Material added 25 Dec 05 (Merry X-mas)
Another Mersenne Prime
Mersenne #43 has been announced: 230,402,457 - 1. More details at MathWorld.com and Mersenne.org. It has almost 10 million digits. One odd thing I only learned a few months ago is that when p ± 1 isn't readily factorable (which I'll call a nearby unfactorable prime), ten thousand digits is considered amazing for a primality proof. Much of the progress in nearby unfactorable primes has been made in the last few months, under listings Generalized Lucas, Generalized Repunit, Lehmer Number, Fibonacci Number, Elliptic Curve, and Lucas Number. These numbers all have interesting algebraic qualities that lead to a proof. Other prime numbers currently seem unprovably prime, these are the Probable Primes.
Serhiy Grabarchuk greetings
Serhiy, who now runs Age of Puzzles, sent in his annual holiday puzzle. This time, it's a field of Christmas Trees.
Updated Programs
Some of my favorite programs have been recently updated. Browser Firefox 1.5, Image processor Irfanview 3.98, Office software OpenOffice 2.01, live linux DVD Quantian 0.7.9.1 , and postscript viewer Ghostscript 8.53. I just found out about geometry program OpenEuclide, which I'll need to add to my list of vector programs. Some of my favorite independent games have been Journey to Rooted Hold, Mount & Blade, and Professor Fizzwizzle. My new favorites of the year are art programs TpX and Inkscape, TeX editing program WinEdt, and text editing program PSPad. Always feel free to recommend a program to me.
A Heptomino Diagram Too Wide
I've learned that wide diagrams don't work well on older browsers (why not try Firefox?) and my new site layout. Peter Esser found that the 196 one-sided heptominoes could fit into 14 lovely rectangles. I had to resize the original GIF (with Irfanview), which makes the lines a bit fuzzy. It would be nice to have this in a vector format. Updated: Jeff Epler -- "I resized this in The GIMP with a multi-step process. My strategy for resizing was to first use "value propagate" to make the black lines two pixels thick. I also replaced the two tones of green with a single cone. Then I rescaled the image to 50% nearest neighbor." [Oh, so that's how to do it!]
Another Mersenne Prime
Under "Mersenne Exponent Test State" at PrimeNet, there is a new unverified Mersenne Prime. It is probably getting tested right now.
MathRec, RecMath, and American Scientist
At RecMath.org, the Circle Packing contest is in its last month. The next contest will almost certainly be Prime Generating Polynomials of various orders. For example, one order-4 polynomial I found is n4 + 29 n2 + 101, which generates primes for n=[0,19]. Almost certainly, that will be trounced, perhaps by you. A variety of new Sudoku types are at MathRec.org. In related news, the American Scientist magazine did a nice article on Sudoku, and kindly mentioned me. I also liked the recent Crazy 8's Sudoku, and latin square puzzles.
Kinematic Models for Design - a Digital Library
Do you like odd mechanisms? KMODDL is a collection of mechanical models and related resources for teaching the principles of kinematics--the geometry of pure motion. The core of KMODDL is the Reuleaux Collection of Mechanisms and Machines, an important collection of 19th-century machine elements held by Cornell's Sibley School of Mechanical and Aerospace Engineering. For example, here is the Eccentric Spur Gear.
King Arthur's Treasures
A million dollars is being given away in the King Arthur's Treasures contest. I don't know much about it.
A Trott Constant
Michael Trott notes that 0.273944195 7392716171 7145915272 4285919273 7251877298 8198629191 7383755281 7177418196 9461917382 83625161549 = 0 + 2/(7 + 3/(9 + 4/(4 + 1/(9 + 5/(7 + 3/(9 + 2/(7 + 1/(6 + 1/(7 + 1/(7 + 1/(4 + 5/(9 + 1/(5 + 2/(7 + 2/(4 + 2/(8 + 5/(9 + 1/(9 + 2/(7 + 3/(7 + 2/(5 + 1/(8 + 7/(7 + 2/(9 + 8/(8 + 1/(9 + 8/(6 + 2/(9 + 1/(9 + 1/(7 + 3/(8 + 3/(7 + 5/(5 + 2/(8 + 1/(7 + 1/(7 + 7/(4 + 1/(8 + 1/(9 + 6/(9 + 4/(6 + 1/(9 + 1/(7 + 3/(8 + 2/(8 + 3/(6 + 2/(5 + 1/(6 + 1/(5 + 4/(8 + 5/(9 + 3/(6 + 4/(7 + 1/(9 + 2/(5 + 8/(9 + 4/(9 + 8/(9 + 1/(5 + 1/(7 + 2/(7 + 3/(9 + 1/(9 + 6/(7 + 6/(9 + 2/(8 + 1/(9 + 7/(9 + ... Does this number go on forever, with the numbers identical on each side? More info can be seen at Trott's Constant.
Clueless Puzzles
A series of HTML tricks can make a puzzle. Iris Game is one of these. DeathBall is another.
Janus Sliding Number Puzzle
Frank Potts took a typical 4x4 sliding block puzzle, and split it on two levels. Eric Solomon made a java version out of it, and together made the difficult Janus puzzle.
Gartriage - A train puzzle
Jean Van Laethem and Aymeric du Peloux made a puzzle game called Gartriage that involves using a train to move cargo to the correct loading station. There are many excellent puzzles here.
Lovely Functions
A lot of lovely functions, done in a style of Days of Christmas, are shown at mathematik kalender. A paper about these Unidentified Figurative Objects is available at arxiv.org.
A long Palindromic prime
Harvey Dubner has found the new palindromic prime 10^140008+4546454*10^70001+1. Caldwell maintains a list of the top twenty palindromic primes.
The 2005 Putnam Exam
The questions of the 2005 Putnam exam, which was held Dec 3, 2005, can be seen at the Harvard math site. They also have all the Putnam tests back to 1938, in case you'd like something to solve during a holiday flight. For more Olympiad level mathematics, try the MathLinks Forum.
Pentomino Oddities
In case I haven't mentioned it before, George Sicherman's Pentomino Oddities page is well worth a look.
Super Duper Games
A good Abstract Strategy Game site is SuperDuperGames.org, which has Abande, Alien City, Archimedes, Blam!, Blockade, Branches Twigs and Thorns, Byte, Cannon, Dugi, Fortac, Frames, Froggy Bottom, Generatorb, Homeworlds, Impasse, Linear Progression, Macadam, Magneton, Martian Chess, Motala Strom, Numica, Pikemen, Praetorian, Pulling Strings, Slings & Stones, Sprawl, Tanbo, Terratain, and Wizard's Garden.
Here's a game played with 12 cards, numbered 0-11. The cards are distributed randomly into two visible groups of cards, Group A and Group B, with the cards being redealt if the Group A cards total less than 21. Players alternate turns. Each turn involves swapping a card in Group A with a lower valued card in Group B. Whoever makes the sum of the cards in Group A go under 21, loses. Pretty simple game, right? Playing the game perfectly requires some deep group theory and Steiner systems, described by David Joiner in his paper MINIMOGS and Mathematical Blackjack.
4D Sudoku
Chris Lusby Taylor: The diagram below shows the SuDoku grid divided into its 3x3 blocks, which are then separated and skewed to show them as planes within a 3x3x3x3 hypercube. This diagram should be familiar to anyone who has played 4D noughts and crosses.
From the hypercube, 3x3x3 cubes can be mentally assembled by selecting a particular value for any one dimension. For instance, w=1 gives a cube containing the three leftmost blocks. Any such cube can be sliced in three ways to give a 3x3 block. For instance, w=1, z=3 is the top left block, w=1, x=1 is the leftmost column. w=1, y=3 is also a valid slice, which I have colored green. I have colored blue the set x=3, z=2, and red the set x=2, y=1. The puzzle is to fill the 81 squares so that all such 3x3 slices contain every digit.
SuDoku has previously been extended by coloring the squares. The Japanese call this Colour Number Place. This adds sets of squares, such as the ones I color red, which are all in the same position within a 3x3 block.
With all those extra constraints, we can create a puzzle with fewer given numbers. Mine has just ten. It has a unique solution which can be found by applying normal SuDoku logic, but over six directions rather than three. It is not easy, but requires absolutely no trial and error. (Answer and Solvers.) [Some earlier 4D Sudoku by Guenter Stertenbrink are at magictour.]
Soap bubbles and Numb3rs
The TV show Numb3rs is continuing to do spectacularly. It is Friday's most popular show, and a favorite of many scientific organizations. I can pompously report that on almost every episode, one of my suggestions gets used. With the recent episode Toxin, my contribution was Soap Bubble Theory (the link shows the amazing math 9-year-olds can do). Many excellent sites explain the math in the show, particularly cbs.com, which offers worksheets developed by Texas Instruments and the National Council of Teachers of Mathematics via a team led by Dr. Johnny Lott (University of Michigan). Redhawke.org also does a great job keeping up with the math, along with TV Tome, the Numb3rs blog, Wikipedia, and Numb3rs.org.
New Coin Puzzle - updated
Bob Hearn: I've attached a couple of the latest kind of coin puzzle I've been generating. They both have the added nice feature that there are a few tricky moves required near the end. They take 48 moves (8 vertices), 66 moves (9 vertices), and 118 moves (11 vertices) respectively, and are definitely harder than they look. [[These are wonderful puzzles, well worth printing out. His search also found a 173-move puzzle on 12 vertices, and a 238-move puzzle on 13 vertices.]]
Hidden Points and Matchstick Graphs
Erich Friedman's latest Math Magic contains several interesting problems, including marksmen duels, matchstick graphs, and hidden points. (Warning -- the second link has the answer to this week's puzzle.) In the first figure below, discovered by Erich Friedman, four points are hidden from every point. Also in the star, discovered by Aron Fay, four points are hidden from each point. Erich found an arrangement of 12 points where 3 points are hidden from each point. Can you find this arrangement, or a different arrangement ... or find a construction that hides 5 points from every point? (The first image was drawn with TpX.)
New largest Gaussian Prime
(1 + i)991961 - 1 is prime, with 298611 digits.
Tribonnaci Factor conjecture
Jonathan vos Post conjectures that every integer is a factor of a Tribonacci number. For example, 10 is a factor of 35890, the 19th Tribonacci number, by the method at MathWorld. The conjecture has been verified to 500 -- seems quite chaotic. Can anyone prove this conjecture, or verify it up to some big number? Answer and Solvers.
Oskar van Deventer in Games Magazine, and River Crossing
The latest issue (Feb 06) of Games Magazine has a great article about the many puzzles of Oskar van Deventer. Oskar himself sent me sent me some nice photos recently. Oskar: "Attached are three photos of large hands-on puzzles that are on display in the Delft Science Museum. The person standing on the River Crossing is Andrea Gilbert, the inventor of this puzzle. The two other puzzles are Richard Tucker’s Hayling Island Maze and my own Key Maze, both made by Wim Zwaan. According to Liesbeth van Hees, the creative director of the science museum, the large puzzles are immensely popular with children." Oskar is in the gray sweater below. Oskar's Planet's puzzle is also recently available.
Magic Tours and more
Magic knight tours on the 12x12 board an many other topics are discussed in the latest issue of the Games and Puzzle Journal.
Floretions
Creighton Dement: I took part in the Congress/Bundestag youth exchange program in 1992-93. My main area of work over the last two years has dealt with "floretions" (pronounced floor-RET-ee-ons), named after mathematics Prof. K. Floret.
Cube Sudoku
Steve Schaefer: I've created a couple of new sudoku variants. The most notable one is a cube sudoku that uses all six sides of a cube with 4x4 faces. The "rows" and "columns" run all the way around the cube, so that each type of block has 16 values. This seems like an obvious configuration to me, so it's entirely possible that someone else thought of it first, but I've never seen one that any one else made. That puzzle was made by hand. I've now written a generator, but the first puzzle it spit out is giving me fits as I try to solve it by hand. There are some new solve rules on the surface of a cube and I may not have discovered them all. (Ariadne picked up the slack in the solver that my generator uses.)
Highest Prime Gap Merit discovered
804212830686677669 is at the start of a prime gap of length 1442. The merit of this gap is 34.98 -- the first improvement since 1999.
The 1/7th Ellipse -- updated
Jay Hall noticed that the six pairs of number in 1/7th, .142857... or (1, 4), (4, 2), (2, 8), (8, 5), (5, 7) and (7, 1), all lie on an ellipse. Eric Weisstein added this interesting object to MathWorld as the 1/7th Ellipse. Chris Lomont: "Out of curiousity, I found a lot more of these ellipses. One with more points is the 1/7373 ellipse, 1/7373 = 0.00013653... which gives seven points {0,0}, {0,1}, {1,3}, {3,0}, {3,5}, {5,6}, {6, 3} on an ellipse. To get 8 points on a single ellipse I found the fraction 4111/3030303 works. I'm yet to find more on a single ellipse. I'm unaware of any proof that it cannot be done, although integer points on curves is very much studied."

Gordian's Knot puzzle now in stores
Last year, I wrote a column about modern burr puzzles, and the phenomenal discovery by Frans de Vreugd of a puzzle made of simple pieces that needs 28 moves to remove the first piece. I suggested to Thinkfun/Binary Arts to talk to Frans about making it, and they did! Released as Gordian's Knot, it is now in stores. It's the ultimate fiddly puzzle. Bob Hearn: "It is absolutely wonderful. I saw someone at MIT fiddling with one last week, and had to go and get one right away. It's visually grabbing. And I can't stop playing with it!" Another write-up is at Torturous Burrs. Thinkfun has also released a new puzzle by Ferdinand Lammertink, Flip-Side.
Jordan Curve theorem proved
Jordan's Curve theorem proves that every closed, nonintersecting curve has an inside and an outside. Jordan's original proof, though elegant, was not considered a strict proof. An elaborate computer system by teams in Japan and Poland has developed, after 14 years of work, a strict proof which fills over 3000 pages.
Bridged DiTans and Triamonds
Bernd Karl Rennhak: I have some news about "Bridged Polyforms". That is a class of polyforms I investigated recently. The idea is not completely new, but I pushed the thing a bit further. The bridging method is pretty generic and can be applied to various polyforms. Basic forms are connected with vertex and/or edges, where the vertex points need the bridges, which creates nice sets of shapes. For the basic forms I used squares, equilateral triangles and isosceles right triangles, still more to come. [[Update -- Oskar has made a very nice fractal version of this.]]

Cool Illusion
A very cool optical illusion involving an animation of pink dots is well worth a look. jjartus notes that this illusion is accomplished by retinal fatigue -- for much the same reason, hospital gown green is the exact opposite color of blood.
Knight Tour Sudoku
Dan Thomasson: Here are a couple Knight Tour Latin Square puzzles I designed that should not be too difficult to figure out. After solving the squares, you should be able to draw four separate 16-move closed knight tours in each square. You can see more information about them on my website at www.borderschess.org/LatinKT-Problem.htm.
The State of Unsolved Problems
Erich Friedman has an excellent pointpower presentation of unsolved math problems.
The Donald Knuth Cubigami Puzzle
George Miller wrote up Donald Knuth's fascinating discovery of a unique pattern for folding tetracubes -- the Cubigami.
G6-smooth pairs
Luke Pebody found 17445&17446+165615i, 52487&52488+195264i, 27839&27840+229680i, 283904&283905+160704i are all Gaussian-6 Smooth pairs. For example, 283904 + 160704i = (1+i)12 (3+2i) (4+5i)3 (5+2i), 283905 + 160704i = -34 (1+4i)2 (5+4i) (6+i)2. He suspects the largest G7-smooth pair is 3562857&3562858+2518776i.
Multiplicative Magic Squares
My latest maa.org Math Games column is about multiplicative magic squares. Christian Boyer and Luke Pebody made many record-setting discoveries.
Truchet Automata
Here's a clever flash-driven truchet tiling automata. Is a life-like blinker possible? Answer and Solvers.
Physics Songs
You can't have enough songs about Physics. To meet this requirements, there is physicssongs.org.
No-guess 9-clue Geometric Sudoku
Bob Harris: Have finally found a 9x9 Latin Square Puzzle (a.k.a. jigsaw sudoku, geometric sudoko, dusumoh) that requires no trial and error. It's at http://www.bumblebeagle.org/dusumoh/9x9/index.html. I've only been searching with the "snakey" nonominoes, since previous results suggest these have a higher density of such puzzles (no idea why, though). Will start searching with other nonominoes at some point.
The Wilson Power Fraction Sequence
David Wilson: To my knowledge, for n = 3 through 7, the smallest known integers that are a sum of two nth powers of positive rationals but not of two nth powers of positive integers are:
a(3) = 6 = (17/21)3 + (37/21)3
a(4) = 5906 = (25/17)4 + (149/17)4
a(5) = 68101 = (15/2)5 + (17/2)5
a(6) = 164634913 = (44/5)6 + (117/5)6
a(7) = 69071941639 = (63/2)7 + (65/2)7
internet NOBle puzzle contest
Toshi Kato: A NOBle puzzle should be clever, sophisticated, and worthy of puzzlemaster Nob Yoshigahara. A Puzzle Design Competition started last week, and now the 13 entries are online (Japan access). Only if you try to solve at least one, then, you can vote best four puzzles with ranking. Please select best four in thirteen puzzles for NOBle prize. Using a rating system, the five best solvers will get a special wooden puzzle.
Sudoku Variations in Italian
With permission, Federico Peiretti has translated my Sudoku column to make an Italian Sudoku Variations.
Circle Packing Contest
Al Zimmermann's Programming Contests: Can your program pack circles of size 5 to 50 in the smallest possible space? If so, \$500 in prizes are available for you. Contest runs until 14 January.
Games 100 announced
Amusingly, the 1980 issue of Games that started the Games 100 was also the first issue I was published in. Of note this year is the Puzzle of the Year, Tipover, by James Stephens. Two years ago, it was River Crossing, by Andrea Gilbert.
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 ed@mathpuzzle.com. My mail address is Ed Pegg Jr, 1607 Park Haven, Champaign, IL 61820. You can join my moderated recreational mathematics email list at http://groups.yahoo.com/group/mathpuzzle/.