations to the Hungarian collaborators Zoltán Járai, Gabor Farkas, Timea Csajbok, Janos Kasza and Antal Járai for the record 51779 digit twin primes (here and here), beating the Previous record at 51090 digits set three years ago by Daniel Papps.

Leonid Mochalov's Puzzle Site

Leonid Mochalov has built a new site filled with many new puzzles of his own design.

The Department of Education likes me

Pam Bentley: I am pleased to let you know that your site has been chosen to be included in this month's Digital Dozen, a list of exemplary web sites for educators selected by the Eisenhower National Clearinghouse (ENC). The list is published each month at ENC Online (enc.org).

ENC is funded by the United States Department of Education and administered by The Ohio State University. ENC collects both physical and virtual resources useful to math and science educators. As part of our mission to serve educators across the nation, each month we choose a dozen sites to highlight.

The sites we select for Digital Dozen must have current and accurate math and/or science content. They must support school improvement efforts and have useful multimedia features or helpful navigation. ENC provides a brief review of the sites chosen and a direct link to the sites. As a result of receiving greater exposure on our web page, you may notice higher traffic to your site than usual.

Material added 22 Aug 05

Magic Sudoku: Alexandre Owen Muniz:When I followed your recent link to a sudoku article, I was reminded of another puzzle featuring a 3×3 group of 3×3 squares. I had defined a 'magic' 45-omino as a polyomino within such a grid, where each row, column and main diagonal contains 5 cells within the polyomino, and each 3×3 square contains a number of cells within the polyomino equal to a number in a magic square at the corresponding position. I thought that it ought to be possible to create a puzzle combining sudoku and a magic 45-omino, and here it is (here's a handy PDF Version):
: The rules of standard sudoku apply: every row, column, and 3×3 square must be filled in with all of the numbers from 1 to 9. Both of the main diagonals must also contain every number between 1 and 9. Also, each number inside the polyomino (blue cells) must be no larger than the number of blue cells in its 3×3 square. Conveniently, this number is the given in that 3×3 square. There is a unique solution. Answer and Solvers.
New Hard and Tiny Sliding Block Puzzles: James Stephens: I've posted a new page on PuzzleBeast called A Dozen Irritating Sliding Block Puzzles. [Lots of unexpectly complicated tiny puzzles.]
Tangled Polygons: Peter F. Esser: Connecting the sides of triangles, dominoes or hexagons by colored paths you get a lot of different pieces. I have just finished some constructions with these sets.
Queens in a Cube: Arrange 121 queens in a side-11 cube so that no two queens attack each other. Answer here. In a related problem, what is the minimal number of rooks needed to cover or attack every cube in a side-8 cube? Send Answer.
Burr Puzzle Program at SourceForge: Andreas Röver: I want to inform you about an update to my burr solving software. The program is currently capable to solve burr type puzzles made out of cube shaped units. To solve means it can find assemblies and also disassemble them as long as they don't contain rotations. You can download it on its webpage. [Ed - For non-free, top-notch programs of this type, visit Bill Cutler's site.]
New Primality test used for Fibonacci Prime U(37511): John Renze is the first to prove a new Fibonacci Prime in 4 years, by using the Coppersmith-Howgrave-Graham method, and the lattice reduce methods in Mathematica.
Fourteen Queens on a 7x7 board for the Heawood Graph: Puzzle 1: Place 13 queens on a 8x8 chessboard so that each queen attacks exactly 2 other queens. After seeing Erich Friedman's Math Magic for large cycle graph with chess pieces, I wondered about other graphs. Puzzle 2: Arrange 10 queens on a 5x6 board to make a Petersen Graph (solved by Daniele Degiorgi). Geoff Exoo noted another nice solution. Puzzle 3: On a 6x6 board, with QQQ_QQ as the top row, put 5 more queens on the board to make a Petersen Graph. Erich then solved all cubic graphs up to 10 nodes. Geoff Exoo solved the Heawood Graph. Puzzle 4 : Arrange 7 white and 7 black queens on a 7x7 board so that with any pair of like-colored queens, both are attacked by a single queen of the other color. Guenter Stertenbrink looked at the Knighted Alfil -- a knight that can also jump 2 spaces diagonally. Puzzle 5 : Arrange 10 Knighted Alfils on a 4x4 board to make a Petersen Graph. One particular case is unsolved, so I'll make it a $20 challenge. 20$ Challenge Puzzle 6 : Using 20 queens, on a board of any size, make a Dodecahedral graph. Whoever does it on the smallest board within the next week wins the prize. Send answer.

Games and Puzzles Magazine Index: Eric Solomon: You may be interested to know that an Index, started by David Parlett and completed by me, to the first series of Games and Puzzles magazine is now available on my web site. This may be useful to those fortunate enough to own a set, and perhaps to those who carry out research using sets held by libraries. [I have a set.]
Programming Contest - Picture Packing: I learned of the Programmer of the Month site, which has regular programming contests. Another contest site is recmath.org.
Happy Birthday Hamilton: William Rowen Hamilton, who discovered the quaternions, and was so happy that he scrawled on a bridge, just celebrated his 200th birthday in a big way. Liam O'Neill: "Thanks for your query,we had a wonderful day,the weather held out and we had a excellent attendance. Our Minister for Science did the honours ably assisted by Professor David J. Fegan of the Royal Irish Academy. Hamilton and Quaternions were much lauded, birthday cake and a presentation of a painting of the location of the 'Holy Grail' to the Royal Irish Academy by the local community,a wonderful speech by Professor Fegan plus a flotilla of boats to transport the v.i.p.'s to the hallowed site. Among the gathering were distinguished visitors from Japan, North America and the UK who were thanked by our committee for their generous support. This celebration had multifaceted support, The main sponsor was the Royal Irish Academy, great credit is due to
Niamh Morris Logistics and Rebecca Gageby PR ably assisted by Vanessa." Among other things, Ireland minted a 10 Euro coin, more details at http://www.science.ie/home/index.asp, http://www.hamilton2005.ie/, http://www.ria.ie/.
Orthogonal Discrete Knots: Hew Wolfe: An orthogonal discrete knot is a path through the lattice of integer-valued points and orthogonal edges between them, which never visits the same point twice and ends where it starts. Two knots are equivalent if, when they are realized with pieces of stretchy string, we can move one around to look like the other. (Also, we consider mirror images to be the same knot.) Naturally we are interested in knots which are not trivial, that is, equivalent to a simple circle. More at the orthogonal knot website.

Material added 06 Aug 05

PQRST 14: PQRST 14 has started. If you'd like to compete, you have until 13 August 2005 to solve the ten puzzles.
Tenth Planet (not counting Sedna, Quaoar, Varuna, Ixion, Chiron, or Orcus): A Kuiper belt object larger than Pluto has been discovered. For now, it is called 2003 UB₃₁₃. Jeff Bryant used some Mathematica to make an animation of its orbit.
Ambigram Maze: Brett Gilbert has made a maze of MAZE as an ambigram, for his sister's birthday.
Andrew Clarke's The Poly Pages at recmath.org: The idea behind recmath.org is to provide permanent storage for good math sites. Andrew Clarke's Poly Pages now joins Al Zimmermann's Programming Contest.
Books: Serhiy Grabarchuk's New Puzzle Classics is now available. All sorts of good puzzles, almost all of them completely original. I bought a copy for myself, but also got a complementary copy. So, a little contest. Whoever writes me the most interesting math puzzle-type letter next week wins the book. Write me. Clifford Pickover's A Passion for Mathematics has been also released. Serhiy also sent me the first volume, Paperfoldings, in his Age of Puzzles project. Write him at serhiy.g (at) gmail.com if you'd like to buy a copy. Here's one puzzle he highlights by Kunihiko Kasahara. Glue the free edge to the opposite side of the strip, then fold to make a gray cube. Then, only using folds, make a white cube.

Infinity Magazine: Tarquin Publications has launched Infinity Magazine. Andrew Griffin: It is aimed at "Recreational Math" enthusiasts of whatever age. The editor grew up with Martin Gardner and is trying to aim it at a similar reader to the Scientific American: math literate individuals who may or may not be educators. I now have a copy of the first issue -- very nice. Contributors include David Mitchell (collapsible cubes), Martin Watson (edge-matching puzzles), Robert Reid, David Singmaster, John Sharp, Snezana Lawrence, Fred Armitage, David Wells, and Richard Ahrens.
Vector vs Raster: My latest Math Games column is mostly about vector programs. I counted 29 of them on the first pass. After being Slashdotted, I'm revising the column.
Button Puzzle: The Button Game comes with a large prize ... there's a cute trick behind it.

Material added 30 Jul 05

Odd Symmetry Update: George Sicherman and Michael Reid have found many new solutions for the Odd Symmetry problem. The Colonel has made a new page about them, the Polyomino Oddities page. The solutions are fascinating. Can you find an improvement, or solve one of the unsolved cases?
2005 Nob Yoshigahara Puzzle Design Competition: The results of the 2005 Puzzle Design Competition have been posted. A dazzling collection of 57 puzzles was on display, which you can now see on the site. The winners are Donald Knuth and George Miller, for a polyform folding puzzle; Hirokazu Iwasawa, for a sliding block puzzle; Markus Götz, for Crazy Elephant Dance; Panayotis Verdes, for a 6x6x6 Rubik's Cube; Shiro Tajima, for a new puzzle box design; and Akio Yamamoto, for the metal puzzle Radix.
Punt Mazes: Andrea Gilbert has posted three sets of new mazes, including the Sokoban-inspired Punt Mazes.
Noncrossing Knight Tour Cubes: Back in 1771, Vandermonde figured out how a knight could make a tour of a 4x4x4 cube (below). In 2003, Guenter Stertenbrink figured out a magic cube knight tour.; Derek Bosch asks : Is it possible to construct a 3-dimensional knights tour for a 4x4x4 cube, such that none of the paths intersect? My initial searches have found tours up to 61 cubes out of 64, but I hope to find a complete solution. Send Answer.
Lego 21 Square Dissection: Jim Reed: I thought you might be interested to know that free-lance lego sculptor Eric Harshbarger has created a Lego mosaic of a 21 square dissection for the Mathematics Department of Auburn University.
Periodic Table of 2x2 Game Topologies: Dave Goforth, as a part of his book The Topology of the 2x2 Games, has made a Periodic Table of Games (PDF).
No Touch Tilings: At Math Magic for July, Erich Friedman has received many solutions for various types of no-touch tiling.
Pentomino Triplets -- Complete Solution: Vasyl Tsvirkunov: Some time ago I did a complete analysis of an old pentomino puzzle -- making three identical shapes from the single pentomino set. I tried to find if anybody did this before me but could not see traces anywhere. There are about 20 different solutions mentioned in different books, starting with Golomb's classic book but that's about it. My program found 339 distinct solutions. It took about a day of computations on relatively fast PC, and that's after a lot of tricky optimizations. Technically, it is brute force analysis as the program tried all 2.8 billion icosaminoes. A few solutions are particularly interesting. There is only one symmetric solution -- I found it by hand many years ago, apparently I was not the only one who found it as it is mentioned in G. Martin's Polyominoes. There were a couple solutions with 1x1 holes. I knew of solutions with 2x1 or two 1x1 holes. To my surprise, there are solutions with much larger holes. The gem is solution with hole in the shape of P-pentomino -- I have serious doubts if anybody would encounter it by searching manually.
All solutions in one file. A strange hex number in square brackets is the solution identifier. This is actually a leftover from one of the previous version of my program but I kept it in place because it is an easy way to uniquely describe and identify almost every solution (BTW, I should probably use the word "configuration" instead of "solution" as some of those "configurations" may have multiple "solutions" -- I never checked that). It is valid for configurations that fit in 8x8 square (all except 3) and it is the smallest 64-bit number with bitwise representation of occupied squares -- I hope this description makes sense, it is easier to explain by drawing pictures. The enumeration algorithm is from http://kevingong.com/Polyominoes/ParallelPoly.html. [Here's a pentomino triplet solvable by logic. Update-- Helmut Postl also solved this problem.]

Material added 18 Jul 05

Odd Symmetry: George Sicherman took a look at symmetric figures made from an odd number of unsymmetric polyominoes. Below, you can see some solutions he found for hexominoes, there are seven unsolved cases. Can you use a lower odd number of pieces for any of the figures below, to make a figure with mirror, diagonal, or rotational symmetry? In a similar vein, use 5 y-pentominoes to make a symmetric figure. Answer and Solvers. For an update on the general problem, see Polyomino Oddities at Sicherman's site.

Vector drawing: My next column will be about vector-drawing software. I already know about Acrylic, Autocad, Cinderella, CorelDraw, Euclidraw, Flash, Freehand, GEUP, Geometer's Sketchpad, Illustrator, Inkscape, jPicEdt, kseg, Mayura, OpenOffice, PSTricks, Sketsa, TeXCad, TpX, WinFIG, Xara X, and yEd. If you have a favorite program for vector (not raster) artwork, please let me know about it.
The Condorcet Die: Jonah Ostroff sent me an interesting set of dice: Red: 111477, Green: 225555, Blue: 333336. "As we learn from some well-known voting problems, competitions that are easy to analyze with two people can become pretty complicated with three. Specifically, we know that the Condorcet Candidate (the one who can beat each of the others in head-to-head competitions) is not always the one who wins in the conventional one-vote system. Applying this to the latest Math Games column, and in particular Oskar's seven dice, I'm not entirely convinced that the diagram provides a winning strategy. A can certainly beat B and C, but are we sure he can beat them both at once?" Can you find the Condorcet Die, and the true winner? (A Google search on Condorcet gives many great sites.) Erich Friedman sent a similar observation, with the dice A: 333444, B: 111166, C: 222255.
Printing Error: Bryce Herdt: I have a small puzzle. Just put the digits 1-9 into the grid below so that the equations in each line, following order of operations, are true.

+   +   =27
+   +   +
+   +   =80
+   +   +
+   +   =31
=   =   =
21 9   79

I know those are all plus signs, but some of them are supposed to be multiplication signs, and look this way due to a printing error. At least, that's my story. Answer and Solvers.
2005 TESSellation contest: Jeff Tupper: Our 2005 TESSellation contest ended last week. Notable entries can be seen at the site.

Material added 08 Jul 05

Oskar's Dice: M. Oskar van Deventer has created a very interesting set of seven dice. Two opponents each pick a different die from a set. Then, you pick a die from the same set, and over ten rolls, you beat both of them most of the time. Can you come up with a set of dice that has this property? Oskar did, and I devote my latest MAA Column on tournament dice.
New format for MathWorld: If you visit MathWorld, you'll see the equations look different, and various other things. Two of the huge projects that have been added are MathWorld Classroom, and Interactive Math Examples.

Material added 04 Jul 05

Al Zimmermann Programming Contest at RecMath.Org: Sam Byrd: It is my pleasure to announce that the latest installment of the Al Zimmermann Programming contest has now begun. For contest information head on over to:

http://www.recmath.org/contest/

Ed Pegg Jr: With the assistance of Sam Byrd, Jean-Charles Meyrignac, Al Zimmermann, Guenter Stertenbrink, Pascal Zimmer, RecMath.Org has been launched. I intend for it to be a permanent locale for good recreational sites, such as Al Zimmermann's contests. Unlike MathPuzzle, it will have a board of directors. If you'd like to be considered for the board, or to have your site permanently stored, send me a message. The first prize contest is Primal Squares.
New Polyform Site: Bernd Karl Rennhak: I am a frequent reader of your mathpuzzle pages, looks like you might be interested. Have a look at http://www.logelium.de/
The main subject of the site is puzzles, classical and new ones. The material was sitting on my desk for month if not years already, until I found some time. The now published chapters present some work with Stomachion tiles, and for example answer the question, why all solutions have tile positions fitting on the base grid. And can you build a decagon from the Stomachion tiles ? ...
The other chapters are concerned with kite tiles and others called Doms (maybe you remember), especially with the tile set DiDom.
If you find the site content interesting, I would be grateful to make it a bit more known by a few words on your pages. [Ed - An excellent, beautiful site, with new results I haven't seen before. Lots of illustrations, with easily web-translatable German text.]
New Packing Pages: Erich Friedman has added packing pages for L's in Squares, Squares in L's, L's in Circles, and Circles in L's.
Flat Polycube Touching Problem: Karl Scherer has found a flat 16-omino such that 12 copies can be arranged where all 12 touch each other. Is there a smaller non-flat polyomino with the same property?
Magic Square of Squares: Christian Boyer has summarized the knowledge of this problem. Ivars Peterson discusses the article in his latest column.