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Updates added 22 January 2004

John Boozer and Guenter Stertenbrink both found gorgeous embeddings of the F64 graph.  I finally tracked down the persons responsible for notating the second graph as [23,-11,-29,25,-25,29,11,-23]8.  In addition to HSM Coxeter and Roberto Frucht, Joshua Lederberg - the winner of the 1958 Nobel Prize in Physiology and Medicine -- did a large study (pdf) of graphs notated in a similar way.  Doctor Lederberg's complete works are available online.  Another graph that deserves a nice picture is the Hoffman-Singleton graph.

Jens Kruse Andersen:  Hans Rosenthal and I have found a prime gap of 1001548 between two probabilistic primes (prp's) with 43429 digits.  The logarithm of the primes is 99997 so the gap is 10.02 times the "typical" gap by the prime number theorem.  I think that regardless of relative size, this is the first known prime "megagap" with identified (probabilistic) primes as gap ends.

Xah Lee has put together a page on Algorithmic Mathematical Art.

Dick Hess modified a puzzle first posed by E.K. Chapin in 1927.  You have 2 mugs, a water supply, and a packet of instant coffee which when dissolved in one cup produces coffee of strength 100%.  Your task is to fix coffee as requested.  You may fill or transfer liquid and may, at any time, empty the entire coffee packet into a mug.  Send Answers.

  1. With mugs of capacity 5 and 3 cups, fix 5 cups of coffee equivalent to 2 packets of instant coffee in 15 cups.
  2. With mugs of capacity 5 and 3 cups, fix 1 cup of coffee at a) 20% strength b) 10% strength c) 5% strength d) 1% strength
  3. With mugs of capacity 5 and 4 cups, fix 4 cups of coffee at 16% strength
  4. With mugs of capacity 7 and 5 cups, fix 4 cups of coffee at 16% strength
  5. With mugs of capacity 10 and 3 cups, fix 1 cup of coffee at 8% strength
  6. With mugs of capacity 10 and 7 cups, fix 1 cup of coffee at 8% strength

Material added 13 January 2004

The Wolfram Functions Site has been vastly expanded.  More than a hundred volumes worth of function information.  A highlight is the 10,000+ visualizations of every function, in many different ways.  Code for all visualizations is provided.  My latest MAA column talks about it more.

Harvey Heinz has updated his page about magic cubes.  I was completely unaware of Frankenstein's Cube.

My last MAA column of December 2003 talks about Cubic Symmetric Graphs.  I'm quite proud of my version of the Coxeter Graph, there -- I made it the logo for the Mathpuzzle Yahoo Group.

Erich Friedman has updated his page of Sequential Domino Packings. His solution for 18 dominoes might not be minimal -- can you fit these into a 65x65 square? Send Answer (if one exists) Smallest packings for both 23 and 24 sequential dominoes might also be very tricky to find.  Robert Reid, Erich Friedman, Minami Kawasaki, and myself have all been busy finding solutions -- see Sequence A005842. Solutions for 16, 21, 34, 46, 54, 56, 60, 62, 63, 65, and 76 can be found in this zip. Minami Kawasaki has put together a page about sequential squares.  

Consectutive Squares

Francis Heaney: I review the new Jell-O Checkers snack packet on my blog today. It has many flaws, and I was hoping someone would step forward to help analyze how they affect the game.  [Ed - I expect a Zillions expert will crack this in hours. Send Answer.]

Cihan Altay: PQRST 08 Puzzle Competition starts on January 17th Saturday at 20:00 (GMT+02). You'll have one week to solve and rate 10 puzzles.

 Yasutoshi Kohmoto: 2004^6 = 3959307^3 + 1393389^3 + 1494^3 = 3848682^3 + 1980119^3 + 27889^3

Karl Scherer has greatly expanded his WireWorld results, with help from Nyles Heise.  Pentomino Odd Pairs (by Livio Zucca) is well worth a look -- can you find a shape made with an odd number of either I or V pentominoes?

A noted game and puzzle creator has recently been harassed for having an almanac.  He'll be attending the MIT Mystery Hunt this week.  Warm-up puzzle.

Material added 1 January 2004

Erich Friedman:  Start with 2003.  Chop it up any way you like: 20,0,3. Square the pieces and add them together to get a new number 400+0+9=409.  Repeat until you get 2004.  It's easy to find a path that takes 7 steps: 20:0:3 --> 40:9 --> 1:6:81 --> 6:59:8 --> 35:8:1 --> 12:90 -> 8:2:44 --> 2004.  Find a 6 step path from 2003 to 2004.  might make a good new yearspuzzle.  this took 7 steps.  Do it in six steps.  Answer and Solvers.

Emrehan Halici: "Im sending 3 problems which Id prepared for the new year. I hope you like them. 1.  Move 2 matches to get 2004.  2.  Move 3 matches to get 2004.  3. Move 4 matches to get 2004."  He also sent a table on how to make the numbers 1-50 using the digits in 2004. Answer.
Emrehan Halici matchstick puzzles

Livio Zucca has created a page with various pentomino and tetromino challenges. For example, find a shape that can be split into either I or T tetrominoes, but none of the others. 

George Jelliss has release Issue 30 of the Games and Puzzles Journal.  The bound volumes he offers are a nice prize, and many of the puzzles there are quite nice. I especially like problems 64 and 65.

Packing squares in squares.  Robert Reid sent my a nice Christmas Present -- squares 1-27 packed into the smallest possible square.  I played around with the problem of 1-25 in the smallest square. One of my best efforts is shown here.  I don't have a proper 8 packed in here.  However, I could fit in 3 4x8 rectangles and everything else into the 75x75 square. Paul Cleary sent me a solution for 1-25 packed into a 76x76 square, but he believes (as I do) that 75x75 must have a solution. Erich Friedman is presenting a slightly different take on Squares in Squares this month at Math Magic.

Reid 1-27  Pieces of 8

A puzzle by Marek Penszko of Poland.  It's a division problem. Answer.

Diablo puzzle

Dave Millar: I was inspired by the room puzzles by Erich Friedman, and made one of my own creation. 3 room, all of 8 squares. From each number, you can see that number of spaces in any direction; up, down, left, or right. Answer.

Room Puzzle

Material added 23 December 2003

Cletus Emmanuel has found that (2^148330+1)^4 - 2 is a probable prime. That makes it the largest known probable prime on the Henri Lipchitz Probable Prime List. Seventeenorbust.com reports that 53592^5054502+1 is now the 4th largest proven prime.

Dave Millar has sent me several nice puzzles. In the Pento puzzle, rearrange the given double size pentominoes so that the squares in the grid represent the number of smaller pentominoes in each one. (Answer)  I also like this logo he did for me. In the Bird puzzle (solution), the bird must be divided into 5 identical shapes, not necessarily of the same size.

Pentomino Puzzle

Colonel Sicherman, on the Logical Hats problem: "After one logician identifies his number, can the other two always identify theirs immediately?  I have not found a counterexample." Answer.

NetLogo 2.0 is available for download.  This is the programming language of Turning Turtles fame.  There are many excellent and instructive programs built in.

NetLogo's Sunflower

I'm trying to learn a little TeX.  I didn't make much headway until I tried out a combination of the winedt shell and MiKTeX.

My latest two Math Games columns are Domino Graphs and Superflu Modeling. Here's a rainbow PDF version of the Petersen Graph I did as an experiment, after getting some kind words for the first article.

Erich Friedman: Find positive integers A, B, C, D, and E all less than 100 so that A2 + B2 + C2 = D2 + E2  and  A3 + B3 + C3 = D3 + E3. Answer and Solvers.

Johan de Ruiter: "Last night I was wondering whether any integer can be written as a linear combination of a finite number of noninteger squareroots of integers where all coefficients are integers. Maybe it's trivial, but I wasn't able to find a solution yet."  I wasn't able to find a trivial proof either, beyond proofs for 2 or 3 square roots. Is there a clever impossibility proof?  Answer.

The square of 40081787109376 starts 16065496578813...   how does it end?

Material added 13 December 2003

My latest Math Games column deals with Integer Sequences.

Des MacHale of University College Cork found a way to fit squares of size 1-24 in a 43115 rectangle. All rectangles smaller than 5491 have been shown to be unsolvable by Patrick Hamlyn. Thus, the necessary excess required for a rectangular packing of squares 1-24 ranges from 14 to 45 (below). I thought this might make a nice puzzle -- treating the blue areas below as holes, divide the 43115 rectangle below into 24 squares.  Slide your mouse over the image to see the solution. For 1-22, rectangles smaller than 5372 are unsolvable.  (Patrick Hamlyn: The squares 1-22 into a rectangle: I just finished searching 53*72, no solutions after 589.5 hours.) For 1-23, 6171 might be possible, or maybe 6270, but probably something larger is needed.  As a warm-up puzzle, here's a beauty: Divide an 1111 square into 11 squares of size 2-5. See my Square Packing column for more.

24 Squares
Des MacHale's packing of squares 1-24 in a 43115 rectangle.

Nyles Heise (nylesheise at yahoo.com) fit a 32-bit WireWorld multiplier into a 2293 rectangle. Nyles Heise Multiplier  Yes, that is a very tiny multiplication program. You can see his notes, or his MCL representation. (Two Updates:  Input 1's, and Output 1's) You can download MCell from Mirek's site. In hexadecimal, the below calculates EF4E75E7 EFA03229 = DFFFFFFFFFF7FFFF in 8116 cycles.  See the Output 1's file for an expanded MCell version.

#SG ..............................................................................................
#SG ..OOOOOOOOOOOOOOOOOOOOOOO.O.O...O.OOOO.OOOOOOOOOO.OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO..
#SG .O.......................OOO.*OOOO....O..........O..........................................O.
#SG ..OOOOOOOOOOOOOOOOOOOOO...O.o...O.O..OOO..O...OOO..OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO..
#SG .......................O.O.....O...O..O..*.O.O....O...........................................
#SG ..OOOOOOOOOOOOOOOOOOOOO..O....O..OO..OOOO*o...OOO..OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOo*OOo*OOo..
#SG .O.......................O...O..O...O.O....*.....O..........................................*.
#SG ..OOOOOOOOOOOOOOOOO...O..O..O...O...O...OO..OO.*..*oOO*oOO*oOOOOOOOOOO*oOO*oOO*oOO*oOOOOOO..O.
#SG ...................O.O.O.OOO....O....O.O..O...O*o.........................................*.O.
#SG ..OOOOOOOOOOOOOOOOO..O.O..O...O..O....OOO.O.O..*.O*o.O*oOO*oOOOOOO*oOO*oOO*oOO*oOO*oOOOO..o.O.
#SG .O...................O.O..OOOOOOO....O.*...OOOO.....O...................................O.O.O.
#SG ..OOOOOOOOOOOOOOOOO..O.O.O....O...*...o.....O......o..*OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOo*O..O.O.
#SG ...................O.O.O.O.O.OO.Oo.O.OOO..OO.......*.o....................................O.O.
#SG ..OOOOOOOOOOOOOOOOO..O..OOOOO..OO.O...O..o.O.oO..o.O..OO*oOOOOOOOOOO*oOOOOOO....Oo*OOo*OOO..o.
#SG .O...................O...*.O.O..OOO....O*...*..*ooO.........................O..O............*.
#SG ..OOOOOOOOOOOOOOOO...O...o....O..O.............o.o.OOOOOOOOo*OOOOOOo*OOOOOOO....*oOOOOOOOO..O.
#SG ..................O..O..O.*....O.O.....*OOo*OOO...........................................O.O.
#SG ..OOOOOOOOOOOOOOOO..OO...O..OOO..O....o.........o*OOo*OOOo*OOo*OOo*OOo*OOo*O...O...o*OOo*O..o.
#SG .O.................O.OO....O.....O...O....OOo*OO............................O.O.*.O.........*.
#SG ..OOOOOOOOOOOOOOO..O...O..OOO.O.OOO.O..O**.......O*oOO*oOO*oOO*oOO*oOO*oOO..oooo...O*oOOOO..O.
#SG .................O.O..OOO..O.OOO.O..*.O..o......O.........................*..O..*.........O.O.
#SG ..OOOOOOOOOOOOOOO..O.O.O.OO.O.O..o.O*OO.O.O......o*OOo*OOo*OOo*OOo*OOo*OO..oOOO.O..o*OOOOO..o.
#SG .O.................O.O..........*.O.*.OO...O.............................o...O..O.O.........*.
#SG ..OOOOOOOOOOOOOOOOO...OOOOOOOOOO.O......*oO.*oOO*oOO*oOO*oOO*oOO*oOO*oOO*.....*o...OOOOO*oOO..
#SG ..............................................................................................

CHESS MAZE by Mark J. P. Wolf (mark.wolf at cuw.edu): "Ive always admired the rich possibilities in the simplicity of the chessboard, and tried to capture this in the Chess maze.  How the maze works:  Starting with the White Queen at a8, capture your way to the White King at h1.  Pieces capture as they do in chess (except pawns, which can capture one square diagonally in any direction), and pieces can only capture pieces of the opposite color.  However, once a capture is made your piece becomes the type of piece that was just captured (and moves accordingly on the next move) and all moves must end in a capture.  Pieces that are captured are removed from the board, so the number of pieces on the board gradually decreases.  To keep track of which pieces are removed, I recommend either crossing out the captured pieces, so as to indicate which squares are empty and can be passed over later in the maze.  (For an easy warm-up, try capturing the black queen in five moves, or the black pawn at a1 in nine moves).  A unique feature of the maze is that as pieces are captured and removed, new pathways open up that were previously unavailable, making the maze fairly difficult to work backwards." Answer and Solvers. Picture of Solution. Doug Orleans created an applet  for this puzzle.

Mark Wolf's Chess Maze

Brian Silverman did a Google search on 5-digit numbers, and discovered that 17839 is the most unpopular number on the internet.  I'm trying to help the number, since 17839 19813 237877 11893969 = 1021 - 9.  It's also a factor of 35! + 11.. Does this number have anything else going for it?  Send 17839 factoids.  Livio Zucca is searching for pento-tetra-tri solutions. Jorge Mireles has expanded his page on Poly2ominoes. Peter Esser has found a way to pack the Sliced Heptiamonds into a rectangle.

Sliced Heptiamonds

Jim Propp recently gave a talk to the theory group at Microsoft Research entitled "Random walk and random aggregation, derandomized", which Microsoft has made available to the outside world.  Watching the video/demo is probably the quickest and most pleasant way to find out the current state of knowledge about these models.

Slouching Towards Bedlam won the Interactive Fiction Competition. The solutions I received for Borromean Rings are great. 

Material added 4 December 2003

RSA-576 has been factored by the programming team of J. Franke, T. Kleinjung, P. Montgomery, H. te Riele, F. Bahr, D. Leclair, Paul Leyland and R. Wackerbarth.  Institutions involved include Bonn University, the Max Planck Institute, the Experimental Mathematics Institute in Essen, CWI, NFSNET, and Microsoft Research. For this development and application of the GNFS algorithm, they will split $10,000.00.  NSFNET (Number Field Sieve Net) just happens to be recruiting, if you'd like to join the effort to factor 2^811 - 1. They recently factored 2^757-1.

188198812920607963838697239461650439807163563379417382700763356422988859715234665485319
060606504743045317388011303396716199692321205734031879550656996221305168759307650257059 =
398075086424064937397125500550386491199064362342526708406385189575946388957261768583317
472772146107435302536223071973048224632914695302097116459852171130520711256363590397527.

Material added 2 December 2003

A few years ago, Michael Shafer was visiting here.  Michael: "Yes, yes, it's all true!  I came across mathpuzzle.com some time in 2000 and bookmarked it to check out the puzzles you came out with every week as well as the interesting links.  www.mersenne.org was one of them and you can see what happened.  Thank you for leading me to Al Zimmerman's contests, the WPC qualifiers, and Theodore Gray's periodic table and fun with sodium as well. The occasional challenges are also fun to spend a few minutes (or sometimes more) pondering.  Keep up the great work!"  If you check out the mersenne.org link, you'll see that Michael Shafer helped to discover the world's largest prime, 220996011-1.  Many congratulations, Michael! You can see more at mathworld.wolfram.com.

More record setting news from Lance Gay: " I just saw your new Square Packing page at maa.org.  I have improved solutions to 198, 205, 206, 253, 258, and 259."  My next goal is to correct my graphic of primitive quilts.

198-21 {{106,92},{12,13,19,48},{2,9,1},{14},{92,16},{19},{9},{3,11},{28},{20,10},{58},{48}}
205-21 {{112,93},{22,23,48},{93,16,3},{13,12},{11,12},{11,11,1},{29},{13},{19,3},{64},{48}}
206-21 {{113,93},{23,25,45},{93,17,3},{19,7},{5,20},{12},{12,5},{7,26,3},{68},{19},{45}}
253-22 {{141,112},{27,28,57},{1,9,16,1},{1},{29},{112,23,7},{16},{16},{32,7},{2,84},{25},{57}}
258-22 {{142,116},{25,33,58},{1,12,12},{116,27},{4,4,25},{12,20},{4,8},{31},{27,1},{84},{58}}
259-22 {{142,117},{25,34,58},{117,31,19},{10,24},{15,14},{28,3},{12,84},{9,9},{16,2},{14},{58}}

Material added 1 December 2003

Two more new columns at my MAA column, Math Games. First, a treatise on Multistate Mazes. Next, a synopsis of the current knowledge of Square Packings, in particular the case of Mrs. Perkins's Quilt.  For the latter, Richard Guy and I tried to create a list of Primitive Optimal Quilts, and already a number of mistakes are apparent. Can you find the three quilts that aren't primitive, and the primitive quilt that can be derived from one of them?  Yesterday, Richard K Guy learned, demonstrated, then stressed to me how important the primitive quilts are for solving the general problem. A correct list of primitive quilts up to size 100 is needed. Write me if you discover a patch in the solution for Mrs. Perkins's Quilt.  I can well imagine a high-schooler picking one of the primitive squares, tracing through the build-up process, and becoming permanent listed in the problem's history.  A great resource for this is squaring.net.

Slightly too late for the multistate column is Perl Code for Logic Mazes.  As a possible gift, Robert Abbott has about 20 copies of each of his wonderful maze books left -- see his site for information. Even raarrer, William Waite has ten copies of his Camera Conundrum puzzle.  It's an incredibly clever secret-opening box.  It won at the last International Puzzle Design competition.  Speaking of that ....

The 4th IPP Puzzle Design Competition has started.  Established three years ago to promote and recognize innovative new designs of mechanical puzzles, the annual IPP Puzzle Design Competition will be held in conjunction with IPP24 in Tokyo. The competition is open to designs made public between July, 2002 and July, 2004. Entry Deadline: June 30, 2004. Judging at IPP: July 30 - August 1, 2004. Awards at IPP: August 1, 2004. Complete rules and information are at the IPP Puzzle Design Competition web site, part of John Rausch's PuzzleWorld. The third competition completed in August with judging and awards at IPP23 in Chicago, USA. It was a great success, with 52 designs participating. Lee Krasnow's Clutch Box won the Puzzlers' Award; and  Mineyuki Uyematsu's Cube in Cage 333 won the judging committee's Grand Prize.

Another gift possibility -- I have a domino cards game out, Auf & Ab.  The link will take you to Funagain Games.  It's basically a nicely improved set of double-9 domino cards, perfect for playing my game Ups and Downs.  If you want a book, I'm still fascinated by Mathematical Constants by Steven Finch. Easier reading is Dudeney's Amusements in Mathematics, which was my first puzzle book.

Routewords is an interesting combination of graph theory and wordplay. Ross Eckler tackled this problem back in 1980 -- find a non-planar word. Also at Word Ways, some of the challenges are worth a look.

Inspired by Bill Cutler's solution, Fan Chung and Ron Graham have done a detailed combinatorical analysis of the Archimedean Stomachion. It turns out the solutions have a fantastic amount of symmetry.  Archimedes could have found all of this easily ... and he may have!

The Stomachion Graph

Dick Hess: The Logical Hats Puzzle.  Logicians A, B and C each wear a hat with a positive integer on it such that the number on one hat is the sum of the numbers on the other two.  They can see the numbers on the other two hats but not their own.  They are given this information and asked in turn if they can identify their number.  In the first round A, B and C each in turn say they don't know.  In the second round A is first to go and states his number is 50.  What numbers are on B and C? Answer and Solvers. Jonathan Welton: I was delighted to see an old puzzle of mine doing the rounds (Logical Hats Puzzle). This was originally published in the Sunday Times magazine  as  puzzle number 1814, and was reprinted in a collection of these puzzles,  Brainteasers by Victor Bryant in 2002 - highly recommended if you like tough puzzles.

A 2000 year old icosahedron is available for auction at Christie's. More glimpses of ancient math are stored at the Vatican.  I recently refound a long-lost link -- Monty Hall's take on the Monty Hall puzzle.


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