MathPuzzle.com

Material 19 August 2010

Fields Medals: NPR does a segment on the 2010 Fields Medals, with Julie Rehmeyer from Wired/Science News.
US Qualifier for the World Puzzle Championship: On Saturday, August 21, the US Qualifier for the World Puzzle Championship will be held. Good luck solving my puzzle, if you try it. The Practice test is available right now.
Penrose Staircase: Also on Wired, puzzlemaster1 Mike Selinker and puzzlemaster2 Eric Harshbarger put together the article Neverending Stories, refering to the movie Inception. Mike asked Warner Brothers for permission to use an image from the film, so they asked Chris Nolan for an image, and he went frame by frame through the scene to pick one for the article. Famous director assisting with a recreational math article -- wow.
Math Puzzle on Futurama: Tonight's episode of Futurama (Mind-Switcher) involved mind-swapping. After the first experiment, though, they found that once two bodies had swapped minds, they could never swap minds again. Then other swaps occurred. Puzzle: how many new bodies are required to get all the correct minds back into the correct bodies? A full solution is given right in the episode (the Futurama writers all have math degrees).

Material 9 August 2010 (an update to the update)

God's Number is 20: It takes 20 or fewer moves to solve any Rubik's cube position. With significant help from the GoogleNet, Morley Davidson, John Dethridge, Herbert Kociemba, and Tomas Rokicki prove that God's Number for the Cube is exactly 20. More details at cube20.org.
An article on OEIS: Julie Rehmeyer wrote the article The Pattern Collector, which talks about Neil Sloane and oeis.org.
Two from Arxiv: Counting Links and Knots in Complete Graphs by Loren Abrams, Blake Mellor
We investigate the minimal number of links and knots in complete partite graphs. We provide exact values or bounds on the minimal number of links for all complete partite graphs with all but 4 vertices in one partition, or with 9 vertices in total. In particular, we find that the minimal number of links for $K_{4,4,1}$ is 74. We also provide exact values or bounds on the minimal number of knots for all complete partite graphs with 8 vertices.
Circle Packing for Origami Design Is Hard by Erik D. Demaine, Sandor P. Fekete, Robert J. Lang
We show that deciding whether a given set of circles can be packed into a rectangle, an equilateral triangle, or a unit square are NP-hard problems, settling the complexity of these natural packing problems. On the positive side, we show that any set of circles of total area 1 can be packed into a square of size 8/pi=2.546... These results are motivated by problems arising in the context of origami design.

Material added 9 August 2010

Computer Update: This is my first try at using Adobe CS5 on a brand new computer. It's fairly easy to get a bona fide supercomputer for 1K-2K these days, so I've been doing a number of big runs on square packing. Part of the new system is two hard drives, one Windows 7 64-bit, the other Ubuntu Linux. For analyzing billions of planar graphs (with help from plantri), and general fun, I prefer the Linux system. It's definitely easier to find and install a lot of good free software on the Ubuntu system, just by selecting it. Windows (compared to Linux) definitely has the edge on running commercial software (like Adobe, Autodesk, and Crossword Compiler). On the other hand, I had to download a scad of software to make sure my Windows system stays clean (CCleaner, AVG, Spybot, Ad-Aware). FWIW, Mathematica runs just fine on either operating system.
"You have two minutes to make a maze I need a minute to solve.": That quote is from Inception, which I consider a must see for any visitors of this site. Here's another Christopher Nolan quote: "I always find myself gravitating to the analogy of a maze. Think of film noir and if you picture the story as a maze, you don't want to be hanging above the maze watching the characters make the wrong choices because it's frustrating. You actually want to be in the maze with them, making the turns at their side, that keeps it more exciting...I quite like to be in that maze." His logo for Syncopy Films is a maze. The Penrose stairs are explicitly mentioned.

Hofstadter, Godel, Escher Bach (discussing Escher's Concave and Convex, p. 106): "Achilles: What happens if you then find a picture within a picture which you have already entered, and you take another swig from the pushing-potion? Tortoise: Just what you would expect - you wind up inside that picture-in-a-picture."

Hofstadter (p. 338): Suppose a friend who has borrowed your car telephones you to say that your car skidded off a wet mountain road, careened against a bank, and overturned, and she narrowly escaped death. You conjure up a series of images in your mind, which get progressively more vivid as she adds details, and in the end you 'see it all in your mind's eye'. Then she tells you it's all been a joke! In many ways, that is irrelevant. The story and images lose nothing of their vividness, and the memory will stay with you for a long, long time.

Hofstadter (p. ix): Little Harmonic Labyrinth. This is based on the Bach organ piece of the same name. It is a playful introduction to the notion of recursive - i.e. nested - structures. It contains stories within stories. The frame story, instead of finishing as expected, is left open, so the reader is left dangling without resolution.

I was reminded of a fractal maze while watching this. I made the new one below -- copies of the brown frame are copied inside itself. Following the fractal path, get from S to F, on the correct level. It's okay to spend more than a minute on it.

Some Recent Purchases: Here are some of my recent purchases that I've liked: , Oskar&Bram's Gear Cube, Gems of Geometry, Dig-it, Blokus Trigon, Thinkfun Z-Knot,, Prof Layton and the Unwound Future, Sudoku Masterpieces (a fantastic exploration of variants by Wei-Hwa Huang and Thomas Snyder), and Anti-virus (by Oskar and PuzzleBeast, superb puzzles, finally available in US). I've also put together a list on Amazon, Games for Math Fun.

The Most Beautiful Geometry Book from the 1800's: Byrne's version of Euclid's Elements replaced most of the words and verbiage with images in red, yellow, and blue. It was an exacting work, nearly impossible to print, and ultimately bankrupted the publisher. Surviving copies sell for thousands. Gorgeous book, now available as a reprint.

Five Trillion Digits of Pi: Alexander J Yee and Shigeru Kondo have calculated 5 trillion digits of Pi.
P!=NP: Vinay Deolalikar claims to have a proof of P!=NP, with copies of his paper at win.tue.nl or at scribd. "I am pleased to announce a proof that P is not equal to NP. The proof required the piecing together of principles from multiple areas within mathematics. The major effort in constructing this proof was uncovering a chain of conceptual links between various fields and viewing them through a common lens. Second to this were the technical hurdles faced at each stage in the proof. This work builds upon fundamental contributions many esteemed researchers have made to their fields. In the presentation of this paper, it was my intention to provide the reader with an understanding of the global framework for this proof. Technical and computational details within chapters were minimized as much as possible."
A Book made from Books: Eric Angelini tells me about an unbooking of the book Napoleon.
Squares made of Squares: Rectangles made entirely of squares can be wonderful problems.

Bill Gosper: "Search the Web for 'squared rectangles'. If I were king, one of these diagrams (undimensioned) would appear daily in the newspaper puzzle pages." rectarith12.pdf or Googebra.htm (with Solving techniques)

I made a few thousand puzzles involving Squared Rectangles. In each case, the dissection is "nowhere neat", in that no two squares share a full edge. Also, all of the squares in each dissection have a size less than 100. I give two different types of puzzle for each dissection. MondrianPuzzles Here's my best nowhere-neat dissection of a size 35 square.

Puzzle Party: Brian Pelcher has written up some details of the Hakone Puzzle Party. Also, the results of the Nob Yoshigahara Puzzle Design Competition are available.
Sawing a Man in Half Trick: Here's a new take on a classic trick -- saw a man in half -- I have no idea how it's done.
A Proposal to Use Arabic Digits: Fibonacci (or Leonardo of Pisa) writes me about a new method for representing numbers -- with Arabic numerals instead of Roman numerals. The first edition came out in 1202, and none are known to survive. The second edition was written 1228, and there are 13 partial copies, and 2 complete copies. All this comes via David Singmaster, who took some great photographs.
: Also, he sent me something called the Rabbit's problem, with a curious sequence of numbers. Click on either picture for a larger version.

Nine Cool Points: To promote some demonstrations, I wrote a blog entry, Nine Cool Points on the Complex Plane.
Modernist Cuisine: The most advanced cookbook series ever prepared, Modernist Cuisine, by Nathan Myhrvold, delves deeply into the science of cooking, and what is possible. The photographs at his site look amazing.
Huge Ice Sheet: Greenland is now 260 sq km (100 sq miles) smaller, due to a large ice island breaking away.

Material added 25 June 2010

New Glider in Game of Life: Andrew J. Wade has constructed a spaceship with period (5120,1024)c/33699586. Announced at Game of Life News, the ConwayLife forum, the Game of Life Status Page, New Scientist, BoingBoing, and Slashdot. Golly runs it.
Two MG Puzzles: Carlos Penedo: Here are two easy puzzles for remembering Martin Gardner:
PUZZLE × 6 = GARDNER
PUZZLE × 6 = GOODBYE
Martin Gardner Issue of College Mathemetics Journal: Submissions are sought for the January 2012 issue of The College Mathematics Journal which will be devoted to the mathematics of Martin Gardner. In order to avoid duplication of effort, authors are strongly encouraged to write the editor of CMJ (cmj at oberlin.edu) describing their proposed paper. Final submissions must be received by February 1, 2011 to be assured of consideration for this issue.
Point Covering Problem: Naoki Inaba, who runs the fantastic Naoki Project (with hundreds of new puzzle types), gave me a puzzle in person at G4G9, and I'm finally getting around to transcribing it. You have a hundred identical coins. What is the minimal set of points that the coins cannot cover? Some thoughts on this -- you could spread out 101 points. You could also pick 99 points randomly in a small rectangle, and at least one of the points would be in the triangle between 3 coins. Send Answer.
Point Problem #2: From Joshua Socolar: There are 6 points. Any triangle of 3 points has an area ≤ 1. What is the maximal area for the hexagon defined by the 6 points? Send Answer.
Flash Anzan: 20 four digit numbers flash in front of you in 15 seconds. Can you instantly give the sum? In Japan, this has become a popular sport. Originally, it came out of clubs that were very, very good at using an abacus. After awhile, the abacus became a mental object, in a technique called flash anzan. There are also various youtube videos of flash anzan. Here is more anzan, and still more. [Update - Joseph Cooper: I saw the recent entry on your site about anzan software. I have just released a highly functional anzan practice app on Sourceforge: Little Soroban. under the GPL 3.0. It allows the user to: 1. Select the time interval between summands (to a decimal value), 2. Select the number of digits. 3. Select the number of summands.]
Polycube Symmetries: What are all the different symmetries of a polycube? George Sicherman has identified 33 distinct polycube symmetries.
Hirsch Conjecture Counterexample: From the abstract (arXiv:1006.2814) by Francisco Santos: The Hirsch Conjecture (1957) stated that the graph of a d-dimensional polytope with n facets cannot have (combinatorial) diameter greater than n-d. That is, that any two vertices of the polytope can be connected to each other by a path of at most n-d edges. This paper presents the rst counterexample to the conjecture.
Rhombus Puzzles: From 8 August 2002: Oyler has sent me a zip file with 20 number logic puzzles by Rhombus, from The Listener. Here is list of known errors. "They all appeared between 1960 and 1980. I've typed them in rather than scanning them as some of the photocopies are not very good. I was out with a magnifying glass for some of the puzzles in the Rhombus3 folder to try and discriminate between c and e and also 3 and 8 in some puzzles so be warned. I think they are all correct. I haven't myself got round to trying them as yet!! The numbers after the title refers to the number of the puzzle in The Listener Crossword series. Rhombus's last puzzle was All Square ( 2547 ) and appeared on 8th May 1980." If you have trouble with DOC files, openoffice.org has a good substitute program. [New -- Scott Marley sent a correction.]
Cast H and Other 3D Puzzles: Oskar has a new commercial puzzle out; Cast H. It's a very nice 2-piece maze, but not nearly as hard as it's rated. Other recent Cast puzzles are Cast Square and Cast Marble. 378% More Oskar is at the YouTube OskarPuzzle Channel, and also at the Shapeways Oskar shop. Shapeways recently featured Oskar in their column Puzzles, puzzles, and more puzzles. In addition, there is the Shapeways 3D Puzzle page. One of Oskar's puzzles was recently used in a wedding proposal.

Square Packing: I've made more progress with the Mrs. Perkins's Quilt problem. I've found lots of interesting papers... for example, a paper by Ian Gambini with square packings on cylinders. Erich Friedman adds a different one -- pack 33 unit squares into the smallest circle. "An easy problem solved for the first time." (by Maurizio Morandi). More square packing is in the June 2010 Math Magic.

100 Strategic Games: I found a blog by Walter Joris, who wrote one on my favorite books.
Outer Billiards, Arithmetic Graphs, and the Octagon: By Richard Evan Schwartz (arXiv:1006.2782): Outer Billiards is a geometrically inspired dynamical system based on a convex shape in the plane. In the case of the regular octagon, the case we study, the arithmetic graphs associated to periodic orbits are polygonal paths in R^8. We are interested in the asymptotic shapes of these polygonal paths, as the period tends to infinity. We show that the rescaled limit of essentially any sequence of these graphs converges to a fractal curve that simultaneously projects one way onto a variant of the Koch snowflake and another way onto a variant of the Sierpinski carpet. In a sense, this gives a complete description of the asymptotic behavior of the symbolic dynamics of the first return map.
Blue Ball Machine: How many balls in the machine?

Material added May 23, 2010

Martin Gardner: October 21, 1914 – May 22, 2010: Martin Gardner died yesterday in Norman, Oklahoma, at a retirement home (near S. Canadian Trails Dr and Chautauqua Ave). At age 95, he still had a fairly active life, working on a number of books, accepting many visitors, and corresponding with many people. For example, David Blaine recently visited him, then gave a private magic show for the other residents. So far as I know, his neighbors never found out why high-level mathematicians and magicians kept stopping by. Various remembrances of him are already online: Scientific American, Discover Magazine, Boing Boing, Wired, SlashDot, Huffington Post, Associated Press, You-tube, and the Richard Dawkins Foundation.

On Friday, I was showing David Talbot of Technology Review around the Wolfram office. His son was with him, so we focused on the periodic table we have there. Near the end, he asked me of what mathematical books I would recommend for his son, and I told him to get Martin Gardner's Mathematical Games. I had no idea that it would be the last time I gave him a strong recommendation while he was still alive.
Don Knuth Metapuzzle: Don Knuth - The Wikipedia page for MASYU has an example 10×10 puzzle with 23 clues. My problem for you is to show that only 20 of those clues are needed. (Solution - Alan O'Donnell: Numbering as a large chessboard, the white circles at E9, C5 and J3 are not required, so as to leave a uniquely solvable Masyu.)

New Erich Friedman Puzzles: Erich Friedman: Some of the new puzzle types since your last update are Aztec Math, Gap puzzles, One of Each, Chess Pack, Triple Letter, Chess Loop, Weight Equations, Precise Pointer, Birdwatching Puzzles, and Color Strip. [Two of his latest Math Magics: Packing Two Shapes and Strata and Ringed Tilings are also well worth a look.]
Mrs. Perkins's Quilt: I've been busy lately studying the Mrs. Perkins's Quilt problem, with Richard Guy and Stuart Anderson (squaring.net). At Wolfram Demonstrations, I've put together a demo with the first thousand solutions, many never seen before. In the weeks since I've posted it, though, we've made about 20 more improvements.

I Pentomino Exclusion: Naoki Inaba: For a given polyomino P, find a polyomino [without holes] Q such that P can't be covered with Q under the following conditions. You have enough Q and you can rotate or turn Q, but Q must not overlap with other Q. For example, If P is "I" pentomino then the attached picture is a solution of Q. I don't know solutions of the case that P is "U", "V", and "Z" pentomino.

Oskar Interview: Oskar van Deventer was recently interviewed by Shapeways.

Material added April 10, 2010

8000 Euros in Prizes for Magic Squares Results: Christian Boyer has announced some new Magic Square Prize Problems, in a recent press release. He's picked some interesting hard problems here. His main site is multimagie.com. For more on this topic, my favorite site is the Harvey Heinz Magic Squares site.
OEIS.org: The new location of the Online Encyclopedia of Integer Sequences http://oeis.org/classic/ is now available, along with various goodies. For example, sounds of the Recaman sequence, or the OEIS Movie. NJA Sloane has turned over all rights to OEIS to the new OEIS Foundation, which is looking for help and donations.
Some Packing Problems: Bram Cohen asks if 16 4×5×6×7 cuboids will pack into a 11×11×11×11 box. A nice problem of his with interesting solutions is to pack 8 7×9 and 8 8×11 rectangles in a 35×35 square.
DNA Fractal Globule: The DNA molecule is about 2 meters long, but fits in a very tiny space. There is now strong evidence that it packs into a fractal globule, similar to the hilbert curve.
Hyperbolic Mappings: Conformal mappings of hyperbolic geometry is a fascinating talk by Vladimir Bulatov.

Shuffling with ordered cards: I liked the paper Shuffling with ordered cards by Steve Butler and Ron Graham.
Single Tile Aperiodicity found: For a long time, whether a single tile exists that tiles only aperiodically has been an open question. The paper Aperiodic hexagonal tile by Joshua E. S. Socolar and Joan M. Taylor gives a solution.

Fermat Number Factoring: When Richard K. Guy turned 80, John Conway bet him $20 that there would be no new factorization of a Fermat number within the next 20 years. The last complete factoring was of F₁₁ in 1988. Now 93, RKG was quite happy with recent breakthroughs in factoring: F₁₂ has 17353230210429594579133099699123162989482444520899 · 2¹⁵ + 1 as a factor. March 27, 2010. Michael Vang.
F₁₄ has 1784180997819127957596374417642156545110881094717 · 2¹⁶ + 1 as a factor. February 3, 2010. Tapio Rajala.
F₁₉ has 8962167624028624126082526703 · 2²² + 1 as a factor. July 18, 2009. D. Bessell & Woltman.
F₂₂ has 3853959202444067657533632211 · 2²⁴ + 1 as a factor. March 26, 2010. David Bessell.
F₅₂ has 81909357657279 · 2⁵⁴ + 1 as a factor. March 15, 2010. Cedric Vonck.
The color-changing card trick: The color-changing card trick at quirkology.com is amazing, definitely a must-see. If you are able to see the trick, you can follow up by solving the perceptual whodunnit.

Trick Art Museum: Even more visual tricks can be seen in japan, at the Takao Trick Art Museum.
Math Movies: Burkard Polster: As I mentioned, I have been collecting math-related movies. In particular, here is a list of mathematical movies that we've collected. Hope you find something of interest to you there.
Nine Point Cubic, a Cubic Curve Gallery, and Triangle Cubics: My demonstration Nine-Point Cubic lets you move around 9 points to find the cubic equation that goes through those points. There are 45 basic types of cubic curve, which are compiled at the Gallery of Cubic Plane Curves by Steven J. Wilson. These are explored further in the gorgeous site Cubics in the Triangle Plane.

Approximation Feed: Kenneth Hammond: I have anapproximation Twitter feed where I post, once per day, an approximation that I find using Mathematica the night before. Two good ones that have come out of this experiment so far are K*716^(1/5) = 9.9999999723... (where K is Khinchin's constant), and the pandigital one shown here. I'd like to encourage others to join in by posting their own expressions on Twitter and hashtagging it with "#approx" so that it can be found on the search page. I believe there are some wonderful minds out there that will surely best--if not outright eclipse--all of these.
Mixed Polyhex Compatibility: George Sicherman: My Mixed Polyhex Compatibility now includes tetrahex-hexahex pairs. Some of them look highly improbable!

Random (blog): Some of the events of G4G9 are detailed by Neil Bickford at his Random (blog).
Facebook: Just in time for a South Park episode that attacks people getting too sucked-in to Facebook, I've gone and joined Facebook.

Material added 22 March 2010

Brain Busters: I've been doing the Brain Buster in the Japan Airlines in-flight magazine Skyward for a few years now. I put together a Brainbusters PDF that has most of these puzzles. Do feel free to mail me comments.

Grigori Perelman Wins Millennium Prize: From Slashdot: The Clay Mathematics Institute has announced its acceptance of Dr. Grigori Perelman's proof of the Poincaré conjecture and awarded the first Millennium Prize. Poincaré questioned whether there exists a method for determining whether a three-dimensional manifold is a spherical: is there a 3-manifold not homologous to the 3-sphere in which any loop can be gradually shrunk to a single point?
Gray Matter: Theo Gray's science experiments are now all online at Popular Science, under Gray Matter. The magazine has also put the entire 137-year Popular Science online.
Beaded Polyhedra: John Gowland: I do rugs and flat beadwork, and think of it as three dimensional graph paper. However, I am impressed with this 3D polyhedra beading; she got third prize at the TOHO (a type of bead) competition she entered.
Record 73-digit Factor of 2^1181-1 Found: From Number Theory List: We are pleased to announce that on the occasion of its 25th anniversary, ECM smashed through the 70-digit barrier by finding the 73-digit prime factor 1808422353177349564546512035512530001279481259854248860454348989451026887 of 2^1181-1 (thereby completing the factorization of 2^1181-1). It used a Playstation cluster.
Nob Yoshigahara Puzzle Design Competition: Nick Baxter: This year is the 10th annual Nob Yoshigahara Puzzle Design Competition. The entry deadline is May 31. The official web site (puzzleworld.org/DesignCompetition/) has all the particulars, and a history of the past entries.
Pi Day at Google: Google used the following logo for Pi day:

Material added 26 February 2010

A Fresh Look at Zome: A long time ago, I had a Zome System challenge to build a Petersen Graph with 15 blue struts of the same length. David MacMahon claimed the prize with an impossibility proof. Today, there is now a Zome Puzzle Kit available. There are also new lengths -- supershorts, extra longs, and half-lengths. There is also the excellent vZome software by Scott Vorthmann. In vZome, one can play with hypotheticals like the oranges and purple struts, which have not been manufactured. Brian Hall found a way to make the complete graph K9 in Zome, using these hypothetical struts. With the current system, K7 is possible in two different ways. Zomepad software is also available, currently in a reader form. Various papers and sites such as Mathematics of Zome (PDF by Tom Davis), and A Look at Zome (site by Chris Henrich). There's also the nice Metazome site.

4×4 Kakuro: Johan de Ruiter (of Puzzle Picnic): Bram de Laat (reigning Dutch sudoku champion) wondered about the existence of 4x4 kakuro puzzles. I found there are 5 essentially different ones (although not too much different) and they are particularly evil. Kakuro rules: Write a digit from 1 through 9 in every box. The numbers above the diagonal line represent the sum of the horizontal digit combinations, the numbers below the diagonal line give the sum of those vertically. Within a combination, no digit occurs more than once. [Ed - Nice find. Interactive versions are available at Puzzle Picnic.)

Diceagons: Tony Madison II: I wanted to turn you on to a brand new online puzzle which I have created. The game is called Diceagons and it is a very challenging math puzzle which involves polygons and dice. This is the first of a series of online games that I will be developing for my website Findagame.
Not a Wake (3.14): Mike Keith: Not A Wake is, quite simply, the longest Pilish text (in which the number of letters in successive words is required to follow the number of digits in the number π) ever constructed, and the first book-length work written entirely in Pilish. Divided into ten sections of 1000 digits, each written in a different style, its words "spell out" the first 10,000 digits of the number π = 3.14159265358979323846... [Ed - Certainly a record-setter.]
New Orleans Saints = Winners, last season: The Football Pool Problem, which gets its name from a lottery-type game where participants predict the outcome of soccer matches, is to determine the smallest covering code of radius one of ternary words of length v. For v = 6, the optimal solution is not known. [Ed - So, six games, with win, lose, draw. Find a set of 65-70 tickets that will always match at least 5 of the games. It's been unsolved for 30 years.]
A New Polyomino Oddity: George Sicherman: A pentomino compatibility is a shape that can be covered with two different types of pentomino. An odd pair is a set where the number of pentominoes used is odd. So, what's the smallest figure coverable with an odd number of X or T pentominoes? Here's a 137-piece solution.

Omnificent English Dictionary In Limerick Form: The OEDILF (Omnificent English Dictionary In Limerick Form) is a project to write limerick definitions of every word.

Material added 16 February 2010

Stacked Decks in Texas Hold'em Poker: Ben Joffe: I've been running simulations to try to solve ideal stacks for Texas Hold'em poker. The decks have the property that no matter which way that are cut, the same player will win the hand. I've solved it for 2, 3 and 4 players, and now trying to solve it for 5, see the results: benjoffe.com/holdem. [Ed-Neat! For example, the below is his solution for four players. The dealer will always win, for any cut.]

Puzzle Fun Online: One of the best defunct puzzle magazines was Puzzle Fun, which was run by Rodolfo Kurchan from 1994-2000. Rodolfo has now put the entire series of Puzzle Fun Online, and has started making new issues, all available for free.

Morpion Solitaire Updates: Christian Boyer: A new update of www.morpionsolitaire.com is now online: look at the February 2010 Morpion news! With a very new record from Japan. And with a scan of Bruneau’s “historic” letter of 1976 including his handwritten grid of 170 moves, which is still today the world record.

A New Keen Approximation: Gerson W. Barbosa: Is the following interesting enough to be a Keen Approximation: e - atan(e) = 1.49999892344176761360? [Ed - I think so. A nice find in the realm of almost integers.]
Triangle Dissections: From arxiv.org: An enumeration of equilateral triangle dissections. We enumerate all dissections of an equilateral triangle into smaller equilateral triangles. We conrm W. T. Tutte's claim that the smallest perfect dissection has size 15 and we nd all perfect dissections up to size 19.
Lots of digits of Pi: A new Pi computation record has been set by Fabrice Bellard - 2699999990000 decimal digits. The BBC did a news report. Of note, this was the first pi record set by a single personal computer.
The Christmas Star Rises: Don Knuth: Sudoku for Christmas. Here's a type of "jigsaw sudoku" that has a holiday message. [Click the link for Don's explanation.]

Factor of F_14 = 2^(2^14)+1 Found: The first factor of Fermat Number 14 has been found by Tapio Rajala, as part of GIMPS. The factor is 1784180997819127957596374417642156545110881094717 · 2^16 + 1.
Recursive Tilings: Herman Haverkor: This paper defines the Arrwwid number of a recursive tiling (or space-lling curve) as the smallest number a such that any ball Q can be covered by a tiles (or curve sections) with total volume O(volume(Q)).
Losing as Little as Possible: Vittorio Addona, Stan Wagon, and Herb Wilf (from arxiv.org): Suppose Alice has a coin with heads probability q and Bob has one with heads probability p > q. Now each of them will toss their coin n times, and Alice will win iff she gets more heads than Bob does. Evidently the game favors Bob, but for the given p, q, what is the choice of n that maximizes Alice’s chances of winning?
Thirteen Spheres: Oleg R. Musin, Alexey S. Tarasov (from arxiv.org): The thirteen spheres problem is asking if 13 equal size nonoverlapping spheres in three dimensions can touch another sphere of the same size. This problem was the subject of the famous discussion between Isaac Newton and David Gregory in 1694. The problem was solved by Schutte and van der Waerden only in 1953. A natural extension of this problem is the strong thirteen spheres problem (or the Tammes problem for 13 points) which asks to find an arrangement and the maximum radius of 13 equal size nonoverlapping spheres touching the unit sphere. In the paper we give a solution of this long-standing open problem in geometry.

Material added 9 February 2010

Two Decs, No Recs, 365 Solutions: I asked George Sicherman to extend his 9×9 results shown on 5 February in Two Shape Irregular Sudoku Squares. The rules: make a 10×10 square with two different decomino shapes, such that there are no internal rectangles of two or more pieces. It turns out there are exactly 365 solutions, an entire years worth. A twodec-norec Zipfile of pngs is available, as is a twodec PDF file. There is also a large twodec picture available, showing all 365 solutions at once (click on smaller image below). George wonders "I wonder how sparse a sudoku you can make with these." I'm wondering the same thing. Is there a good program available for working on irregular sudoku? Solutions by Bruce Harold.

Handy Programs: Looking for a handy math program? My current list of regularly used free programs includes Burr Tools, Golly, IrfanView, Jenn, SeifertView, Simon Tatham's Puzzles, Sudoku Susser, VLC Media. Some of my favorite free programming languages are Cygwin, GAP, Lua, PARI, Perl, Python, MiKTeX. My list of favorite purchased programs includes Mathematica, Crossword Compiler, Textpad, Geometry Expressions, Adobe CS4, E Editor, Stella 4D. Are there any good programs I should know about? Let me know.

Material added 8 February 2010

Orthogonal Game of Hip Update 2: Combining various entries here. The Game of Hip, by Martin Gardner, is played on a 6×6 board. Red and Blue alternate placing a stone of their color on the board. The first person to complete a square of their color, in any orientation, loses. Ignoring rotations and reflections, there is a unique tie game. Josh Geifer wondered what the largest solution would be if only orthogonal squares were considered. Both he and I managed to find several 10×10 solutions with no orthogonal squares, but none of them seemed extendable to 11×11. As a sidenote, Josh mentioned he's a son of TV writer Lewis Greifer, who wrote The Prisoner episode "The General" under pseudonym Joshua Adam, from his and his brother’s first names. We showed an 11x11 solution. William Rex Marshall computer-found a 12×12 drawn position with 72 counters of each colour, with rows in binary (71, 2716, 2505, 3186, 2846, 1461, 1699, 3129, 1356, 870, 1745, 3215). WRM followed up with a computer-found 13×13 drawn position, with rows in binary (427, 5346, 4663, 1925, 5420, 7321, 2548, 1119, 3890, 2665, 6467, 5080, 2702). I noticed a couple of repeating shapes in that solution, so I experimented with that shape, and found the below 14×14 position, with rows in binary (10724, 9551, 3882, 8825, 14611, 5064, 2718, 7764, 1266, 12839, 10129, 5436, 15529, 2533). This particular design cannot be extended to 15×15. Is there a drawn 15×15 position? Let me know.

New Orleans Saints = Winner, last season.: I've had a few accolades for anagrams, recently, at anagrammy.com. I also set a record for the two longest well-mixed pair of single source anagrams, with omnidirectional antennas, Daniel Constantine Marino. Some of the recent anagramming winners were:
Scarecrow, Tin Man, Lion, Dorothy = Mind, rosy heart, crown, location. (Ed Pegg Jr)
McDonalds Restaurant chain = Standard lunch to Americans. (Andrew Brehaut)
Skeletons in the cupboard = Bones unlocked their past. (Tony Crafter)
Director James Cameron = Set major cinema record. (Meyran Kraus)
Triple chocolate square = Atherosclerotic plaque. (Ed Pegg Jr)
Golden Anniversary = Dear Granny's in love. (Dharam Khalsa)
Righteous indignation = Idiot shouting in anger. (Paul Lusch)
A crisis on Wall Street ~ will start a recession. (Ellie Dent)
If love isn't here ~ then life is over. (Meyran Kraus)
Economist = :( :( :( emoticons. (Ed Pegg Jr)
"Astro Boy" = Say "Robot." (Adie Pena)

Material added 5 February 2010

Two Shape Irregular Sudoku Squares: I asked George Sicherman if he could find all ways to make a 9×9 square using only 2 enneominoes such that the 9×9 squares had no subrectangles of 2 or more pieces. He cranked up his programs and quickly sent me all possible solutions. The next question would be which of these have valid Sudoku solutions.

Eternity II Remains Unsolved: The two million dollar Eternity II puzzle remains unsolved. Eternity II is a very, very hard edgematching puzzle, similar to MacMahon squares. There is an Eternity II Yahoo group with much discussion. (And a Yahoo MathPuzzle Group, of course).
Math Dice: Eric Harshbarger: Ed, thought you might get a kick out of these math dice I created. I'd probably sell a set of 3 for $3.00 (discounts for large orders). [Ed - One puzzle from Eric's Logolog blog: make a 4×4 word square with 16 different letters. The link gives a list of all the solutions, all of them using obscure words.]

183-digit Prime-period Oscillator in Game of Life: Adam P. Goucher: The oscillator repeats every 2^607-1 generations. It is accomplished by a p8192 base loop (2^13) followed by 506 period doublers (2^506) and 44 period quadruplers (2^88) and a 1 generation glider advancer (-1). This corresponds to the Mersenne prime M607, which has 183 digits. This method can be extended to yield any Mersenne prime oscillator, including the recently discovered 12-million-digit megaprime. To avoid timing issues, the glider advancer is asynchronous. The input glider is allowed to arrive at any time (apart from a small 'window' where it collides with the clock glider). The oscillator (requires a cellular automata program like Golly to run).
Seven Staggering Sequences: Neil Sloane's paper Seven Staggering Sequences [PDF] is available on arxiv.org.

Material added 4 February 2010

Magic Tile: Roice Nelson: MagicTile is a new Rubik's Cube analogue which extends the original to regular polygonal tilings spanning all three constant curvature 2D geometries (spherical, flat, and hyperbolic). In this abstraction, Rubik's Cube follows from the special case of a tiling of squares on the sphere. A heptagonal tiling of the hyperbolic plane leads to a 24-colored puzzle based on Klein's Quartic! [Ed - This is an amazingly versatile and beautiful program.]

Math Magic - Spaced Out Polyominoes: The latest Math Magic has a very interesting challenge based on spaced-out polyforms. George Sicherman has already found some amazing solutions. Recent previous episodes have gotten beautiful results in sparse tilings, unique packings, and 3² + 4² = 5² tilings. George also recently found some vastly improved Odd Pairs.

More Erich Puzzles: Erich Friedman has also made several new puzzles for for his Puzzle Palace. These include Knight Tour, Latin Square, Line Segment, Chess Avoidance, and Chess Loop.

Polyhedral Calendar: Jean-Charles Meyrignac: I don't know if this is a new idea, but I found a free hexahedral calendar to do yourself. [Ed - Very nice. Two 6-pentagon stars, put together flat and with a rubber band weaving between the corners, will pop up into a dodecahedron, a fact mentioned by Martin Gardner.]
Multimagic Update: Christian Boyer: Various news, new enigmas, and recently developments on additive-multiplicative magic squares is available at multimagie.com.
Oskar's Cube on iPhone: M. Oskar van Deventer: Oskar's Cube (mechanical version) can also be played on iPhone now. The design and programming was done by Albert Leung and Mikheal Kruk, presented as Amazing Cube Maze at Whifflebird. It can be bought at iTunes as Amazing Cube Maze (link launches iTunes). [Ed - Object to App.]

Subway Shuffle as Puzzle: Bob Hearn: Just thought I'd drop you a line to let you know that my Subway Shuffle game has now been physically produced, by Popular Playthings. It was re-themed, and is now called "Athena" [Ed - App to Object.]

Material added 31 January 2010

Prime Curios: The book Prime Curios by Chris K. Caldwell and G. L. Honaker Jr. is well worth a look over at Amazon. Might be worth a look at Barnes and Nobles (Prime Curios), too, due to Amazon de-listing publishers, recently (Update: Amazon has relisted the publisher). Another source is the publisher, CreateSpace (Prime Curios). You can also visit the author's site - Prime Curios. Opening the book at random, 14593 is listed as the largest prime factor of 12345678, and 16033 is the first prime both followed and preceeded by 20 composite numbers. All sorts of interesting prime facts, as you might suspect.
Skypuzzles.com: Sky Williams: I only have like 100 hits, ever, compared to your 4 million. How's Google #1? Anyways, at Skypuzzles.com, I have a number of my puzzles that have been in Games Magazine. [Ed - A very interesting collection of puzzles.]
Crossword Sudoku: George Sicherman: A recent issue of Games Magazine included four Crossword Sudoku puzzles by Ken Futamura. This is the cleverest twist on sudoku I've seen yet! You use the clues to fill in the words, then fill in the rest of the matrix as a sudoku with letters instead of digits. They're almost as much fun to construct as to solve. Here's a Crossword Sudoku that I made in Ken's honor.
ACROSS. 2. Buck Rogers enemy Killer ______. 4. Suffix with mino-. 5. 2nd Amendment org. (abbr.) 8. Sound of hesitation. 9. Counterfeiter. 11. A giant word.; DOWN. 1. N.Y. baseball player. 2. A character in Street Fighter. 3. A kind of line. 6. A kind of sleep (abbr.). 7. What Crossword Sudoku puzzles may find. 8. Educ. inst. near DFW airport (abbr.). 9. What one may have with Crossword Sudoku puzzles. 10. In the morning (abbr.).
Four Million: Chris Lusby Taylor: I've just noticed that, at the bottom of your home page you claim "Yes, over 3 million" visitors. Actually, you are now over 4 million! Congratulations and many thanks for maintaining this wonderful site which never ceases to stimulate and amuse. Best wishes for 2010. [Ed - many thanks to you and all.]
Polynomial Plot: John Baez: "A while back, my friend Dan Christensen drew a picture of all the roots of all the polynomials of degree at most 5 with integer coefficients ranging from -4 to 4." Ed - That's the start of a fascinating column on Polynomials Roots, which has a fantastically detailed set of pictures.
Five Configurations: Arrange points and lines so that exactly 5 lines go through every point, and 5 points are on every line. Here's a new answer.

Material added 18 January 2010

Planetary Art: Each of the below is a close-up picture of some planet. Each picture is linked to the original source.
2010 MIT Mystery Hunt: The latest MIT Mystery Hunt just finished -- "The coin was found by Metaphysical Plant at 5:50 AM Sunday, January 17."
Fractal Pinwheel: Natalie Priebe Frank And Michael F. Whittaker wrote a very interesting paper called "A Fractal Version Of The Pinwheel Tiling."
$pinwheel fractal$
RSA Composite 768 factored: The 232-digit number known as RSA-768 has been factored by a large team.
2010: I sent a 2010 puzzle to Will Shortz for National Public Radio, and it got a lot of response from people like Joe Becker, Emrehan Halıcı, Robert Wainwright, Michael Reid, Alain Zalmanski, Jordi Domènech i Arnau, Juha Saukkola, and John Grobben. Some commentary is a Mazurland.; 2345*6/7 = 2010
2/3*45*67 = 2010
36 * 57 - 42 = 2010
Σ (n^3 - n/3), n = 0...9 = 2010
8*12*67*95 / 304 = 2010
0^7 + 1^9 + 2^8 + 3^6 + 4^5 = 2010
2*3*5*(7+11+13+17+19) = 2010
9+8*7+6*54*3*2+1 = 2010
2700 Billion Digits of Pi: Fabrice Bellard has managed to compute a lot of digits of Pi on a normal PC. All previous records were set on arrays in supercomputers.

Material added 23 December 2009

Christmas Tree Lane: I recently visited Christmas Tree Lane after giving a math talk at Fresno State. I noticed that it's an anagram of thermal resistance, and dressed appropriately.

Erich's Holiday Puzzles: Erich Friedman has put together a batch of Holiday Puzzles.

New Tightest Tetrahedra Packing: Back in August 2009, I put out a demo with a tetrahedra packing with density .782. That record didn't last long. The paper "Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra" by Amir Haji-Akbari, Michael Engel, Aaron S. Keys, Xiaoyu Zheng, Rolfe G. Petschek, Peter Palffy-Muhoray & Sharon C. Glotzer, published in Nature 462, 773 - 777 (2009), has found a packing with density 0.8502671806. The GlotzerLabWiki has a java demonstration and data. The new best packing shows quasi-crystal properties. A write-up is at Science Daily.


Facebook Puzzle Challenges: Derrick Schneider: Facebook has taken a page from Google and set up a bunch of puzzles for aspiring programmer employees. But even if you don't want to work there, the puzzles are available for anyone to try. Some thought-provoking challenges in there that are -- obviously -- heavy on the "write a program to solve this" attitude. Thought mathpuzzle readers might like it. [Ed- Some intriguing challenges here.]
The Prisoner: One of my favorite shows is The Prisoner, from the 1960's. I recently started going through the excellent Blueray set (it was on sale, but pricey again now). I also learned that AMC put the Prisoner series online.
Checkerboard Illusion vs Vacuuming: I quite liked how the Checkerboard Illusion vs Vacuum turned out.

Material added 30 November 2009

Polyform Semi-oddities: George Sicherman: As you were the first to encourage Mike and me in finding polyomino oddities, you may be interested in some polyform semi-oddities. So far I've found only partial impossibility proofs.
Recent Demonstrations (out of 5579): Here are some of the interactive demos that have been posted at demonstrations.wolfram.com recently. Click on an image to go to the demo.

Ignobel Prize Festivities: The festivities of the 2009 Ignobel Prize are featured at this week's NPR program, Science Fridays.

Material added 29 November 2009

Puzzle collector Laurie Brokenshire in Daily Mail: An article about Laurie Brokenshire talks about his thousands of puzzles. I got a chance to try out many of them when I visited him in England, as part of a working trip with Adrian Fisher, the master maze maker.

4D Magic Puzzles: Roice Nelson: The opensource Rubik analogue program MagicCube4D just had a major release supporting a huge number of new 4D puzzles. General duoprisms, the 4-Simplex, the Dodecahedral Prism, and the 120-Cell all have twisty puzzles associated with them now. Even the 4D Rubik's Cube variants were extended, and allow up to 9 cubies-per-side!

Jan Zoon Heptacubes: Jan Zoon: I sent a Christmas card to Kate Jones and she mentioned I should also send the picture to you.

I call the figure Notre Dame. The lower block is composed out of four complete sets of Heptacubes. The White and Black Heptacubes are set in blocks of 31. The blocks are placed in such a way that they alternate each other. They form the lower half of the big block. The upper half of the big block is composed of the same heptacubes but now one set of brown - white pieces and the other set of white - brown pieces.

The middle part is composed of four complete sets of hexacubes. Again two sets one black and one white in cubes of (10x10x10 with one additional tetracube). Here the cubes are also formed out of sub-blocks which are alternately placed in such a way that you can see the way the are built. The next layer is formed by the sets of white-brown and brown-white hexacubes. On top you find the two towers, They are made of four complete sets of pentacubes. For the base all white and brown pentacubes are used. Above them are the white-brown and the brown-white pentacubes. So in total all possible heptacubes, hexacubes and pentacubes are used to make this picture.[click on the image to see a much larger version]

A problem of squares: Bernardo Recamán Santos: Find a four-digit square number which has at least one digit in common with every other four-digit square number. [A nice little problem.]
The Complete Works of J S Bach: One odd purchase I made recently was the Complete Works of J S Bach, for a little over $100. I picked up Beethoven and Mozart, too, while I was at it. After listening to thousands of works, I'm pretty firmly a Bach guy. His Cantanas, which stretches to 60 CDs, are incredible. Almost every Bach work seems to have something marvelous. Beethoven's major works are incredible (perhaps 20 CDs worth) -- but his minor works don't seem as compelling. Mozart's major works are also incredible -- but I find myself returning to Bach's minor works.

Material added 27 November 2009

Old papers: I cleaned out my house over the past few weeks, and came across a birth certificate for my grandmother, born as Cecelia Marion. I learned that her parents were Hormidas O. Marion, and Virginia Greenough. I'd only heard of Hormidas mentioned once by my mom, in relation to a particular gold nugget. As a trapper in Canada, the nugget is one of the things he brought down with him, to sell if he got desperate. He got a job as a dishwasher in Fort Pierre, and gradually earned enough money to buy a large island, which became a profitable orchard. He gave the nugget to Virginia as a present before they married. The island became a large portion of what is now Pierre SD. Her brother was Louis Greenough. He built the fifth car in the world, shortly after seeing the first car at a world's fair. He created the first motorized bus, and was indirectly responsible for some of the first auto safety laws, via cities banning him from driving any of his vehicles into city limits.
Total[({57, 399, 679, 995, 1167, 1293}^k)] == Total[({115, 299, 767, 925, 1205, 1279}^k)]: The above sum of powers works for k=1, 3, 5, 7, or 9.
Tito Piezas: Jaroslaw Wroblewski found the second soln to the above system! This is: [57, 399, 679, 995, 1167, 1293] = [115, 299, 767, 925, 1205, 1279]. There is also a 6.6 partition such that it is good for k = 1,2,3,5,7,9: [-1205, -767, -299, 399, 995, 1167] = [-57, -679, -1293, 115, 925, 1279] just like the first one found in 2000 by Shuwen. In summary, for optimal multi-grades, there are,
(k,4,4), k = 1,2,3,5 (completely solved as quadratic forms)
(k,4,4), k = 1,2,4,6 (as quadratic forms, and elliptic curve soln)
(k,5,5), k = 1,2,3,5,7 (as quadratic forms, and elliptic curve soln)
(k,5,5), k = 1,2,4,6,8 (with only elliptic curve soln)
(k,6,6), k = 1,2,3,5,7,9 (unknown)
(k,6,6), k = 1,2,4,6,8,10 (with only elliptic curve soln).
Science, Technology, Engineering, and Mathematics: Mythbuster went to the White House, as a part of an event promoting STEM.
The Earth with Saturn-like Rings: I found this video of Saturn-like rings for the Earth quite fascinating.
Sorting Contest Animation: Dick Saunders Jr.: You might like the animations at A Sorting Contest.
National Sudoku Championship: Wei-Hwa Huang and Thomas Snyder, authors of Mutant Sudoku, were featured in a Time video article on the event.
Lottery Comparisons: I recently tried to find lists of what lottery games were played, where. I wasn't able to find any pick-6 type lotteries with 43, 50, or 51 balls.

3D Mandelbulb: A 3D version of the Mandelbrot set has gotten a lot of attention lately. A gorgeous rendering method is used.
Numb3rs reduced to 16 episodes in Season 6: For some reason, Numb3rs has been steadily losing its audience, and CBS has reduced the season order from 23 episodes to 16 episodes.

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