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Material added 13 April 2004

At, a new issue of The Life of Games is available.  Erich Friedman's Strobogrammatic Expressions at Math Magic this month is quite interesting.  At Fools Errand, you can see Cliff Johnson's April Fool's hunt. Cihan Altay will launch the PQRST 09 solving competition on 17 April 2004.  The below is one of Cihan's puzzles.  Divide the object into 2 identical shapes, or 3 identical shapes, or 4 identical shapes.

Cihan Altay puzzle

The 123456 folding puzzle is very nice.  Put it on cardboard, and have fun showing it to your friends.  Craig Kaplan's work on geometric ornamental tiling is fascinating.

I recently had the pleasure to try out the Quantian distribution. This is a math-intensive Linux distribution that runs directly off of a CD.  For any computer, you can put in the CD, reboot, and you'll be running Quantian Linux in a few minutes (keyboard, mouse, and all else are auto-detected).  When done, reboot, remove the CD, and you are back to your current system.  Your hard drive will be untouched. It has many math packages I've wanted to try for awhile.  Several of them, such as LyX and TeXmacs, have nontrivial set-up procedures.  I was delighted to be able to try them all.  GAP, Gnu Scientific Library, Python modules including Scientific and Numeric Python, apcalc, aplus, aribas, autoclass, DrGeo, euler, evolver, freefem, gambit, geg, geomview, ghemical, glpk, gnuplot, gperiodic, gri, gmt, gretl, lp-solve, mcl, mpqc, multimix, rasmol, plotutils, pgapack, pspp, pdl, rcalc, xaos, XLisp-Stat and xppaut. (I'm just scratching the surface.) A free download is at UofW.  It can be bought for $2.29 at I'm still going through the disk.  I didn't know there was this much free math software available.  Xaos has the best real-time Mandelbrot zooming I've ever seen.

There are arbitrarily long progressions of primes.  Eric Weisstein wrote a news article about it.

Material added 3 April 2004

Gathering for Gardner 6 was covered in today's New York Times.

William Gosper has a few copies left of his wonderful circle puzzle.  It's made from beautifully cut, inch thick plastic.  It was made by the geniuses at Puzzle Palace.

The monthly additions to the Puzzle Museum are well worth a look.

Patrick Hamlyn:  "Robert Reid sent me a very nice little manual puzzle which took several attempts over a week or two before I got it. Using cardboard, plastic, neoprene, high-density foam, icecream container lid or just paper: Cut out five straight trominos and three U-pentominos in the same scale. Now cover one set with the other. Real easy right? Everyone I've given it to swears blind it's impossible. Some get it in a few minutes, many (like me) take days or weeks. Here I reproduce Robert's 'spiel':  COVER - UP:  As usual the politicians have bbundered; they are trying to cover-up three twisted facts by five straight lies and they are getting nowhere. Can you show them how it is done? There is only one solution. "   [I got a copy of this as well -- it's a fantastic little puzzle.]

Berend Jan van der Zwaag: "How about Het Ding for unusual material making polyhedra?  It's basically six telephone poles and many metres of steel cable."

Material added 30 March 2004

Minami Kawasaki: "My friend Shigeyoshi Kamakura found the answer 23 (66*66)."   Robert Wainwright: "You may be interested in knowing that the recent Consecutive Square posting (square sizes 1...23 within a square of size k= 66) exhibits the "tightest" arrangement so far discovered. In correspondence with Robert Reid last year, we arrived at a measure of "tightness" based on the following:  Sqrt[k^2 - (SUM 1^2 ... n^2)]/n.  In otherwords, the square root of the exposed area divided by the size of the square enclosure. By this measure, Reid's n=43 (within a k=166 square) held the record with a value of 0.25686+.  Your recent posting has a corresponding value of 0.24595+."  This is an unexpected correction for core sequence A005842.  The conjecture that squares 1-26 can't fit into a 79x79 square seems weak, now.  See also sequence A0081287

Material added 22 March 2004

Bryce Herdt:  It's good to have something new for you. First, I have found a solution to the proportion SEVEN/NINTHS = 7/9. I checked several fractions before finding a doubly-true one. Has anyone else ventured into this arena?  Answer.

Boris Alexeev, who discovered the very nice 2004 = 2004 a few weeks ago, has won second place in the Intel Science Talent Search.  For his study into Automata, a frequent topic here, Boris has earned a $75,000 scholarship.  Congratulations, Boris!

Erich Friedman:  722855133 has the peculiar property that all of its length 3 substrings are divisible by 19.  722/19 = 38, 228/19 = 12, and so on.  For hand solvers -- find the largest number so that all length 2 substrings are distinct and divisible by 19.  For computer solvers, find the largest number so that all length 4 substrings are distinct and divisible by 19.  Is there a pattern to this?  Can any number higher than 19 be used?  Answer.

I took a look at the Asmus Schmidt method for complex continued fractions for a Math Games column.  Perhaps someone smarter than me can figure out that code that generates each generation.  Circles tangent to three others is Apollonian packing, and Eric has code for that.  The other part is a circle with 3 points picked on the diameter, and one must find three circles tangent to those given points and each other.  Likely, one of the Triangle Centers allows an easy answer.  Another tact is to try looking at the Moebius transforms of all circles in given generation, using the 8 special matrices Asmus Schmidt picked out.  I'd like to see generation 8 or so.  Send Answer.

The following is a little puzzle by Oyler.  Answer.

Simon Plouffe did an interesting analysis of approximations to physical constants

Here is a peek at a Math Games column I'm working on.  I built a Great Rhombicuboctahedron with supermagnetic spheres. Can you guess how their poles are aligned?  I'm looking for unusual materials for making polyhedra.  If you know of one, send me a note.  I'll also be attending Gathering for Gardner 6, and hope to have a full report written by Monday.

Great Rhombicuboctahedron

Material added 12 March 2004

<>John Gowland has put together a crossnumbers site about his many great logic puzzles.  Be sure to click on his Puzzles link, near the bottom of the page. It's all great stuff.  Speaking of logic, the Retrograde Analysis Corner recently got a large update.

Nyles Heise: "In response to the $20 challenge posted on 6 July 2003, I've built several wireworld implementations of a Turing Machine.  I'm including two of them here.  The Turing machines do unary multiplications.  I set out to do the exact algorithm found in A. K. Dewdney's "The New Turing Ominibus", but backed of slightly by leaving the original numbers on the tape along with the product.  The first design has 6 states plus halt and 4 symbols.  It is basically the Nich Gardner approach.  That is, it has specific latches to simulate the symbols on the tape.  The symbols are read from and written to the tape location under the R/W head.  The shift register is shifted either left or right one symbol.  It's set up to do 4 = 2 X 2.  The second design has 9 states plus halt and 2 symbols.  This is Karl Scherer's approach where the tape is simulated as a loop in which electrons pass by the control mechanism and get modified as required. It's set up to do 5 X 4 = 20.  Quite detailed descriptions of both machines are included in the .mcl and viewable by clicking the blue information button."  Well, that will win it.  You'll need Mirek's Cellebration to run these.

WireWorld Turing Machine

Michael Rios sent me an interesting problem.  What is the largest possible area of a quadrilateral with sides 1, 2, 3, and 4?  Answer.  If you can figure that out, try solving (2,3,4,5) and (3,4,5,6).  See a pattern?  A general solution eludes me.

Oskar van Deventer has added the Tunable Maze to  You can tune this fractal maze while you are solving it.

Tunable Maze

George Sicherman is looking at the Polyiamond Exclusion problem (which apparently has never been considered).  He would especially like to hear from anyone that can conclusively solve the straight polyiamonds of length 3 and 5.

Eric Weisstein has started a page of Integer Sequence Primes.  If you happen to find a prime beyond what we have listed there, send us details.  Also, if you find a vast amount of nothing over a large range, send us details about that, too.  Anything new will likely go into the Probable Prime database.  Also, the Aspenleaf Concepts site lists many prime result lists.

Dick Hess has done some research into Mixed Doubles tournaments.  Two separate problems are of interest.  (1)  Schedule a group of 4n doubles players on n courts so they play as many rounds as possible without opposing or partnering the same player twice.  It's OK for partners in one round to be opponents in another round.  (2)  The same as above except there are 2n men (designated by odd numbers) and 2n women (even numbers) that always play mixed doubles.  The results below will give you a flavor of this interesting problem.  I look forward to hearing from you and thank you for any help you can offer. If you can improve these solutions, please let Dick Hess know (ri(hislastname) at  Also, write me.

For 8 players there is a 3 round solution that holds for both problems and even allows married couples (1 with 6, 2 with 5, 3 with 8 and 4 with 7) to participate without ever playing with or against each other.
1: 1 2 3 4  5 6 7 8
2: 1 4 5 8  3 2 7 6
3: 1 8 7 2  3 6 5 4
For 12 players the situation is more interesting.  I ran exhaustive search computer programs and believe 4 rounds is maximum for men's or ladies' doubles but only 3 rounds is possible for mixed.
Men's or ladies' doubles:
1: 1 2 3 4  5 6 7 8  9101112
2: 1 3 5 7  2 9 610  411 812   
3: 1 4 6 9  2 8 712  3 51011
4: 1 71012  2 3 8 9  4 6 510
Mixed doubles:
1: 1 2 3 4  5 6 7 8  9101112
2: 1 4 5 8  3 6 912  7 21110
3: 1 611 2  312 5 4  710 9 8
For 16 players there is an elegant solution that allows us to schedule 8 married couples in 7 rounds so that spouses never play as partners or opponents.  The same solution works for men's doubles as well.
1: 1 2 3 4  5 6 7 8  9101112 13141516
2: 1 4 5 8  3 2 7 6  9121316 11101514
3: 1 6 914  3 81116  5 21310  7 41512
4: 1 81510  3 61312  514 9 2  71611 4
5: 11013 6  31215 8  5 41114  7 2 916
6: 11211 2  310 9 4  51615 6  71413 8
7: 114 712  316510  9 615 4  11 813 2
For 20 players the best I can do for mixed doubles is the 7 rounds shown below.  I have yet to explore the case for men's or ladies' doubles.  Nine rounds is the theoretical maximum but my search program has only looked at a small fraction of the full set of possible solutions.
1: 1  2  3  4   5 6 7 8  9101112 13141516 17181920
2: 1  4  5  8   3 2 7 6  9121316 11101720 15141918
3: 1  6  9 14   3 81116  5 21310  72019 4 15181712
4: 1  8  7 18   31015 4  5161120  9 21714 131219 6
5: 1 16 17 10   3141320  518 9 6  7 41512 11 819 2
6: 1 18 13  2   3 61910  5 41716  7121114  9 81520
7: 1 20 15  6   31217 8  5101914  7 2 916 11 41318

I put together a column on Number Games as part of my Math Games series.  One thing I mentioned in the previous column was Artlandia's SymmetryWorks.  Artlandia has just released Version 3.

Material added 29 February 2004

A new puzzle-maze by Oskar van Deventer, Lab Mouse, is now available courtesy of Henley Mob.

Jan Kristian Haugland has put together a series of Amazons puzzles, along with numerous other constructions.

The Metamath site has a free set of handy mathematical symbols, 3000 proofs, and proofs set to song.  Lots and lots of formal proofs in HTML-rendered abstract mathematical notation. A sample: 5 ((R1A) ⊆ xA ℘(R1x) → ∀x(R1A) ⊆ xA ℘(R1x)). 

A new Kuiper Belt object has been discovered. "2004 DW" is estimated to be 1650 kilometers in diameter.  Pluto, also in the Kuiper Belt, is 2320 kilometers in diameter.  This is the largest solar system object to be discovered since Pluto.

Mathematician John Rainwater published an extensive number of papers, undeterred by his own nonexistence.

Henry Segerman has put together a very nice illustration of the Poincare Disk.  You can also look at his Cantor Set, Entropy, Dodecahedron, Fractal, Mandelbrot, and other Autoglyphics.  His Colloquium Posters are quite nice as well.

Material added 22 February 2004

National Public Radio had a nice piece about Persi Diaconis and his coin flipping machine.  (His home page) If you flip a quarter and catch it in your hand, there is a 51% chance it will land on heads.  The report ends by saying that a coin at a football game has different odds, since it is landing on the ground, but the odds haven't been figured out. 

I had a chance to listen to Bill Gates during his High Campus Tour.  UIUC was his first stop, and he's going on to Harvard, MIT, and Carnegie-Mellon.  He talked about some of the computer science he's involved in.  Bill Gates: "I've been getting a lot of interest mail recently. I don't know if you are getting some of these same offers.  The most interesting ones say that for just dollars a month, they will pay all my legal bills. I'm not sure if I was the intended recipient for those.  A few days ago, my 7 year old  came in and woke us to say that she'd been trying out the computer.  'It's amazing!' she said.  So I said 'Okay. Keep using it.' But she went on. 'No, no, no! We Won! We won money, dad!'  I didn't want to say something flip like 'We don't need more money' so I got up, and of course it was one of those come-on type things, and there's my seven year old thinking she's won some kind of amazing contest. I had to explain to her that it was someone trying to get her to go to that website.  We have a lot of work in getting the computer to model our interests. What is worth interrupting us for?"  A great talk.  If you'd like to hear it, write me, and I'll give you the address for a recording.

In 2001, I published Hamlyn's Challenge.  At the time, I was planning to have a major prize contest, with the hexominoes sold in a nice cube.  But I never figured out how to get the 37 pieces made in a nice bakelite material at a price anywhere near what I could afford.  Barry Rawson just solved what the contest was going to be.

Hamlyn's Challenge

Though I didn't get the hexominoes made, the R.G.B Round-up puzzle is made in bakelite, so it has that great feel, in addition to being a great puzzle.

Material added 14 February 2004

Erich Friedman suggests the annual puzzle of writing the current year with a minimal number of one digit:  2^(22/2) - 2x22 = 2004.   (333+3/3)x(3+3) = 2004.  ((6+6+6)/6)x666 + 6 = 2004.   4^(4+(4+4+4)/(4+4)) - 44 = 2004.   What are your best efforts for 0-9?   Answers and Solvers.

Evan Variano and Paul Chaikin did a study on random packings.  Randomly placed spheres pack with an average density of 64 percent.  Randomly placed M&Ms pack with an average density of 71 percent.  The maximum density for both shapes is 74 percent.

I put together a column on Guillochè patterns as part of my Math Games series.

MathWorld has been redesigned.  Also, A New Kind of Science is now available online.

Jorge Mireles has put together an applet for studying snake-like length defined grid spirals.  He's made lots of discoveries with these.

Length 5 Spiral

Giovanni has put together his own page on polypolyomino solutions.  Colonel George Sicherman extended the problem to polypolyiamonds.  Both nicely extend the Livio Zucca polypolyomino pages.

Phillippe Fondanaiche found a very interesting solution for packing squares of area 1-100.

Greg Dye pointed out to me some Flash Optical Illusions at .

Andrea Gilbert recommended the Quantum TicTacToe applet.

A nice write-up about Tetris inventor Alexey Pajitnov is available at BBC News Online.

Material added 2 February 2004

Mary Youngquist Hazard has passed away.  She was the supreme solver of the National Puzzlers League, solving every puzzle published for several decades.  I remember one transposal puzzle with the base haplomitoses/omphalosites that only she could solve. Back in 1978, when I was much younger, she wrote to me with some praise for one of my first efforts. In addition to being a top-notch solver, she composed many puzzles and poems. Her most famous puzzle is a poem:

Winter Reigns

Shimmering, gleaming, glistening glow--
Winter reigns, splendiferous snow!
Won't this sight, this stainless scene,
Endlessly yield days supreme?

Eying ground, deep piled, delights
Skiers scaling garish heights.
Still like eagles soaring, glide
Eager racers; show-offs slide.

Ecstatic children, noses scarved--
Dancing gnomes, seem magic carved--
Doing graceful leaps. Snowballs,
Swishing globules, sail low walls.

Surely year-end's special lure
Eases sorrow we endure,
Every year renews shared dream,
Memories sweet, that timeless stream.

--Mary Youngquist--

Material added 1 February 2004

The American Mathematical Society has a lot of gorgeous mathematical art at Mathematical Moments.  Well worth a look.  I especially like the polyhedral origami with dollar bills.  Also quite nice is their series of columns -- What's New in Mathematics.  I also found +plus magazine there.  In the online Notices, I rather liked the opinion article Fleeced, on the high cost of journals.  The Bulletin is also online.  I liked the article about the proof of Catalan's conjecture.

I highly recommend everything at George Miller's Puzzle Palace.  There are various designs there by Oskar van Deventer, Donald Knuth, Archimedes, Greg Frederickson, Dic Sonneveld, Frans de Vreugd and myself that you won't find anywhere else.  George makes limited batches of excellent puzzles, and he's offering the extras. If you appreciate well made, well designed puzzles, this and Kadon are the places to go.

There is indeed a method for fitting squares of size 1-37 in a 133x133 square.  I've commission Kate Jones to make an artistic realization, one for myself and one for the winner, Berend Jan van der Zwaag. The solution he found by hand is below. Kate's version will look much better, if you're interested in one, visit Kadon Enterprises and write to them about Sequential Squares.  Plenty of other square packing problems are mentioned in my Square Packing column.
37 Squares.

Cihan Altay sent me the below puzzle mere seconds after I posted Dick Hess's puzzles.  AnswerSolving Program.
Cihan Altay's Degree

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