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Material added 21 November 04

Robert Abbott (, whose house is fine after going through the eyes of hurricanes Jeanne and Frances, recently did an excellent interview with Eric Shamblen ( See Robert's site for information on David McCullough's book on the history of mazes.

Bryce Herdt: I had is a puzzle I thought up while looking at a cracker. The cracker was circular and had 21 holes arranged in a square grid minus the corners. I wondered if there was a hexomino with unit squares the size of the grid, three copies of which could cover 18 holes orthogonally. I found one that happens to be a path hexomino. It's a nice puzzle, and I'd like to see if there are more solutions, too. Answer and Solvers. This problem led to Patrick Hamlyn's investigations, above.

James Stephens: Hi Ed. There is a new puzzle on the PuzzleBeast web site called the Bulbous Blob Puzzle. It is a project Oskar van Deventer and I collaborated on that has been in the hopper for quite some time. The challenges turned out nicely I think. Thanks again for hooking me up with Bill Ritchie for Tipover. The project has been a blast and it's great fun seeing it hit the stores. [Ed-I like Tipover a lot as a physical puzzle -- I would like to get a real copy of Bulbous Blob as well.]

James Propp has put together a page on the mysterious router-rotor model, and other studies of quasirandomness.

Henri Picciotto has a school course called Geometric Puzzles in the Classroom. His books on SuperTangram activities have gone out of print, so he now offers them freely. Also, he is selling tetratan sets there for $3, to make them more affordable for teachers.

Google has launched Google Scholar. I'll have to see how well it compares to CiteSeer and

f(f(x)) = x4 - 4*x3 + 8*x + 2. What is f(x)? If anyone knows of a good discussion of problems of this type, I'd like to see it. Answer and Solvers. From these techniques, I especially like the fact that f(x)-x divides f(f(x))-x for all polynomials.

My latest Math Games column is about Golomb Rulers, Sparse rulers, and Costas Arrays.

Allen Brady has determined Busy beaver S(3,3)≥92649163. I talk about the Turing machine Busy beaver problem in one of my Math Games columns. There are also details at MathWorld and Wikipedia.

A question going around: Write 56! as a product of 56 positive integers, so that the smallest is as large as possible. One solution is 15 × 16^3 × 17^3 × 18^8 × 19^2 × 20^12 × 21^9 × 22^5 × 23^2 × 26^4 × 29 × 31 × 37 × 41 × 43 × 47 × 53. Is that the best solution? The highest possible would be Floor[56!^(1/56)] = 21. What values of n! have particular nice factorizations into n integers? Answer and Solvers.

Jack Brennen found an interesting series in 899643225×2n-1, for n for 8 to 25. This gives 18 semiprimes in a row. Can anyone find 19 in a row?

899643225×28-1 == 173×1331263963
899643225×29-1 == 2347×196257917
899643225×210-1 == 53×17381786083
899643225×211-1 == 397×4640980667
899643225×212-1 == 821×4488354019
899643225×213-1 == 2351×3134784049
899643225×214-1 == 1531×9627534029
899643225×215-1 == 19×1551553115621
899643225×216-1 == 71×830408709769
899643225×217-1 == 11×10719821526109
899643225×218-1 == 193×1221948567743
899643225×219-1 == 101×4670021258899
899643225×220-1 == 532663×1770996473
899643225×221-1 == 499×3780939055301
899643225×222-1 == 9008023×418890713
899643225×223-1 == 19427×388467306037
899643225×224-1 == 122953×122758360583
899643225×225-1 == 5205467×5799098797

Dale Walton sent me several interesting dissections. Click on any of the images below to see a larger version. Dale Walton: For these dissections, the basic principal is that any scalene triangle may be broken into 4 similar triangles with areas in a geometric progression in three ways. For the areas to recur as the process is iterated, the sides should be integer powers of a base number n, where the base number satisfies a relationship such as n^0 + n^j = n^k where j and k are integers. I have seen this kind of relationship expressed in different form in some of the postings. One of the ways is 4 equal triangles. The other ways each contain a pair of equal triangles and one smaller and one bigger triangle. How much bigger and smaller depends which edges are parallel, and on the sequence of powers used to define the edges of the triangle (These power relationships can be studied independently of the base number). Then, for a given sequence, j's and k's can be chosen to allow the sums along the edges to add up to edge lengths in the same series, which in tern determines the base number. When this process is used, substitutions can be repeatedly made without exceeding a finite number of triangle areas, - and if the substitution is systematic, a pattern analogous to an "L" system is created.

Material added 11 November 04

I got mentioned in the November 9 Google Blog. Wow -- they called me a wimp! Hmmm ... I'll expect the Google staff to solve a few questions that came up at our recent Mathematica conference (posed by Michael Trott). (Wei-Hwa Huang wrote the 29 October GoogleBlog).

  1. Find a regularization procedure for divergent integrals in Rayleigh–Schrödinger perturbation theory with V[x] = λ*Exp[x^4] for the harmonic oscillator basis.
  2. Find functional equations for p(t; q) = Product[1 - t*qk, {k, 1, ∞}] with t=exp(i p/q π) and small integers p, q.
  3. Find the simplest finite energy, propagating, rational function solution (in x, y, z, and t) of the free field Maxwell equations.
  4. Find a proper spatial discretization for the numerical solution of the Schrödinger equation for potentials where wave packets return in finite time from reflection at ∞ (like V(x)=-exp(x)).
  5. Find a closed knotted field line of a current-carrying trefoil knot.
  6. Find the nth approximation of the half-space free particle path integral in closed form.
  7. Find the implicit polynomial form of the Poynting vector magnitude for the Lienard-Wiechert potential of a accelerated point charge.
  8. Find the spectral properties of vibration patterns in n-dimensional Hilbert curves (n=2,3,4,5,6,…).
  9. Find the qualitative and quantitative properties of geodesics on triple periodic minimal surfaces.
  10. Find the number of mutually unbiased bases in small nonprime dimensions (n=6 for a start).

About a year ago, James Stephens at PuzzleBeast introduced me to a puzzle he was working on: The Kung Fu Packing Crate Maze. I thought it was a great series of puzzles, so I told Thinkfun about it. They worked with Oskar van Deventer and George Miller (, and now they've released the puzzle set as Tipover, which you can now buy. At both sites, you can try out java versions of the puzzle.

01-Nov-2004: The Optimal Golomb Ruler with 24 marks has been found via a 4-year computer search by 41,805 people donated computer time to this experiment. OGR24 is the following: 9-24-4-1-59-25-7-11-2-10-39-14-3-44-26-8-40-6-21-15-16-19-22. A total of 555,529,785,505,835,800 rulers were checked. This page has more info on Golomb Rulers. Some more details can be seen in my Golomb Rulers column.

What's amazing about this sequence? 3,9,27,19,26,16,17,20,29,25,13,8,24,10,30,28,22,4,12,5,15,14,11,2,6,18,23,7,21,1. Answer: See section 4. Note that this sequence is simply the powers of 3 modulo 31. The differences, Mod 31, are all different. Can the same thing be done with sequence of length 32, or 33? This is a famous unsolved question. Some more details can be seen in my Golomb Rulers column.

You throw a ball over a flat field as hard as you can. Easy: what upward angle maximizes the distance away that the ball lands? Harder: what upward angle maximizes the distance that the ball travels? Answers. Don Reble sent a DVI file.

Eric Weisstein and I wrote up Seven Mathematical Tidbits. An interesting coin-removal game is analyzed at

John Baker developed some magic hexagons, and wonders if they are new. I don't recall seeing them myself. George Sicherman: I think I saw some magic hexagons long ago in a 19th century book of mathematical recreations. It may have been by Kraitchik.

Material added 3 November 04

Martin Gardner celebrated his 90th birthday on October 21, 2004.

Some of the items I'm recommending as possible gifts: Auf&Ab by me, Super Scrabble, Hercules Puzzle by Jean-Claude Constantin, Lunatic Lock by Gary Foshee, Free the Key by Oskar van Deventer, Yin and Yang by Doug Engel, T-Party by Perry McDaniel, Dovetail Burr by Junichi Yananose, Elk/Clef/Star Brass puzzles, Tantrix, Star by Ea Ea, River Crossing 2 by Andrea Gilbert, Einstein Brainteaser by Robert Rose, Third Degree by Bill Cutler, Mechanical Key Ring by Rocky Chiaro, Lockout Brainteaser, Lucky Clover by Oskar van Deventer, Oskar's Maze by Oskar van Deventer, Oskar's Cubes by Oskar van Deventer, Mathematical Constants by Steven Finch, The Million Word Crossword Dictionary by Stanley Newman, Mathematica Guidebooks by Michael Trott -- and the ultimate free stocking stuffer, the 2004 Particle Physics Booklet by PDG. Some programs I like are Mathematical Explorer by Wolfram Research, Quantian, Crossword Compiler 7, and WinEdt. is another puzzleshop to order from.

Dave Millar: Happy Halloween Ed! I've enclosed a link puzzle. The idea is to connect n and n with a path of n spaces.
1 in a box means to fill only that box with the color of the number.
2 in a box means to fill the box with the 2 and the adjacent box with a 2 in it.
3 and so on... in a box means to fill both of the boxes labeled 3(n) and create a path of 3(n) from one box to the other (inclusive, the numbered boxes are part of the path).

I made a super-symmetrical card game with the Hoffman-Singleton Graph for my Math Games column.

A very amusing visual code is hidden within the images of blob1 and blob2.

I heard various stories about political gerrymandering, and found all of the relevant maps at It doesn't look as bad as I had heard. Bob Harris: The congressional district maps are interesting. I think you underestimate the gerrymandering though. Take a look at Georgia. Districts 1 and 12 have unjustifiable arms jutting into other regions. Henry Baker: More cool election maps that distort the US map to account for voting amounts instead of land area:

Another recent find was the Computational Geometry Algorithms Library.

Material added 21 Oct 04 (Happy 90th birthday, Martin Gardner)

Kirk Bresniker: For a^3 + b^3 = c^2, with gcd(a,b,c)=1 and a<b<100000, if a and b are prime, then for a<b<100000, c has a factor of 12. Is this always true? Another fascinating picture he found was (a^3+b^3) - floor(sqrt(a^3+b^3))^2. Answers and Solvers.

11^3 + 37^3 = (2 * 2 * 3 * 19)^2 8929^3 + 28703^3 = (2 * 2 * 3 * 2 * 2 * 7 * 37 * 397)^2
1801^3 + 56999^3 = (2 * 2 * 3 * 5 * 7 * 32401)^2 44111^3 + 48817^3 = (2 * 2 * 3 * 2 * 2 * 7 * 11 * 3847)^2
2137^3 + 8663^3 = (2 * 2 * 3 * 3 * 5 * 4513)^2 57241^3 + 87959^3 = (2 * 2 * 3 * 5 * 11 * 44641)^2
6637^3 + 86291^3 = (2 * 2 * 3 * 2 * 2 * 11 * 31 * 1549)^2

Nick Baxter: "I updated the US team web site ( this morning with good news--Team USA came back with the team title. I've got more notes and pictures to add over time, but feel free to put something on your site as soon as possible; I know some people are dying to hear."

I wrote a Math Games column about the Riemann Hypothesis and ten trillion zeta zeros. Chris Caldwell pointed me at a zeta zero animation.

I've started an Editor's Pick at the Mathematica Infocenter.

The many different levels of P vs NP complexity can be seen at the fascinating site The Complexity Zoo.

Material added 16 Oct 04

Sometimes, news just pours in. This has been a big week for math and puzzles. First off, the World Puzzle Championship is happening in Opatija, Croatia right now. You can peruse the participants at the Feniks site, the enigmatski magazin of Croatia. At their site, you can see a PDF of the various puzzle types (not the puzzles themselves, apparently not yet released).

The US team was sponsored by Google, which recently released the math-heavy Google Labs Aptitude Test. I had a fun time solving all the problems with Mathematica. You can see the story and solutions at the Mathematica Google Aptitude story I cowrote with Eric Weisstein.

For awhile, I've been watching Zetagrid for them to hit the trillion mark -- to verify the Riemann hypothesis through the first trillion zeros of the Riemann Zeta Function. They are at 932 billion, so I thought I'd do a story about that next month. On 13 October, Xavier Gourdon and Patrick Demichel announced that they had used a more efficient technique by Odlyzko and Schonhage to find the first ten trillion zeroes of Zeta. His PDF paper on the topic is available.

Michael Trott's Mathematica Guidebooks have finally been released: Graphics and Programming. Symbolics will come out later. These are the most significant books for recreational mathematics to come out in the last decade. Why? He provides programs for visualizing more than a thousand wonderful things within mathematics, many of which will be new to you. If you don't believe me, you can read some other opinions of the book. As a sampler, you may also peruse dozens of items he couldn't quite fit into the book.

Gardner mentions the problem of making a torus with 48 equilateral triangles (a ring of 8 octahedra). John Conway improved this solution to 36 triangles, and called the solution unique (3 octahedra and 9 tetrahedra). It was rediscovered by Nikita Borisov (who made pictures). I tried to find it myself before the pictures were available, and found a different solution (2 octahedra, 12 tetrahedra). Are there better solutions? Is there a third configuration for 36 triangles?

Stewart R. Hinsley has made some very nice fractal tilings, including one of the Rauzy tile.

Keith Tyson has made a large puzzle for the London Times, involving 13 cubes -- A Work That Requires a Different Form of Investment.

Kevin Hare can further the bounds on a hypothetical odd perfect number if c301 = sigma((sigma(11^18))^16) can be factored. He has set up a page for tracking factorization efforts.

Gordon Royle found a 7-coloring of the Hoffman Singleton graph. From any vertex, there is a unique path of length 2 to any other non-adjacent vertex. There are no cycles of length 3 or 4 in this graph -- it's girth is 5. It can be proven that any graph with n vertices and more than n Sqrt[n-1]/2 edge has a girth less than 5. There are only 4 possible graphs that have this maximal number of edges: the pentagon, the Petersen graph, the Hoffman-Singleton graph, and a conjectured graph with 3250 vertices. Whether the last graph actually exists is one of the great unsolved questions in graph theory.

The Hoffman-Singleton graph.

Material added 11 Oct 04

You may download Prove or Disprove. 100 Conjectures from the OEIS by Ralf Stephan. Ralf is also maintaining a status page of the conjectures that have been proven.

MathWorld T-Shirts are now avaiable. In the same news story, Eric Weisstein welcomes me to my new job -- helping him with MathWorld. In other news, the Periodic Table Table that I assisted Theo Gray to make has won the Scientific American Sci/Tech Web Award.

Michael Rhoads has released Dream Travel and the18th Rule, a CD built entirely with the Rule 18 Cellular Automaton. You can compare it to the sounds of Automatous Monk, or Pascal's Triangle. Quite nice. Michael stopped by to visit me today, and showed me how he combined Mathematica with CSound to make the CD. If this new series of CDs is successful, he may follow up with a CD of Mathematical Constants.

I have Chaos Tiles back in stock! Probably not for long -- when the 15 remaining cases are gone, that will be the end of non-periodic dominoes. $100+shipping for a case of 12, $60+shipping for a mangled case of 12 (the domino pieces are still fine), and $15+shipping for individual games. You can also get my game Auf&Ab from FunAgain Games, made by the kind people at Franjos.

I turned the 666 puzzle into an MAA column. In a previous MAA column on WireWorld, I asked if an actual working calculator could be built. Mark Owen replied with his WireWorld Computer.

The 10th Annual Interactive Fiction Competition has started. For some beautiful paper folding patterns, take a look at Spidron by Daniel Erdely. Eric Harshbarger recently had a Puzzle Party. Cletus Emmanuel has found a new Carol Prime....(2^175749 - 1)^2 - 2.

Here is a sequence of sums of powers with no common factors. 11^3 + 37^3 = 228^2, 56^3 + 65^3 = 671^2, 57^3 + 112^3 = 1261^2, 312^3 + 217^3 = 6371^2, 433^3 + 242^3 = 9765^2, 877^3 + 851^3 = 35928^2. Can anyone find more cases of cube+cube = square where none of the numbers have a common factor?

Brian Trial: A Professor Yu Li has a site with some interesting circle packings using consecutive integer radii 1 - n. Scroll down to the bottom for the*.gif files, and accompanying *.txt files have coordinate lists.

Pierre-François Culand: Your popular and very interesting web sites are partly or essentially dedicated to sliding block puzzles. I’m not a specialist of puzzles nor of mathematics, nevertheless, a few weeks ago, I started to develop my own Windows sliding block solver program. My modest goal was only to get a convenient tool to help me to design, to play with, and to solve my own sliding block puzzles… and also, simply, to enjoy the fun of programming. The first result of this work, SBPSolver V1.6 is available as a small Windows freeware at I would be very grateful to you if you would be nice enough to have look on it and let me know your specialist opinion about it. [Good enough to mention here. I also like Taniguchi's Sliding Block solver.]

Glenn C. Rhoads: Joseph Myers found a 16-hex that is 10-isohedral. This seems to be the new record for the largest isohedral number. I'm attaching a postscript file which shows a fundamental domain (of size 20) illustrating how it tiles the plane. As a visual aid, I've shaded half the tiles so that both the set of clear tiles and the set of shaded tiles contain exactly one tile from each of the 10 transitivity classes. The clear and shaded regions are equivalent with respect to a 180 degree rotation plus a translation. [To view postscript files, get Ghostview for free.]

The Little Blue Book -- Particle Physics Booklet -- will soon be sent for free to all the people that have requested a copy.

Material added 29 Sep 04

The Doppleganger Maze, by Brian Smith. "You and a doppleganger(DP) start on the square marked S. You and DP take turns moving up, down, left, or right. After your move, DP will make a move in the opposite direction unless there is a wall blocking him, in that case he will not move. You and the DP can occupy the same square and do not block each other's movement. Your goal is to get yourself and the DP to occupy the square marked F (Either after your move or DP's move)." Answer and solvers.

The DoppleGanger Maze, by Brian Smith

On 3 January 1657, Fermat made a challenge to the mathematicians of Europe and England. He posed two problems involving S(n), the sum of the proper divisors of n: 1. Find a cube n such that n + S(n) is a square. 2. Find a square n such that n + S(n) is a cube. Frenicle de Bessy found ten solutions to the first problem within 2 days. Can you top Frenicle? Answer and Solvers. From the same era, The Emperor of France offers a kilogram of gold to the first person that can explain Chladni plates.

Prime Number Races by Andrew Granville and Greg Martin is quite good: . This is the best demonstration I've seen on the relationship between prime numbers and the Riemann Zeta Function zeroes.

The Collatz Conjecture is a notorious problem. Dan-Adrian German approached the problem by converting it to a fractal, and then analyzing the fractal. I like this technique. I'd like more people to look at it.

GarE Maxton has released Conundrum III, for the ultimate puzzle comprised of brass, pewter, white brass, bearing bronze, silicon bronze, aluminum bronze, steel, stainless steel, cast iron, aluminum, magnesium, titanium, tungsten carbide, monel, copper, oxygen free hard copper, neodymium iron boron, zinc, inconel, silver, and gold.

Erich Friedman put together his page of Polyform results for Math Magic. The Knight Covering of a 31x32 board seems quite interesting.

It is known that the first 144 digits of Pi equals 666. Ilan Honig points out that the first 146 digits of the golden ratio equals 666. That led me to wonder -- given a random irrational number, what is the probability that 666 will be hit exactly by taking successive digits? Answer and Solvers. Picture. is a new site devoted to games, and has started things of with the 100 Best Games of All Time.

Can K10, the complete graph on 10 nodes, be divided into 3 Petersen Graphs? If you can solve that, can you divide K50 into 7 Hoffman-Singleton graphs? The last one is an unsolved problem in graph theory. Enrique Garcia Moreno Esteva: It turns out that, definitively, K10 has no edge decomposition into Petersen graphs. In order to settle this, I wrote a Mathematica program. The program is fairly easy to write (two hours with lots of typos, 15 minutes without them) if you know what to do and how to use Mathematica/Combinatorica. It is quick and dirty, but it only takes 20 minutes to run on a three year old PC. It was also a very good exercise that gave me quite a bit of insight into the kinds of things and functionality that need to be added/improved/changed to have a strong Mathematica based graph theory package.

Chaim Goodman-Strauss has put up some excellent talks on tilings. Those who like tilings can consider the Aperiodic Summer School next year.

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