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Material added 30 September 2003

Robert Abbott: "I've been working on a new series of mazes, called "Eyeball Mazes."  They are at this address: http://www.logicmazes.com/eyeball.html."  Of course, just like anything Robert Abbott does, these are very cool, and are highly interesting to study. So, take a gander at Eyeball Mazes as the Puzzle of the Week. (It's one of the year's most popular puzzles.  Solvers.)

Patrick Hamlyn and Andrew Clarke are the world's top polyformists, but they were perplexed by a seemingly simple (to them) task -- making a rectangle with the 227 balanced, non-holey octominoes.  Andrew Clarke found the below non-solution.  Patrick found over 77 million different ways to place 226 of the pieces.  Was there a way to prove a solution is impossible? Andrew Clarke: "Classify the balanced pieces as even if they can be divided up into dominoes with an even number in each direction and odd if they are 1-3 in direction. For those which cannot be divided into dominoes add a domino so that the new shape can be divided into dominoes. In this division remove one domino that is in the same orientation as the added domino and then classify as before. If my count is correct there is an odd number of odd pieces and, since any rectangle will be even, no rectangle can be made."   Beautiful.

Jorge Mireles recently posted a solution to the ILOST tetromino problem, which is a shape that can be covered with any of the tetrominoes (I L O S and T). Michael Reid knows of a shape with area 32 that can accomplish the same task. Can you find it? Solved by Clinton Weaver and Bob Wainwright.

The 2003 Interactive Fiction Competition has started.  Feel free to vote for your favorites.

I've been playing around with Graphs recently.  I learned from H S M Coxeter's Beauty of Geometry that the Desargues graph can be labeled in an interesting way -- the vertices are all the ways of choosing two or three numbers from {1,2,3,4,5}. The edges connect vertices that differ by adding or subtracting a number.  Can you find the labelling?  Another graph I looked at recently was the Foster Graph. With a computer search, I managed to find a nice embedding. The Foster Graph has an interesting characteristic polynomial: (x-3)(x²-4)⁹(x)¹⁰(x²-6)¹²(x²-1)¹⁸ -- this level of factorability is exceedinngly rare -- it indicates how much symmetry this graph has.

Material added 26 September 2003

Eric Solomon:  The Telescope maze offers a new puzzle of some interest.  It is rather similar in some ways to Andrea's tilt mazes, and the first 15 levels are fairly easy.  The mazes are based on 'telescopes' which push and pull a ball across an orthogonal  grid with the object of getting it into a hole.  I think there must be an efficient algorithm based on the minimum  and maximum extension points of the telescopes for solving any maze of this type.

Karl Scherer has made a grand page about WireWorld.

Geoff Morley sent me good data for Mrs. Perkins Quilts.

Material added 15 September 2003

Square-free numbers have no repeated factors.  One thing I learned from Steven Finch's new book, Mathematical Constants, is that the probability that a number is squarefree is 6/Pi² ~ .607927. Months ago, I pointed out my Neglected Gaussian page to Steven.  That had a nice result on the GCD of two random numbers.  After looking through his book, he wrote back to me, and asked if I saw the last page ... an extra page after the index that talks about the Collins-Johnson result.  He liked the result so much that added it the day before he sent the book off to the printer. It's a great reference book, and is loaded with lots of the wonders in mathematics.

ListPlot[FoldList[Plus, 0, Table[If[SquareFreeQ[n], 1, 0], {n, 1, 100000}]]/Range[100001], PlotStyle -> PointSize[.001], AspectRatio -> 1/7]

The above is the percentage of square-free numbers for regular digits.  The Collins-Johnson result implies that the probability that a random Gaussian integer is squarefree is 6/Pi²/Catalan ~ .663701.  Here's a plot for the 100000 smallest (by Abs) Gaussian Integers.

ComplexCircle = Take[Sort[Flatten[Table[a + b I, {a, -200, 200}, {b, -200, 200}]], Abs[#1] <; Abs[#2] &], 100000];
ListPlot[FoldList[Plus, 0, Table[If[Max[Map[Last, FactorInteger[ComplexCircle[[n]]]]] == 1, 1, 0], {n, 1, 100000}]]/Range[100001], PlotStyle -> PointSize[.001], AspectRatio -> 1/7]

The publishers of The DaVinci Code have started up a themed puzzle set.

Codefun.com has some beautiful combinations of genetic code (with 20 amino acids) and the icosahedron.

If A+B = REACQUAINTS, and B+C = RESUSCITATING, what are A, B, and C, if A&C are related? Solved by Bryce Herdt. A similar piece of wordplay: {flea, Dr. Seuss, cabin, duster} and {self-assured, business card, disturbance, defaulters}.

The preorder for The Fool's Errand sequel is somewhere on Cliff Johnson's site.

I've been looking at Turing Machines again recently, and I've decided it's time to settle Sigma(5).  The last time it was seriously attacked was in 1989, on a 33MHz machine.  Another solvable problem might be to resolve the 11 unresolved cases for Paterson's Worms.  If you would like to assist with either of these, let me know.

On 27 September, I'll be attending a small puzzle party in the San Francisco area.  I'm allowed to invite a few people, so write to me if you'd be interested in attending.

Robert Wainwright sent me some solutions he found in 1993 for the Mrs. Perkin's Quilt problem.  {29,24,24,16,13,13,7,6,6,5,5,4,4,3,3,1}, {52,39,39,28,24,24,15,9,8,7,7,6,6,5,4,4,1,1}, {91,63,63,52,39,39,24,24,15,9,8,7,7,6,6,5,4,4,1,1}.  Here is the solution to the side-154 square, in 20 squares.  Can it be beaten in a non-trivial way?

Robert Wainwright's solution for 20 squares.

Material added 11 September 2003

I did some experiments with 51-star flags.  You can see my favorite effort here.  You also might like a tighter packing.

Andrea Gilbert has made a number of additions to clickmazes.com, including 4D mazes, parallel-universe mazes, and counter-step mazes.

Material added 5 September 2003

Erich Friedman's Math Magic this month discusses one of my favorite unsolved problems -- pairwise touching polyforms.  I'd love for someone to give a definitive answer for the 4x4x4 box, or the 2x5x5. If you can solve or extend anything there, please send Erich your findings. What is the smallest box for 9?

Gordon Bower -- "You mentioned liking close approximations on this week's Mathpuzzle page update (and have posted a variety of them in the past.) This, and the diophantine inequality puzzle of the week before, reminded me of a fun little exercise I indulged in in the spring of 2000. The question of the week for the now-defunct student math club, of which I was an advisor, was "is there a power of 2 that begins with ten sevens in a row?" (Yes, there are infinitely many of them beginning with any finite string of digits you want.) But no-one bothered to try to actually FIND such a number. The next week I had to give a short presentation explaining how I proved that 2^40193336864 = 7.777777777996 x 10^12099400021. (No harm in me revealing the answer, since the fun is in discovering the method of finding the solution for any sequence of digits you want.) It makes a fun party game (if you are a sufficiently warped-minded mathematician) to find an n such that 2^n starts with your girlfriend's birthdate or suchlike and surprise her with it. Must warn you that mine was not overly impressed by it, though." [Can you figure out the technique?  Answers and Solvers. Bodo Zinser sent a very nice DOS program]

A long unsolved problem, due to Ramanujan, involves the parity of partitions, in the partition function. Here's a picture of it, using the Mathematica code ( MultipleListPlot[{FoldList[Plus, 0, Table[2Mod[PartitionsP[n], 2] - 1, {n, 1, 10000}]], FoldList[Plus, 0, Table[Mod[PartitionsP[n], 3] - 1, {n, 1, 10000}]], FoldList[Plus, 0, Table[Mod[PartitionsP[n], 4] - 1.5, {n, 1, 10000}]]}, PlotJoined -> True, SymbolShape -> {None, None,, None}, PlotStyle -> {Hue[.01], Hue[.35], Hue[.65]}, AspectRatio -> 1/5]; ).  For even/odd parity, in red, the plot shifts up one for odd, and down one for even.  The unsolved part -- is there a simple formula for Mod[PartitionP[n], 2]? At the moment, the value of Mod[PartitionP[10^100], 2] seems unknowable.

Running totals of PartitionP parity, in Mathematica 5, then processed through Irfanview.

Jorge Luis Mireles and Micheal Reid invite people to submit solutions and improvements to their polyform equivalency charts.

A photorealistic painting of a puzzle can be seen at the Steve Mills gallery.

Puzzle hunts have a number of good sites.  You might enjoy the Golden Tickets at T-Hunts.com.

Material added 25 August 2003

I picked up the Futurama Season 2 DVD (I'm a fan of the show), and I happened upon the following dialog between Bender and a robot named Flexo.
  Bender:  Hey, brobot, what's your serial number?     Flexo:  3370318
  Bender:  No way!  Mine's 2716057!       (Everyone laughs)
  Fry:  I don't get it.    Bender and Flexo:  We're both expressible as the sum of two cubes!  Wahoo!
  David X Cohen (audio commentary):  I invite the zealous viewer to check that claim.  It is true!  There's a trick to it, the numbers are all integers, but it's a little tricky.
  That sounds like a great little challenge.  As a warning, if you search for the numbers, you'll find the answer.  If you'd like to try trickier numbers, here a a few more numbers that are the sum of two cubes:  6669, 7222, 119041, 2716343, 2741256, 3370087, 6017193, 6742008, 9016488, 16776487, 24375176.  Can you figure out all the tricks?  Answer and Solvers. 

On a similar but unrelated name, beautiful graphical images can be made with Fractorama, which gives code samples for making fractal variations of many types.

Jerry Slocum, with the help of an international team, has tracked down much origin and history for The Tangram Book.  A precursor was the Sei Shonagon's Wisdom Plates, shown below (the pieces make a square in two different ways.) The Tangram was invented between 1796 and 1802 in China by Yang-cho-chu-shih.  He published the book Ch'i ch'iao t'u (Pictures using seven clever pieces).  The first European publication of Tangrams was in 1817. The word Tangram itself was coined by Dr. Thomas Hill in 1848 for his book Geometrical Puzzles for the Young.  He became the president of Harvard in 1862, and also invented the game Halma.  Jerry tracked down the sets owned by Poe, Napoleon, and others.   Martin Gardner: "This will surely be the classic reference on the topic for many decades to come."

Sei Shonagon's Wisdom Plates, from 1742

A new large prime, 1176694¹³¹⁰⁷²+1, has been discovered by Daniel Heuer. This Generalized Fermat prime has 795695 digits, and is currently number 5 on the list.  The continually updated list is maintained by Chris Caldwell at The Largest Known Primes page.

Andreas Gammel has long been working on the Teabag Problem -- "What is the maximum volume that a teabag can hold?"  You can look at Andreas' page, which describes the Inflatulator. 

Michael W Ecker has recently updated his page for his magazine, Recreational & Educational Computing.  He's continually published it since 1986, and it has long been associated with the Journal of Recreational Mathematics.

Karl Scherer, the master creator for Zillions of Games, has put together a WireWorld explorer. 

Larry Brash's Anagrammy page is well worth a look if you haven't seen it.  I particularly liked an extremely apt anagram found in June by David Bourke ... ?? ~ RECONSIDERABLE. (Answer)  Another beauty, by Joe Fathallah, is "Rats and mice ~ in cat's dream." 

Andrew Glassner has a very nice page, one part of it is the series of Graphics Gems, which has been made available for free. 

I like close approximations.  One I found recently is 95^(1/12) ~ 19/13. Comments.

Previous Puzzles of the week are here.