Fair Dice, Iamonds,Eternity, Mazes, Atom Stability, Dissections, Kites&Bricks,Triangles, Retrograde, Pentagons, Prize Problems, BOOKS, and Partridges. 

From Serhiy Grabarchuk: Neo Matchstick Snake. You have
a 2x2 square and a pile of matchsticks. (Every matchstick has its length
equal to 1.) The object is to form the longest possible snake exactly
within the 2x2 square (including its border). Matchsticks must not cross
each other, and a final snake must not have selftouches even at a
point. It can be closed in a loop. Angles between the adjacent
matchsticks can be 45, 90, 135 or 180 degrees only. It's easy to find
Snake12 or Snake13, but there is even a longer snake. Jeremy Galvagni
and Bob Wainwright both matched my own length
15 solution. Roger Phillips found a beautiful symmetrical
length15 solution. Jim Lewis Melby, Clinton Weaver, Susan
Hoover, Michael Dufour, and Brian Trial sent length 14 solutions.
If one sticks to multiples of 30 degrees, I couldn't get past
Snake15. I managed to put a degree30 Snake13 entirely inside a
1.95 sided square. Can you find it?
A lot of new small sliding block puzzles
are available at Puzzlebeast.
Al Zimmermann runs one of the best
programming contests. In his latest, to win $500, you'll need to
find polyhedra
with optimized surface areas.
An extensive site about boardgames is the Games Journal. I was
introduced to it through a 1 2 3
part essay on game systems by Ron HaleEvans. I'm also quite
pleased with my subscription to Abstract Games magazine.
And I'm constantly amazed by the new things invented with Zillions of Games.
Last week: "I recently learned the
importance of the numbers 299792458
and 9192631770.
If you wish, you can read the official paper on units."
Several of you, including myself, were slightly nonplussed about the
international standard of weight being based upon a casting made by George Matthey in 1879.
I was happy to learn of an ongoing project to replace the old standard
with a perfect sphere
of silicon.
I recently came across a site devoted to
logical fallacies, Stephen's Guide
and at the Nizkor
project.
Anyone near Hampton Court in England
shouldn't miss the Puzzle Palace
run by Adrian Fisher, now through April 27th.
Bathsheba Grossman now has 4 different
very reasonably priced Polyhedral
Sculptures available. A different type of mathematical art is
described at the Termesphere
site. The art of simple programs is well represented at the Texture Garden. I'm helping
to put together a gallery of programrelated mathematical art for the NKS
2003 conference, please contact
me if you have anything that might be appropriate. Netlogo 1.2 has been
released  I'd love to hear an
opinion of this on anyone in high school, or past high school, for
that matter. Ralph
the Triangle has written many interesting papers about the
mathematics of tiny mixers and perfect pyramids. Darij Grinberg
has a page on Triangle
Geometry proofs.
CINQUANTE
ET UN + ONZE  SIX = CENT UN + QUINZE  SOIXANTE is a true statement
in French, and a twosided anagram.
material added 17 March 2003
material added 16 March 2003
material added 11 March 2003
I fixed the link to the ISEE3,
so I may as well point out the Shape of Space again as a
separate type of manifold. Stephen Hawking weighed in on this as
well in an episode of the Simpsons:
"Homer, your theory of a donutshaped universe is intriguing."
According to a recent New
York Times article, Homer may have been right.
material added 9 March 2003
A conference devoted to Mathematical Art
produced many beautiful objects and sculptures, which you can see at the Bridges & Isama
website. I did some experiments in mathematical art myself while looking
at the marvellous Nova
Plexus by Geoff Wyvill. If you don't mind rubber bands, you
can make one of these with 12 pencils and 12 small rubber bands  it's
quite attractive. If you don't like rubber bands, you can always
use more
pencils.
One good pencil puzzle is to arrange 6
pencils so that they all touch each other. With more work, 7
pencils can all touch each other. With even more work, 8 variously
sharpened pencils can all touch each other. Can you figure out
how? Send answer.
A nice result, from http://www.ktn.freeuk.com/9f.htm
A fiveleaper is a type of generalised knight that makes moves of length
5 units, with coordinates either {0,5} or {3,4}. In Variant Chess, GP
Jelliss made the following observation: “Since the fiveleaper has four
moves at every square of the 8×8 board it follows that in every
closed tour the unused moves are also two at every square, and therefore
form either a tour (is this possible?) or a pseudotour (i.e. a set of
closed circuits). The question of whether such a double tour is
possible was in fact answered in the affirmative by Tom Marlow in a
letter to me of 17 November 1991:
5leaper
double tour #1 . 5leaper double tour #2
20 47 62 55 06 21 46 63 . 46 37 60 11 56 49 04 63
31 42 57 50 11 34 29 44 . 39 24 31 52 27 40 23 32
02 59 16 09 52 03 36 17 . 54 15 06 21 34 29 16 13
13 22 39 26 19 14 23 38 . 57 08 03 64 19 36 59 10
54 05 28 45 64 61 56 07 . 26 41 50 45 38 25 42 51
51 48 35 30 43 58 49 10 . 47 28 61 12 55 48 05 62
32 41 24 37 12 33 40 25 . 20 35 30 53 14 07 22 33
01 60 15 08 53 04 27 18 . 01 18 43 58 09 02 17 44
I found an article
about the fantastically complicated journey of the International
SunEarth Explorer 3 to be fascinating. I knew about the satellite
years ago, back when I worked at NORAD. Basically, the Lagrange
point between the Earth and Sun is unstable, but there is a stable orbit
around it.
I also very much
liked a long discusion of why Boron
could provide an ideal fuel
source for cars, without pollution. For the less serious, you can
peruse silly
molecule names.
Gary J. Shannon has
posted his investigations in a WireWorld like, logicgate rich CA at his site.
material added 2 March 2003
Robert Reid sent me his efforts for
packing 16, 17,20, 22, 41, 43, 45, 49, 51, 64, 65, 76, 90, and 94
consectutive squares into a square. In each case, it is impossible
to fit the squares in a smaller square. I redrew two of them. In
general, what are the smallest rectangles than can hold all the squares
up to size n? The squares up to size 42 don't
quite fit into the smallest square.
DNA was discovered 50 years ago. For
most living creatures, 20 animo acids are enough: A (alanine), C
(cysteine), D (asparatic acid), E (glutamic acid), F (phenylalanine), G
(glycine), H (histidine), I (isoleucine), K (lysine), L (leucine), M
(methionine), N (asparagine), P (proline), Q (glutamine), R (arginine),
S (serine), T (threonine), V (valine), W (tryptophan), Y (tyrosine).
A few exotic organisms have 3 other amino acids. In honor of
DNA/RNA, here is a puzzle: aUcgAcgAACaUuGCUaCuGgaUaAuCg. Answer. Once you figure out the
capitalization... what is the longest word which is mostly capitalized?
2 1 
U 
C 
A 
G 
U  UCAG FFLL 
UCAG SSSS 
UCAG YY** 
UCAG CC*W 
C 
UCAG LLLL 
UCAG PPPP 
UCAG HHQQ 
UCAG RRRR 
A 
UCAG IIIM 
UCAG TTTT 
UCAG NNKK 
UCAG SSRR 
G 
UCAG VVVV 
UCAG AAAA 
UCAG DDEE 
UCAG GGGG 
material added 19 February 2003
Robert Abbott let me know about a Washington
Post article about Binary Arts, Bill Ritchie, and Andrea Gilbert's Clickmazes.com. Serhiy
Grabarchuk, who runs Puzzles.com,
filled me in on how that website is joining in. Andrea's plank
maze will be a key demo at the currently ongoing Toy Fair.
Snopes.com is usually good for accuracy,
but I noticed a scientific blunder on this page about
diamonds. Can you figure out what it is? Mark Thompson
matched my answer: I wouldn't call it a scientific "blunder," but I can
see a way of telling whether a diamond was produce from graphite or from
Uncle Fred. If it's from Uncle Fred, it will have a whole lot more
Carbon 14 in it. Perhaps that could be determined nondestructively with
a geiger counter, though the normal method would require destroying the
gem, in order to verify it.
Brendan Owen made a very nice discovery.
Four corners of a cube can be removed to make a tetrahedron.
If a cube is divided into 4 identical pieces so that each gets an
entire corner, you get pieces he calls cubecorners. If four
cubecorners  tetracubecorners  are connected together with
full face connection,one piece that can be made is the original cube.
It turns out there are exactly 27 other shapes that can be made,
and they can fit together to make a cube. After solving all that,
he put them all together in a lovely
applet.
I've spent a lot of time lately trying to
LURN. Left, Uturn, Right, No
turn  these are the directions a Turmite must
choose from, and I wondered what would happen if a turmite could split
as an action... say Left and Right. In addition, I added the rule
that when two turmites met, they annihilated each other. Some
interesting patterns came out from my initial study. Here is my Mathematica notebook, for those that want to
study them (with some help from Eric
Weisstein's MathWorld packages).
My main interest is finding turmites that will grow for a long
time, then selfannihilate.
material added 5 February 2003
Nick Baxter kindly added my Two
Hearts puzzle to his Sliding
Block Puzzle Page, just in time for Valentines day.
Brian Trial: Take a 2 1/2 inch by 4 1/4
inch card and cover it with as many U.S. pennies, nickles, dimes, and
quarters as you can to get the highest dollar amount. Coins must lie
flat, must not overlap or stack onto other coins in any way, and must
lie entirely on the card. For reference, a U.S. quarter has a diameter
of 0.955 inches, a nickle has a diameter of 0.835 inches, a penny has a
diameter of 0.750 inches, and a dime has a diameter of 0.705
inches. Answers. (Noone,
including the problem creator, got the correct
answer.) Now, that's an awfully UScentric puzzle, so here's a more
international version  using the coins of your country, what is the
most money in coins you can place on a 7cm by 11cm card, using the same
rules?
Serhiy Grabarchuk sent me his Stars & Spirals puzzle: Link the six Stars and six Spirals with exactly 5 connected straight lines. Stars and Spirals along your route must alter (...Star...Spiral...Star...Spiral...), and each of them must be visited just once. Joseph DeVincentis, Brian Trial, Aron Fay, Ron Zeno, Dan Tucker, Bob Kraus, Scott Purdy, Brendan Owen, Adam Fromm, Gromit, Robert Reid, Koshi Arai, Shlomi Fish, Ken Duisenberg, Mike McCraw, Earl Gose, Henry Robertson, Clinton Weaver, JeanCharles Meyrignac ,Juha Saukkola, and Bathsheba Grossman sent in the solution.
JeanCharles Meyrignac: After more
than 3 months of computation, I just finished the computation to the c1
solitaire problem on the french board. In the book Ins and Outs of Peg Solitaire from
D.J. Beasley, it is mentioned that it is possible in 21 moves. In fact,
I discovered that it can be done in 20 moves, and the solutions are very
rare (only 280). More Info.
Geometry In Action
Java Gallery, by Clark Kimberling, is well worth a look. I
think of all the courses I took in high school, Geometry was the one I
found most useful, both in terms of proof technique and the general
usefulness of geometric construction.
I have updated the Neglected Gaussian page with many solutions by Fred Helenius and W. Edwin Clark.
material added 26 January 2003
If the edges wrap, can a set of double6
dominoes be placed in a 8x7 rectangle so that all numbers are in
connected groups? The below is one of my efforts  the threes are all
connected if you consider the edges as wrapping, same with the blanks
and sixes. However, the ones and twos are both in disconnected
groups. Can everything be connected? It turns out the answer is yes. The solution below, by Jason
Woolever, is for the harder problem of connecting everything without the
use of doubles. Other answers and solvers.
See also my 2
September 2001 update for a related set of solutions by Roger
Phillips. The dominoconnection number for the torus is thus 7.
Suppose we go to 1x1x2 blocks as dominoes. What is the domino
connection number for a 3D block? If the faces of a block wrap,
it is called a 3torus, or a 3manifold. You can learn more about
these at The Shape of Space.
Without doubles, what is the domino connection number of the
3torus? With doubles? With 2 of each double? In 3D
space, as the number of doubles increases, the domino connection number
would go to infinity (why?), but I don't know if anyone has looked at
how fast. Send Answers.
James Stephens of Puzzlebeast.com has
added a new variety of restricted
sliding block puzzles. Very nice idea, and wonderfully
realized.
The cubicular goodness of the 2003 MIT
Mystery Hunt (acmecorp.com)
can be seen as an offshoot of the main MIT Mystery page. All the
puzzles are there for your perusal.
Tom Marlow notes two interesting squares:
4253907186^2 = 1809572634 7102438596, 5296031874^2 = 2804795361
0423951876. Daniel Scher's Geometry in Motion has
moved. Bob
Kraus has put Soccolot on the ZOG site (interesting
game). Martin
Watson has add lots of great stuff. Ivar's Peterson has had a
number of great columns lately, such as his Dearth of Primes
writeup (I had no knowledge of this).
An even more interesting game is Amazons,
and there is a very nice analysis of it at the More
Games of No Chance page. The full book is available online, but
I've seen so far is good enough to prompt me to buy
it.
For g4g4.com,
I wrote up an article about recent Polyform
discoveries. Comments are
welcome.
material added 15 January 2003
Cihan Altay has started PQRST 4. There are
many clever ideas here, I especially like Puzzle #8. Answers must
be submitted by 18 January.
Robert Reid found a mistake in one of the
puzzles on my old Solution page. "The first puzzle this week is by Scott
Purdy. The thick path travels from A to B, visiting every
dot. Can you remove 7 of the thin lines so that this is the only path from A to B that visits each
dot? In more mathematical terms, for K(n), what is the minimal
number of edges that needs to be removed for a unique Hamiltonean path
between two given points? The case for K(8) is unsolved (as far as
I know). Partial
solution to the general case by
Scott Purdy and Erich Friedman." Okay, seems okay  but
Robert Reid found a solution that removes only 6 lines ... and solved
the K(8) caseby removing only 9 lines. Can you find Reid's sixline
solution? Only Jim Boyce matched Reid's answer.
1 2 3 4 5 6 7 8 = 2003. Add each of
+, , ×, ÷ exactly once to
make the equation true. Answer.
This is by Yoshio Mimura.
I found his page while looking to see if anyone else had noticed the
octal square 177771777177771.
So ... 177771777177771. Twenty years ago, Nob Yoshigahara noticed that 81619 × 81619 = 6661661161. Are there larger square numbers using only two digits? Noone knows. The Mathematician Secret Room has more data for the 3digit square problem. I wondered if I could find new complex squares with that property (I couldn't), or two digit squares in other bases (easy). Nick Baxter looked at rational squares of two digits, and found some interesting solutions. There is enough here to figure out the significance of 34343443434344.
(40457+27469i)^2
= 882222888+2222626666i (20644+3425i)^2 = 414444111+141411400i
(13773+12464i)^2 =
34344233+343333344i (23704+1811i)^2 =
558599895+85855888i
(9998+3334i)^2 =
88844448+66666664i (4999+1667i)^2 =
22211112+16666666i
(3507+2369i)^2 =
6686888+16616166i (16667+16666i)^2 =
33333+555544444i
(1754+719i)^2 =
2559555+2522252i
(2087+106i)^2 = 4344333+442444i
(1667+1666i)^2 =
3333+5554444i
(328+175i)^3 = 5152552+51122225i
(Base 9) 2534^2 =
6666677
(Base 9) 32641^2 = 1181111181
(Base 8) 13240265^2 =
177771777177771 (Base 9) 35577^2 = 1414111444
Robert Henderson has found a remarkable
Latin pentacube solution. With the following division of a 5x5x5
cube into 25 different pentacubes, find a a way to color the cube in 5
colors so that every row, column, stack, and pentacube is comprised of
all 5 colors. Here is the answer.
N N N N M W W F N M C W F [ M D S F Q J D S S S J
C T O O O C T T [ [ C W [ [ M H H F Q M D S Q Q J
B T T O O B B B \ G C W B G G D H F Q G D H H J J
V X X Z \ V V X \ \ Y V X \ G R U U I I R R U I K
Y X Z Z Z Y Y Z P P Y V P P I U U K P I R R K K K
Some prime numbers remain primes when reversed. Are there an
infinite number of them? John Gowland sent me the following crossnumber
puzzle called Reversed Squares, which is based on reversible primes.
Here are some clues, a solving strategy, and the answer.
material added 5 January 2003
Those who have Zillions of Games can try
out Bob Kraus's
very nice Extraction puzzles, now at the Zillions site.
Eventually, some brave soul is going to need to look at all of the
Zillions files that are available, and summarize what is best.
That will be hard, because most of the files are quite good, and
timeabsorbing.
Here are some small fractions that make a
nice approximation to a familiar number. 22/17 + 37/47 + 88/83 !=
Pi. Can anyone find a better approximation with small fractions?
I've noticed the greedy algorithm doesn't work very well at
finding better solutions.
Some things, I've been keeping around.
For example  here's a little puzzle  I can't remember if I
made it, and if I did make it, I can't remember the answer, and I
haven't solve it. So it wouldn't be all that fair to present it,
would it? Well, here it is, anyways. And here is a whole page of other material I
never quite figured out how to present, so here it is, all in one big
batch. A similarly disorganized page is my g4g5
writeup. Comments are
welcome. Answer to 4divide
(Solutions sent by Remmert Borst, Cihan Altay, Paul Cooper, Jonathan
Welton, Jon k McLean, Franz Pichler, Clinton Weaver, Agaeus, Matt Elder,
Joseph DeVincentis, Kirk Bresniker, Jeremy Galvagni, and Juha
Hyvönen) If you like division puzzles, I don't think anyone has
solved all of Mike Reid's puzzles.
The various integersided blocks with
sides 0<a<b<c<7 will fit into a 7x7x15 block, as shown by Erich Friedman.
Warning: Big File! Edward Brisse
compiled all of the Triangle Centers into one big text file. I
won't give a direct link, but you can find it at EdwardBrisse.txt, or
EdwardBrisse.zip, on this site.
material added 31 December 2002
While playing around with Pick's Theorem, I came up with a deceptively tricky little problem. Divide a 5x5 square into 5 regions which have identical perimeters but differing areas. All lines must be straight, and must connect grid vertices. Answers. Another by Livio Zucca. I used a trick in my solution (1)  I was surprised when Joseph Devincentis sent me different solution (2): "This was a very nice puzzle, and a wonderful demonstration of the usefulness of Pick's Theorem. With the theorem I could quickly see that I needed to somehow split up the regions using lines that crossed only 6 of the internal points, with the five regions containing 0,1,2,3,4 of the other internal points. It still took me a while to find the correct perimeter and arrangement of lines to make it work." After that, Daniel Scher and Martin Bernstein sent solutions (3) and (4). Taus BrockNannestad sent a 5th answer, and William sent 20 more (using the 345 triangle trick in a normal grid).
Theo Gray and I managed to collect all 90 stable elements. We have started testing some of the samples. One of the most bizarre  a weird rock I found when I was six years old has turned out to be 38% titanium. On the other hand, a "Titanium" tennis racket wound up having no titanium whatsoever.
Stephen Wolfram has made some of the Historical Notes
from A New Kind of Science
available. I like what he does with WireWorld.
The Retrograde Anaylsis Corner has made some nice new updates.
material added 25 December 2002
(Merry Christmas)
I've updated my Prize
Puzzle page.
NetLogo 1.2 has been
freely released at the Center for Connected Learning and ComputerBased
Modeling, at Northwestern University. The Logo language is
frequently known as the language for "programmable turtles." A
turtle with the instruction set {move 1, turn 90 degrees} would make a
square. There is much, much more, and this releases is filled with
lots of excellently documented programs in art, biology, chemisty,
physics, computer science, earth science, mathematics, social science,
and more. No programming knowledge is necessary, you can just start up
the models and what how they work. With slight programming, the models
are easily modified. It's a wonderful package for learning.
JeanCharles Meyrignac found an old puzzle that involved the 18 ways to threecolor a tromino. How many can be placed in a 7x7 square so that only like color touch? It seems 13 is the answer. In an 8x8 square, he can place all but 1 piece, and isn't sure if all 18 is possible. He wrote a program to find how many of the 40 4colored trominoes could be placed in an 11x11 square, and he found two solutions with 33 pieces placed. He does not know if these can be improved. It's a nice task to split the grids into the different trominoes.
1 1 1 1 1 2 2 2 2 1 1 2 2 2 1 1 1 3 1
1 1 1 2 2 2 2 3 1 1 2 2 2 1 1 3 1 2
1 2 2 1 2 2 2 2 1 3 3 2 2 1 1 1 1 1
3 3 3 1 3 3 3 3 3 2 2 2 4 4 4 4 1
3 3 3 4 4 4 3 3 3 2 1 2 4 4 1 1 1 1
4 4 4 4 4 4 4 3 3 3 1 2 4 4 4 3 1 2 3
3 2 2 2 2 2 2 1 3 1 2 2 3 3 3 3
3 1 2 1 1 4 4 4 4 4 3 2 3 2 2 2 3 3 3
1 2 1 1 1 1 4 4 3 3 2 3 4 4 4 4 4 4
4 1 2 4 4 4 1 4 3 3 2 2 3 4 1 4 4 3
2 2 4 1 1 4 3 1 2 4 2 4 1 2 4 4 1 3
Older Material  29 Oct 02 to 13 Dec 02
Older Material  3 Sep 02 to 21 Oct 02
Older Material  16 Jul 02 to 26 Aug 02
Older Material  13 May 02 to 9 Jul 02
Older Material  17 Feb 01 to 5 Aug 01
Older Material  2 Jun 00 to 11 Feb 01<
Previous Puzzles of the week are here.
