Isosceles Right Triangles in Circles
Erich Friedman has added a new page to his Packing Center
: Tans in Circles
. The page includes diagrams of five new packings, which Eric found this month. He is looking for improvements.
Math Games: Chessboard Tasks
My latest column is on Chessboard and Grid tasks
. I like staying cutting edge on these topics, so I was quite pleased to receive a note from Guenter Stertenbrink a few hours after my column went live, about progress with the Queens graph
. The queen graph has been proven impossible for 2, 3, 4, 6, 8, 9, 10. Solutions have been found for all other cases up to 25. The state of the order-26 queen graph is unresolved. It is hypothesized that the queen graph is solvable for all n>10.
I played around with Goldbach's conjecture
a few days ago, and made a list. 3 and 5 are necessary for 3+5=8. 7 is necessary for 5+7 = 12. 11 seemed to be completely unnecessary, though, so I pulled it out. 13 is then needed for 5+13 = 18 (can't use 7+11=18, since I've ruled 11 unnecessary.) And so on, looking at each prime, and determining whether they are necessary or unnecessary. Jacques Tramu confirmed and extended my results. The following primes seem to be completely unnecessary, up to 60000. 11 17 29 41 59 67 71 73 89 97 103 127 137 149 151 163 173 179 181 191 193 197 223 227 229 233 239 241 257 263 271 277 311 317 331 347 349 353 359 367 373 379 389 397 409 419 433 443 461 463 467 479 487 499 503 541 547 557 563 571 577 587 593 599 607 613 617 619 631 641 647 653 659 661 677. So far, roughly 3/5th of the primes are unnecessary. Can anyone extend this concept within the even numbers under a trillion? Does the distribution continue to be 2 to 3 necessary to unnecessary? You might enjoy looking for the sum for 208 that avoids unnecessary primes.
The Escher configuration of a 3-cube compound
creates 67 different cells. Is this the best possible?
[You might enjoy the Adaptive IQ Test
that Chris helped to put together.]
Kai G. Gauer
An interestingly difficult Kriegspiel problem
by Geoffrey Foster is available. A Kriegspiel applet
Eric Friedman and Adam Landsberg have proven the 3×n
game of Chomp always has a unique first move. See their paper at ftp://ftp.orie.cornell.edu/pub/techreps/TR1422.pdf
Stable Tents contest
Erich Friedman has turned 40, and is doing a contest on stable tents
at Math Magic. You can see the current records
for the problem.
Densest Packings of Equal Spheres in a Cube
Hugo Pfoertner has made a page of the Densest Packings of Equal Spheres in a Cube
Math License Plate Contest
The Math License Plate Contest
yielded some nice plates. (Mine is MTHPZZL, as it turns out.)
Page of Math Errors
The Page of Math Errors
is good material.
For Chess, Go, or organized sports, people often get high quality components to enhance the game. One of my favorite games, Mafia (Princeton rules
, NTNU rules
), is now available as a nice card deck as The Werewolves of Miller's Hollow
In the early 80's, I came across the game of Morpion
in the French magazine 'Jeux & Stratégie'. They gave a 170 move solution that was found by hand. This solution is available at the site. It appears that no progress has been made in 20 years. Do you happen to know about any progress, or could you direct me to a place where I can discuss this game? I am also interested in the general solution, i.e. the maximum number of moves for an arbitrary pattern of N given points. [I don't know of any progress.]
Largest Factor Ever Found by the ECM Technique
A factor of 3466
+1 has been found: 709601635082267320966424084955776789770864725643996885415676682297
. It's the largest factor ever found by the ECM technique
. Other large factors are maintained by the Cunningham project
Good Math Books
I'm always looking for good math books. One recent find was the Handbook of Mathematics
by Bronshtein, Semendyayev, Musiol, and Muehlig. I'd actually seen this book before in a German edition,a language I do not know. As I paged through it, I thought "wow, this is a good book." I just recently found the English edition.
NYT Marks 100th Anniversary of Einstein's Most Famous Paper
The New York Times
devoted an entire page to Physics
on the subject of the 100th anniversary of Einstein's most famous paper.
Multiple-unit Dodecahedral Constructions
If you have too many dodecahedra, you might try making some Multiple-unit Dodecahedral Constructions
starts on April 23rd, Saturday at 20:00 (GMT+02).
Imagine my surprise when I visited MathForge
[ Permanent Link
Brian Prentice: I enjoy reading your Math Games
pages and exploring the ideas that you illustrate. Here is a program that your readers may find interesting: Tiled CA
. This program runs cellular automata simulations on a large number of grids which can be constructed from various shaped tiles. These tile shapes can be triangles, squares, parallelograms, pentagons, hexagons or octagons. A grid editor is included with which new grid definitions may be constructed or existing grid definitions may be modified. Are there any grids for which there is no rule supporting gliders? [Tiled CA
is the most gorgeous Windows program I've seen in awhile.]
Material added 29 March 05
Math Games: A Zillion Connection Games
My latest Math Games column
concerns Cameron Browne's book Connection Games
, as well as the game engine Zillions of Games
. I hadn't looked at the ZOG site
carefully in awhile, and was quite pleased to see the well-organized game list
as I started writing. The column has my usual gaggle of links to other neat places. One site I didn't know about was the complete solution to 7x7 Hex
Game of Life: 17c/45 Caterpillar Spaceship
A new speed has been found for a spaceship in the Game of Life: The 17c/45 Caterpillar spaceship
. To watch it in action all at once, you'll need a 4195x330721 pixel monitor.
It was discovered by Gabriel Nivasch, Jason Summers, and David Bell. See Eppstein's page for other glider news
. You can also see his paper Searching for Spaceships
(PDF) in the online book More Games of No Chance
Friedman's Tangrams Pages
Erich Friedman has starting two new packing pages, Tans in Squares
, Squares in Tans
, and Tans in Tans
. If you can improve any of these, write to Erich. On the Tans in Tans
page, you'll see that I improved on his first result for 3 tans in a tan. I knew it would be a good puzzle, the answer there isn't the best possible, numerically. Can you sort through the simultaneous equations and find the exact solution?
Y-pentomino 12-Fold Replica
One long unsolved problem has been whether the y-pentomino could make a 12-fold replica of itself. Patrick Hamlyn hates unsolved polyform problems. His polyform solving program can usually find solutions in microseconds. Here, it took "116 hours, 296 million 'off-by one' partials, 41 billion piece placements." Still unsolved is why this particular problem was so difficult, and why this particular solution worked.
Sums of Powers
Somewhere, I discuss the x^3 + y^3 = z^2 problem. I recently learned of Dario Alpern's Sums of Powers
page. For an older puzzle, try the following:
has been discovered.
Melvyn Knight once asked about solutions to N = ( x + y + z )*( 1/x + 1/y + 1/z ). For example, with N=103, the smallest solution is x=14156395253, y=-131237206100, z=1736693066100. For a much harder problem, try N=888. Some numbers don't have solutions, such as N=5. Using elliptic curves, it was possible to isolate all the N which might have solutions. After a multi-year search, every number from -1000 to 1000 has been resolved in the Knight problem
You probably know that most of the regular polyhedra
are closely related to the golden ratio
, or Fibonacci constant
. Did you know that the Snub Cube
is related to the tribonacci constant
? It was news to me.
, you can now try out a formula search
, within the world's largest database of functions. For example, if I'm looking for a function with Catalan, Pi, and E, the search points me to http://functions.wolfram.com/Constants/Glaisher/27/0001/index.html
You can also try out function plotting
, for any standard function. Do let me know if you have any comments on either of these. Write me
problem of the month concerns bishops and knights on the chessboard, and is generating interesting results.
Adrian Fisher (the MazeMaker): I was impressed by Col George Sicherman’s delightful 4-piece Foxagon puzzle
, which creates a side-2 hexagon. Here is my follow-up to it, the Goose-agon puzzle
(because in England we have a board game called Fox and Geese) with 9 pieces, which creates a side-3 hexagon.
I rather like Turmite music
Michael Keller has launched the Solitaire Laboratory
The Crank Problem
Karl Mahlburg has solved
an infamous problem by Ramanujan, which deals with Partition Function Congruences
. The question (known as the crank problem) is basically "Why are 5, 7, and 11 special in partitions?"
Colonel's 4-piece Foxagon puzzle
The Colonel's 4-piece Foxagon puzzle
is now available online. This is a fantastic little puzzle I've had on my desk for months. You simply need to make a hexagon with the pieces so that nothing matches.
Andrea Gilbert has gone medieval on us, and challenges with the XL-maze
. Also, she did an interview for PuzzleMonster
David Wilson: I submit three "ancient mathematical documents" that have been damaged and need to be deciphered and restored.
Math Games: Keen Approximations 2
My latest Math Games
column is a vastly updated Keen Approximations
. Below is my current list of the
most amazing approximations. The two winners of my contest are Titus Piezas III and Derek Ross. My favorite here is Golden ratio + Feigenbaum alpha + (7/8)Catalan's constant
~ 0.0000000014117 Still no counterexamples to the Uniformity Conjecture
. Titus wrote a great article --
On Keen Approximations
George Bell has made an ultimate Triangular Solitaire
page. NPR did an audio story about Don Knuth
. NPR also talked about crocheted hyperbolic surfaces
Semiprime Unfactorable Number?
Awhile back I wondered if anyone could find an unfactorable number that was provably a semiprime. Don Reble found one
. Phil Carmody strengthened it
Gaussian Moat | Smith Numbers | Pronunciation | Crosswords | Crypto Standards
I just came across a recent paper on the Gaussian Moat problem
Shyam Sunder Gupta has written an extraordinary Smith Numbers
I also came across a Mathematics Pronunciation Guide
The American Crossword Tournament
recently took place, and got on CNN
The NSA has announced new cryptography standards
Rube Goldberg devices are interesting to watch. One particularly nice film is a 2 minute Honda commercial
. Another is The Way Things Go
OpenOffice 2.0 Beta
OpenOffice 2.0 Beta
recently got released. I like it. I'm hoping to try it with OOoLatexEquation
Wolfram Notebook Indexer
If you have Windows, Mathematica, and Google Desktop Search, you can try the Wolfram Notebook Indexer