**material added 11 February 2001**

A circle of dominoes where no neighboring pair of dominoes shares a number.

In the above image, any two dominoes that share no numbers are connected with a line. They are arranged in a circle, with the property that no neighboring pair shares a number. Finding five-fold symmetry wasn't easy. Is it possible to make this graph with three-fold symmetry? Can 3 circles of 5 dominoes be made, so that each circle has the "neighbors don't share numbers" (NDSN) property? A more direct version of the question was sent to me by Douglas West, whose book directly inspired the drawing. His version: "Does the graph contain three pairwise-disjoint 5-cycles?" Send Answer.

Before tackling the problem above, you might want to try a simpler problem first. From the diagram above, remove all the dominoes with a 6, and remove the 1-2 domino. Can you make a nine-circle with the NDSN property? Send Answer.

A few weeks ago I asked, "Hmm, can the 24 interior blocks be distributed so that each of the 24 pieces becomes a hexomino?" Patrick Hamlyn found a brilliant little impossibility proof: "No. Refer to the attached pic. Consider the three outlined empty cells. The left two must contribute a cell each to the dark green and the ...um... dark aqua pentomino. The right-hand one must thus contribute to both the red and the purple."

Dan Hoey wrote a Dots and Boxes analyzer, and made an excellent commentary for my game against Joe. See it at my Dots and Boxes page. The winner is usually the first to make a sacrifice - I find that lovely! The next really interesting game for QDB would be the 3x5 game.

I downloaded Irfanview recently (it's free), and have made it one of my main programs. Another site allows you to make License Plates:

The class of Odette de Meulemeester has started a Pentomino Bridge competition, with €25 going to the winner.

Alex Selby has updated his Eternity page. And here's another view of the Domino graph. Can anyone find a symmetric version of this graph when the dominoes including blanks are thrown in? (No doubles, and I couldn't find one.). Can 3 circles of 7 dominoes with the NDSN property be made? Or 7 circles of 3?

**material added 4 February 2001**

A lively mailing list I've been following is polyforms@yahoogroups.com . Here's a solution recently found by Peter Esser:

I mentioned Mathematical Art last week. Ishihama Yoshiaki pointed me to his excellent http://www.asahi-net.or.jp/~hq8y-ishm/gpart/gp-art.html. Igor Bakshee pointed me to the excellent Graphica books.

I watched the first XFL games last night. The games were scheduled by two mathematicians (New York Times story). I liked learning that Tim Lester, the quarterback for the Chicago Enforcers, is a high school math teacher.

**material added 28 January 2001**

The Tetris Logic puzzle by John Gowland was popular last week. Bob Kraus decided to try making some himself, and came up with six logic tetris puzzles. Wow! Bob's Site hosts the guest Puzzle of the Week. Puzzle #3 is particularly nice.

Joseph DeVincentis beat me at 3x3 QDB.

It isn't easy to introduce hard material in an amusing way. Mathematica Animations do just that.

A nice gallery of mathematical image is at http://www.kfunigraz.ac.at/imawww/vqm/pages/colorgallery/index.html.

Here's the Game of Sprouts Association webpage.

Aram Hakobyan, Roel Huisman, Mark Thompson, Stas Soumarokov, Andrej Jakobcic, Aron Fay, Evgeni Lukin, HappyMutant, Chuck Fallon, Stephen Kloder, Joseph DeVincentis, and Michael DuFour solved Scott Kim's 4x4 Queens problem. His general case problem is still unsolved -- for an n x n board, where each queen attacks exactly k other queens, what is the most queens that can be put on the board?

Joseph DeVincentis, Michael DuFour, Koshi Arai, and Brett Champion solved the 11x11 rectangle problem, divided into areas in the teens. Solutions.

**material added 24 January 2001**

**Dots and Boxes** starts by drawing
a square grid of boxes. After that, players alternate connecting dots (no
diagonal lines). Whenever a square is completed, the player that drew the
final line claims that box with their initials, and moves again. When all
boxes have been claimed, the person with the most boxes wins. My only gripe
against Dots and Boxes (notation here) is relatively
slow game opening. I tinkered with the rules, looking for a fix. **Quick
Dots & Boxes** (QDB) is played in the same way, but each player draws
**two** lines per turn. **The instant that a player draws a third line
on any box, their turn ends**, and play goes back to normal Dots and Boxes.
QDB is a fairly deep game. I *think* the second player can always force a
win *on the 3x3 board*. Let me know.
When I posted this two days ago, I forgot to mention that it was the 3x3 board
I was looking at. Joseph DeVincentis did an analysis
of the 2x2 board. More on Dots
and Boxes.The book was recently reviewed in Scientific American.

Winning move in QDB: A3— . Are there others?

Let me point you to Karl Scherer's website. Karl has produced about a hundred great games for the Zillions engine (the best thing in recreational gaming, ever). I'm still trying a lot of them out. Also, I've been seeing his name a lot in the copies of the Journal of Recreation Mathematics I've been going through. I'm quite impressed by all Karl's done. Why, just today, he has new things available at the Zillions site.

**material added 21 January 2001**

Now that I have a copy of ONAG2, I see the cover is by puzzlemaster Scott Kim. Neat! Here's a nice puzzle by him: Put 6 queens on a 4x4 board so that each queen attacks exactly 2 others. Send answer. This problem can be expanded, of course.

Dwight Kidder solved John Gowland's Double Squares, and said "Incidentally, as an aside, note 72^2_27^2 = 2277^2. (5184_729 = 5184729)" Okay, let _ be the concatenation equation representation. Are there integers a, b, and k such that (a^k_b^k)^(1/k) is an integer, for k>2? Send answer. If you can scribble any results or proofs in a margin, write to me. I pondered trying to fit the Wiles proof into the margin of a handout on Diophantus, but my printer resolution wasn't strong enough. I also took a look at 14 proofs that Zeta[2] = Pi^2/6.

I mangled Erich Friedman's lovely problem last week -- divide an 11x11 rectangle into rectangles with areas in the teens. What is the highest number of different rectangles that can be used? Send answer. He goes on-- "Another (unsolved) question similar to your fill-agree one: take a 2x3 rectangle, and cut holes in the individual squares in every possible way. There are 24 of these possible - arrange them in a 12x12 square so that all the holes are touching.

I'm still loving the Kites and Bricks problem. Koshi Arai asked if the 7x7 could be built with 7 1-2 kites, 7 1-3 kites, and 7 dominos. I looked at extending that to nxn, where the third piece could be a shape of your choice. The 8x8 problem was easy. Is the 9x9 solvable?

I've moved Michael Reid's fantastic series to the Similar Dissection Page.

I caught up on movies, and saw* Castaway*,
*Crouching Tiger Hidden Dragon*, and *The Emperor's New Groove*.
I liked them all, but liked the last one the most. It was a tremendous amount
of fun. I asked if the sequel could be loosely based on *The Emperor's New
Mind* by Roger Penrose. I got a reply back from Disney within ten minutes
-- 10:52PM on a Saturday -- "Thank you for your letter. Unfortunately,
there are no plans at this time for a sequel to this film." Not the answer
I wanted to hear, but at least it was extremely speedy! I wonder if they publish
any math journals.

Puzzle Palace (www.puzzle.gr.jp/index_e.html) has puzzles similar to those found in Mensa Math & Logic Puzzles .

Here's an entertaining maze applet by Sami Silvennoinen.

I recently asked for differing ways of making Pascal's triangle with Mathematica.

**material added 14 January 2001**

I like perusing college bookstores. Usually,
I can count on finding at least ten great books. I now have in my hands Winning
Ways, 2nd Edition, Volume 1. A few seconds after I touched it another
math student said to me, "*I can't believe it's finally out. I've looked
everywhere.*" Anyways, I compared my first edition to the second edition
(Volume Spade -- This book is 1/4 of the full book.). New preface. Modernized
format. The "Extras" are expanded. Amazons
is mentioned. The "References and Further Reading" sections are
expanded. A new section on Blockbusting. Various minor corrections and added
explanations. I'm highly pleased with it. I planned to put a Winning Ways
type problem here, involving Amazons, so I tried to find a site for the game.
I ran across Abstract
Game Magazine (why did no-one tell me?)! Then I found Amazon
Rules, written by me, oddly enough. And Amazon
links, completely me-free this time. I finally found Berlekamp's
article on Amazons.

Another excellent book I found at the bookstore was Introduction to Graph Theory by Douglas West. He's one of the coauthors of another of my favorite books, Mathematical Thinking. I wish all textbooks could be this good.

Dick Hess asked "3 Pythagorean Triangles have the same hypotenuse, and areas A, B, and C. Can A = B + C?" John Robertson informed me that this is equivalent to the unsolved Magic Square of Squares problem. $100 from Martin Gardner, if you can solve it.

A similar problem is at the site of Junichi Yananose - Holey Rectangle. Rearrange the pieces below to get a 6x8 rectangle with 16 circular holes. There are many other puzzles at his site, and some stellations.

Holey Rectangle copyright 1999 Junichi Yananose. Click on image
for a larger version

Another site with spectacular displays is by Vladimir
Bulatov. Take a look at his beautiful Stellation
applet.

HORSELESS CARRIAGE and GREGORIAN CALENDAR are two phrases that can be said without moving the lips. What 22 letter, three word phrase has the same property? It's something you can order at a restaurant. Send answer.

Math news, re the Euler Brick -- There are no perfect integer cuboids with the smallest edge <= 2^31. An exhaustive computer search found 13,401 body cuboids, 7,992 edge cuboids, and 13,044 face cuboids for a total of 34,437 in this range.

**material added 7 January 2001**

Gearing up for the rerelease of **Winning Ways** and **On
Numbers and Games**, I've recently been reading two excellent books on game
strategy. You can get all of these books at **www.akpeters.com**.
Other sites devoted to game strategy: http://www.abstractboardgames.com/,
http://www.gamerz.net/, http://www.zillions-of-games.com/.

**Hex Strategy** by Cameron Browne. Mathematical proofs are
best introduced by purely visual means. I believe my own first proofs involved
tictactoe. Hex Strategy is filled with game analysis, commentary, and puzzles.
It comprehensively lists related games. Many of the strategies given were
new to me. Another section has blank game boards of many sizes for easy copying
and play. Hexy is mentioned.
Beautifully made book.

As usual, I've moved a lot to my Solutions and Commentary page. Many thanks to all who sent solutions.

Harvey Heinz has published a
book on Magic Squares.

Erich Friedman asks for number suggestions.

Dean Howard informed me that The Flanders Panel is a mystery with a retrograde chess problem. This reminded me of a problem I made in 1988. "Things are looking bad for White. He has a king on G3, and a bishop on E8. Black still has all 16 pieces. With a bishop sacrifice, White can force a stalemate in 4 moves. Can you construct Black's position?" John W. C. McNabb sent me this answer. My intended answer is here.

**Cluster Rooks** has been moved
here.

**material added 28 December**

I've been looking more at Rock, Paper,
Scissors lately. Andrzej Nagorko very graciously sent me his prize winning
code Greenberg.c , and has allowed me to post it
here. You can see some of the competition's code here,
at the Roshambo site. One paragraph from the comments in Phasenbott intrigues
me - **"**Phasenbott's metric would be more appropriate in a non-zero
sum Rock-Paper-Scissors game where one simply tallied points for wins. This
game is more interesting from the theoretical standpoint, as there is now
incentive for cooperation and no longer a single optimal strategy. Random
scores an expected 1/3, but cooperating players could do better by alternating
wins, for 1/2. A player wanting to do better than 1/2 would try to exploit
the other player, but not enough that the other player detects that it's worthwhile
to switch into Random mode. The weak player scoring say 2/5 could know that
it's being exploited by the stronger, but still go along with it as if it
refused (by going Random) its score would drop to 1/3. This in my mind makes
for a much more interesting Rock-Paper-Scissors game to study than "Roshambo".
Maybe the next Rock-Paper-Scissors programming contest will feature such a
non-zero sum game. [Hint, hint. :) ]**"**

I'd score this Win = 2, Tie = Loss = -1. Random scores
zero! I'm considering my first Programming Competition -- 1000 games against
all other programs, highest score wins. I'll likely make a few simple programs
myself. For example, always play Rock, until the score drops a certain amount,
then play randomly. Let me know if
you'd be interested in entering.

I did read both The
Planiverse by A K Dewdney, and The
Wonder of Numbers by Clifford Pickover. I enjoyed them both.
The NPR Puzzle of the Week was mine. STATEHOOD UNWON = TWO THOUSAND ONE. Paul
Strauss, United States Senator for the District of Columbia, wrote to thank
me for discovering this.

**material added 3 January 2001**

Michael Kleber reminded me of an old poem on seeing Marek Penszko's comment, so I penned the following: As I was going to St Ives I posted the poem to the National Puzzler's League list. Dean Howard (shrdlu) sent a reply. |
The kits in this fine piece of lore
Are seven to the power four. Each cat takes one, maternally, Or seven to the power three. The sacks, as must be clear to you, Hold seven to the power two. The wives, to have a bit of fun, Pet seven to the power one. And last, our multi-mated hero Has seven to the power zero. When added up, the number shared Becomes precisely twenty squared. So after the subtraction's done You still retain two thousand one. And so St. Ives turns out to be The start of one more odyssey. |

**material added 20 December**

**The** most important books in Recreational Mathematics
have just been reprinted.
*Winning Ways for Your Mathematical Plays* (Berlekamp, Conway, Guy)
and *On
Numbers And Games*, 2nd Edition (Conway).

If you know anything about the X-Files, you'll probably enjoy the launch of the Octium IV chip.

**material added 19 December**

The new demo page of webMathematica is well worth a look.

**material added 17 December**

John Rausch -- The
Puzzle Art of Stewart Coffin is now online.

Lew Baxter -- R(86453) = (10^86453-1)/9 is a Probable Prime Repunit.

Craig Kasper -- The lastest Putnam
competition has been posted.

Mike Henkes -- I've created a new
idea for sliding block puzzles. [very nice]

Bill Ritchie -- FlipIt,
by Nob Yoshigahara, is now available. [good puzzle set]

Martin Demaine -- I have been working on folding
problems. [fascinating]

Nick Baxter -- I've posted rules for the International
Puzzle Design Competition.

Livio Zucca -- I've solved your DSP problem:

C M Shearer has created a page about the master of tiling, Marjorie Rice.

I've gotten some excellent puzzles from Bob Kraus and Patrick Hamlyn. I'm having some system problems at the moment, so please forgive me as I present them as Ascii Puzzles. Bob did some Balloon Balance problems, while Patrick has made a wonderful puzzle with 10 octominoes. I also include Michael Reid's list of unresolved rectifiable polyominoes.

Quite a few people were interested in the Forbes article on Stephen Wolfram. He's recently built a scrapbook page. Page two features a cryptography challenge with some sort of prize. I have no idea what that prize might be. For more Cellular Automata, you can visit my old page. If you have any links I should add that page, mail me and tell me about them. Mirek's Cellabration is something I should add, for example.

The exchanges of the G4G4 convention are well worth a look.

Randomness, games, and brain power. If you've seen
the movie *The Princess Bride* (complete
script), you know about Iocaine Powder -- the most dreadful poison.
A snippet...

**Man in black:** [turning his back, and adding
the poison to one of the goblets] Alright, where is the poison? The battle
of wits has begun. It ends when you decide and we both drink - and find
out who is right, and who is dead.

**Vizzini:** But it's so simple. All I
have to do is divine it from what I know of you. Are you the sort of man
who would put the poison into his own goblet or his enemies? Now, a clever man
would put the poison into his own goblet because he would know that only a great
fool would reach for what he was given. I am not a great fool so I can
clearly not choose the wine in front of you...But you must have known I was
not a great fool; you would have counted on it, so I can clearly not choose
the wine in front of me.

**Man in black:** You've made your decision then?

**Vizzini:** [happily] Not remotely! Because
Iocaine comes from Australia....

Wei-Hwa Huang told me about the Year
2000 Rock Paper Scissors competition writeup at Mindsports Olympiad.
The winner of the 1999 Competition was a program called Iocaine
Powder. I recently tried playing a thousand rounds against Iocaine
-- and got trounced. It's squirmingly creepy to have your random behavior
predicted by a short program.

Here are more of his math quickies.
Similar stuff:

Add two points to make this true:
(18 + 12) (18 - 12) = 18

Add one point to make this true:
= = 11
(Ali Muniz, silly)

Use matches and one penny (the dot on the 'i') to form
the equation: S i X = 3 1 Move one object to make the equation
true. (Daniel Scher)

Carlos Penedo of Neuquen sent me two pages of creative
takes on the 71 puzzle. Page 1, Page
2.

I've finally added Mensa Math & Logic Puzzles by Dave Tuller & Michael Rios ($8) to my Books page. Similar puzzles can by seen on my World Puzzle Championship page.

My little word square puzzle is still topical (December 28 -- and probably
always will be):

ACROSS 1. The person who won the election 2. Smallest buffalo
3. Voter turnout, e.g. 4. Outside inside France

DOWN 1. Strike sharply 2. Preposition 3. Griffins do this
4. Cappelletti, to a kid

I've moved Kites and Bricks
to their own page. Here is a **hard** puzzle (as of 28 December, it's
still unsolved). Build a 5.8 x 6.4 frame, where 1 is the unit side length.
Put 10 short kites, 4 long kites, and 2 dominos into this frame.

In Russia, Paint by Numbers ( ) is known as the Japanese Puzzle, or Japanese Crosswords. There are several russian newspapers devoted entirely to such puzzles, there. Gunnavy Dennis, from the Ukraine, has a website for these puzzles.

My Pythagoras page has been updated. Wei-Hwa Huang's challenge is now at my Solutions page.

At my prize page, I've added a $100,000 prime puzzle, and $50 for a solution to Livio Zucca's Big Pizza Puzzle.

I've also built a page of unsolved puzzles. Actually, I have at least one solution to each of these, but I wonder if anyone else can solve them.

The 12345 Maze had many solvers. I've posted them all at my Solutions page.

The 1-2-3 maze was moved to my Solutions page.

The task of making two identical rectangles with a set of dominoes was solved by Juha Saukkola, Michael Reid, Joe Logic, Joseph DeVincentis, and Dan Hoey. Here are the interesting write-ups.

Cleverwood expanded. One designer they added is Eric Kelsic, a teenager who has designed several new mechanisms.

Entropy has been moved to my Solutions page.

Sliced Tetrominoes. There are 14 ways to remove a 45^{o} triangle
from the tetrominoes so that a connected shape is left. These 14 pieces
make a perfect 7x7 square. Can you find it? Next, make two 5x5 squares
with a triangle missing at the same location. Many thanks to Roel Huisman
for showing this to me.

If you like the ambigrams of Scott Kim, you'll like this cover of the book Angels & Demons. It's by John Landon, the author of Wordplay.

Laser Tank is a nice puzzle game.

Patrick Farvacque showed me some beautiful puzzles.
One puzzle is a Tower of Hanoi variant. Get 3 pairs of differently sized
coins, and draw three squares. Put one of each coin in two of the squares,
with smaller coins stacked on larger coins. One stack is Heads, the other
is Tails. Now then, moving one coin at a time, and never putting a larger
coin on a smaller coin, swap the stacks. How many moves does this take?
Jeffery Gosselin solved this, it takes 25 moves. Stephen Kloder has found
that the general number of moves required is (7/3)*2^^{(n+1)} - (1/6)*(-1)^^{n}
- 3n - 7/2. You can see another Hanoi variant by Junk Kato here.

David Singmaster had several nice problems to share with
me. One is Polyform Towers. You have an unlimited set of all the
cubic polyominoes (if you don't know what they are, visit Andrew
Clarke's page). Select **three** of them, and make the highest
possible tower, stacking them. David Singmaster conjectured that most
of the time, stacking them flat by their longest length gives the highest tower.
I challenged him, showing a tower with the pieces stacked at slants. The
diagonals were longer, but gravity worked against me. I did find three
octominoes that provide a counterexample, though. Can someone find them,
or provide a simpler counterexample? Write
me.

The Dot Maze has been solved by Joseph DeVincentis.

One unusual puzzle I've gotten recently is Danlock.

I found out about the International String Figure Association. I innocently asked its president if he knew how to do cat's cradle, and he showed me some amazing patterns.

Adam Dewbery found a solution to my threecolor puzzle. Basically, build a rectangle so that no two pieces of the same color touch each other. Adam sent me a 5x14 solution. Can you find it? No-one but Patrick Hamlyn has found the 7x10 solution, so here is a hint -- 8 pieces touch the X piece. No-one has solved Hamlyn's Challenge, either.

The Space Visualization Olympiad contains unfolded polyhedra nets which must be solved as a maze. Very nicely done. Can you become the world champion of 3d visualization?

In a recent letter from Wolfram Research: Sarah Flannery, the 16-year-old Irish Young Scientist of the Year who used Mathematica to devise a highly innovative, fast, and secure system of encoding data on the internet, has written about her experience in a recently published book, "In Code"... more here.

For my Puzzle of the Week, I'll concentrate on the works of Andrea Gilbert, another person I met in London. Robert Abbott has added to his Things That Roll page, and has added a very nice maze by Andrea to it. The paper versions are interesting to analyze -- all of the people I've heard from so far solve the maze on paper.

Another fabulous new invention is her Plank Puzzles. There is a lot of depth to this new type of maze, I enjoyed solving all of them.

Significant discoveries have recently been made with the Heesch Tiling problem. A configuration with a surround number of 5 can be seen at the Geometry Junkyard. Mark Thompson also discusses it.

Ignacio Ruiz de Conejo solved Steve Stadnicki's problem of dissecting a [4 5 6] triangle to make a [3 8 10].

Roger Phillips and Dylan Thurston have enumerated 62 order-6 Prime Triangles
(**partitions without subpartitions**). See my Triangle
page. Also, William Watson and Timothy Firman have solved my Integer
Triangle problem. Miroslav Vicher has found all of the order
seven prime triangle partitions.