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Fun Answers
© 2000-2016   Gérard P. Michon, Ph.D.

 Center-symmetrical MARTIN GARDNER 
 ambigram, created by Scott Kim in 1993
To  Martin Gardner who has brought
more mathematics to more millions than anyone else
.
Berlekamp, Conway & Guy  (Winning Ways, 1982)
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   Glider in Conway's 
 Game of Life  

Related articles on this site:

Related Links (Outside this Site)

The Gathering 4 Gardner Foundation.   |   Fans of Martin Gardner (FaceBook)
Game Inventor:  Martin Gardner  (a tribute from  Kadon Enterprises, Inc.)
Committee for Skeptical Enquiry (CSIOP)   |   Skeptical Enquirer
 
Scott Kim, Puzzlemaster
Intriguing Tessellations  by  Marjorie Rice  (San Diego).
James Randi Educational Foundation  (JREF).

Martin Gardner   |   Mathemagician   |   Persi Diaconis   |   John H. Conway   |   Scott Kim   |   Bill Gosper   |   David_Singmaster (1939-)   |   Sol Golomb (1932-)   |   James Randi   |   Donald Coxeter (1907-2003)   |   M.C. Escher (1898-1972)   |   Ray Smullyan   |   Jerry Andrus (1918-2007)   |   Jay Marshall (1919-2005)

 Conway ties a knot  

Video  (courtesy of  Encyclopedia Britannica)
mp4 (116 MB)  |  wmv (113 MB)  |  Vimeo
Mystery and Magic of Mathematics:  Martin Gardner and Friends
 
Alternate title:   Martin Gardner, Mathemagician.
An episode of  The Nature of Things with David Suzuki  (CBC, 1996)  Who is this at 1:07 ? 
 (Please, tell me!)

Featuring, in order of appearance:   David Suzuki 00:48,
Meir Yedid  00:55 (finger-hiding trick "sleight of hand") Herb Zarrow & Max Maven 01:05, Bill Gosper 01:09, Martin Gardner 01:15, Sol Golomb 01:18, Ron Graham 01:27, James Randi 01:42, Jay Marshall 02:09, John H. Conway 02:53, Michael Weber 03:08, Scott Kim 03:54, Persi Diaconis 12:20, Donald Coxeter 29:03, Doris Schattschneider 31:58  and  Marjorie Rice 32:52.

 Gathering for Gardner
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The Lives and Games of Martin Gardner


(2010-05-23)   Martin Gardner (1914-2010).   Eulogy.
Born on Oct. 21, 1914, Martin Gardner passed away on May 22, 2010.

After a brief illness, Martin Gardner died unexpectedly at Norman Regional Hospital at the age of 95.  The precise cause of death is unknown.  His passing was quick and painless.  Martin Gardner is survived by two sons:  James (of Norman, Oklahoma) and Tom (of Asheville, NC).  He is mourned by many friends and countless professional or amateur mathematicians.

The passing of Martin Gardner has urged a few people who had crossed his path to recollect those precious moments:

  • An interview with Martin Gardner on February 28, 1979
    (with Stan Ulam, Ron Graham, Peter Renz and Don Albers) 
    posted in  The Back Bench  by  Tony Barcellos  (2010-05-23)


  Arthur Harold Stone
Arthur H. Stone
(2009-02-04)   Hexaflexagons   (1939)
Gardner's first column in  Scientific American  (1956).

In 1939, Arthur Harold Stone (1916-2000)  was a British doctoral student who had just arrived at Princeton University to study general topology.  Since American sheets of paper were wider than European ones, he was trimming letter-size American sheets to fit British binders.  (The European size was  not yet  standardized as "A4".)  Stone was left with lots of strips of paper to fold and play with.   One day, he stumbled upon a  flexagon  and showed it to some of his fellow students, including Bryant Tuckerman (1915-2002), Richard P. Feynman (1918-1988)  and John W. Tukey (1915-2000).  They formed the  Princeton Flexagon Committee.  Soon, it seems everyone on campus was making and flexing  hexaflexagons.

 Martin Gardner
Martin Gardner, 1960
  In December 1956, the  hexaflexagon  craze got a fresh start when  Martin Gardner  sold an article about it to  Scientific American.  The publisher of Scientific American, Gerry Piel, immediately entrusted Gardner with a monthly column:  Mathematical Games,  premiered in January 1957 and lasted for more than 25 years.   Martin Gardner
Martin Gardner, 1996

 Come back later, we're
 still working on this one...

Wikipedia :   Flexagon


 Solomon Wolf Golomb
Solomon W. Golomb
(2009-01-07)   From Dominoes to Polyominoes
The  12  pentominoes  of  Sol Golomb  (1954).

Polyominoes were devised in 1954 by Solomon W. Golomb when he was a 22-year old graduate student at Harvard.  A  polyomino  consists of N unit squares in the plane, each sharing at least one of its sides with another square.

According to the wording of that simple definition, there is one zeromino  (consisting of an empty set of unit squares)  but there are  no  monominoes  (N=1).  However,  many  people consider a lone square to be a monomino...

Two polyominoes are considered distinct only if they cannot be obtained from each other by rotating or flipping.  There is only one domino  (N=2)  but there are 2 triominoes, 5 tetrominoes, 12 pentominoes, 35 hexominoes, etc.

 1 domino, 2 triominoes, 5 tetrominoes

1, 0, 1, 2, 5, 12, 35, 108, 369, 1285, 4655, 17073, 63600, 238591, 901971, 3426576, 13079255, 50107909, 192622052, 742624232, 2870671950, 11123060678, 43191857688, 168047007728, 654999700403, 2557227044764, 9999088822075, 39153010938487,  ...  (A000105)

 The 12 pentominoes arranged in an
8 by 8 square, with a 2 by 2 hole.  

The set of  12  pentominoes proved to be most endearing...  Those twelve pieces are used in a two-player game  (proposed by Golomb himself in December 1994)  which is played on an  8 by 8  chessboard:  The players alternate placing a piece until one of them is unable to do so  (and is declared the loser).

Hilarie K. Orman proved the game to be a  first-player win.

Pentominoes have been marketed whose thickness is the same as the side of the constituent squares.  Twelve such pieces have a combined volume of 60 cubic units, which can be assembled into 3 types of  cuboids  (2 by 3 by 10,  2 by 5 by 6,  3 by 4 by 5).

 Soma Cube
(2009-01-10)   Piet Hein's Soma Cube
Nonconvex  solids consisting of 3 or 4 cubes.

The 3D equivalents of polyominoes are solids consisting of unit cubes which share at least one face with another cube.

Two such shapes are distinct only if they're not congruent by rotation.

7 distinct nonconvex shapes can be obtained in this way with 4 cubes or less (only one consists of just 3 cubes).  This includes two  chiral  pieces which are mirror images of each other.  Those are the so-called  soma pieces  whose combined volume is 27 units.  They can be assembled into a cube 3 units on a side.

Legend has it that the  Soma Puzzle  was devised by Piet Hein (1905-1996) during a lecture on quantum mechanics by Werner Heisenberg (1901-1976).

Wikipedia :  Soma Cube


(2009-01-18)   The Pentagons of Marjorie Rice   (1975)
Tessellations of the plane by  convex  pentagons.

 Come back later, we're
 still working on this one...

Intriguing Tessellations  by  Marjorie Rice  (San Diego)   |   Links
Perplexing Pentagons  by  Doris Schattschneider  (Moravian College, Bethlehem, PA).
The 14 Types of Convex Pentagons that Tile the Plane  by  Ed Pegg, Jr. & Branko Grunbaum
Recipe  for finding  pentagons that tile the plane  by  Bob Jenkins


   Sir Roger Penrose (1931-)
Sir Roger Penrose
(2009-01-17)   Aperiodic Penrose Tilings   (1977)
Matching the colors of the  kite  and  dart  tiles.

The two Penrose tiles are quadrilaterals consisting of two pairs of equal sides whose lengths are in a  golden ratio :

f   =   1.6180339887498948482...

This yields angles that are multiples of  p/5  and allow various types of pentagonal patterns around any vertex where several tiles meet  (without voids).

 Penrose's Kite and Dart

The convex tile is called a  kite,  the other one is dubbed  dart.  They bear a specific color pattern like the one pictured above.  The colors must match along any side where two such tiles touch.

It's nice to have circular arcs centered on vertices for the inner boundaries between colors but more creative designs can be used, as long as the colors on equal sides enforce the same matching as what's illustrated here.  Another (colorless) alternative would be to enforce the side-matching with jigsaw-type notches.

Penrose Tiles (MathWorld)

 Mirror-symmetrical MARTIN GARDNER 
 ambigram, created by Scott Kim in 1996
(2009-01-05)   Ambigrams
The  inversions  of  Scott Kim  (1981).

Scott Kim is a friend of Martin Gardner who practices an elaborate type of calligraphy where the spellings of words changes when they are rotated or viewed in a mirror.  The example at right has mirror symmetry whereas the title of this page is symmetric with respect to its central point.


 John Horton Conway  (2009-01-02)   Survival in Conway's  Game of Life
Any  computable query boils down to  that  question!

If you learn to be good at a game, you find
what it is you should have been thinking about
.
John Horton Conway  (1937-)

The  Game of Life  (GOL)  invented by John H. Conway  (in 1970)  is a zero-player game.  Once a board configuration is set up, it just evolves according to fixed rules, like life would unfold in a completely deterministic universe.  The point is to discover  life forms  with an interesting evolution...  A very rich catalog was eventually compiled which provided a few components that allow the simulation of any imaginable deterministic computer!

In Conway's game, the board is just an infinite grid of square  cells.  Each cell is either dead  (empty)  or alive  (occupied by a black dot).  The  neighbors  of a cell are the  8  cells which share a side or a corner with it.  There are just two rules which govern the evolution of a configuration from one generation to the next:

  • A cell  survives  if and only if it has either  2  or  3  live neighbors.
  • A cell is  born  [in empty space]  if and only if it has  3  live neighbors.

 Block  Block The so-called  block  is the simplest stable  life form.  It consists of four live cells in a square configuration.  Each of them survives because of its  3  live neighbors and no cell is born because no other cell has  3  live neighbors.

The  blinker  consists of a row of  3  live cells which oscillates between a vertical and a horizontal configuration.  The center cell survives, both extremities die and two cells are born which replace them at a right angle...  Again and again.

 Vertical Blinker  Horizontal Blinker  Vertical Blinker  Horizontal Blinker  Blinker

The most interesting of the  small  life forms is the  glider,  which consists of  5  live cells and moves diagonally one unit in  4  steps:

 Glider  Glider  Glider  Glider  Glider

 Moving Glider  
 Dart Spaceship  Dart Spaceship Spaceships :   The life patterns which move  p  cells in  q  generations are called  spaceships  and are said to be moving at  p/q  times the speed of light  (c).  The aforementioned  glider  moves at  c/4  diagonally.

At right, is the  dart  spaceship which moves  at c/3.  It was discovered by David Bell in May 1992.

Three small spaceships were discovered by Conway  (in 1970)  which move at speed  c/2  (namely:  2 cells in 4 generations)  either horizontally or vertically:  The  small fish  (or  float)  the  medium fish  and the  big fish.  They are also respectively known as the  lightweightmiddleweight  and  heavyweight  spaceships  (abbreviated LWSS, MWSS, HWSS).

 Small Fish.  Medium Fish.  Big Fish.

Gardens of Eden :

One early question about the  Game of Life  was the existence of board configurations which cannot result from the evolution of a previous population.  Such a configuration is known as a  Garden of Eden.  The existence of  Gardens of Eden  can be demonstrated by the following numerical argument:

A population contained in a square which is  5n-2  cells on a side has either no parent or at least one parent fully contained in a 5n by 5n square.

Such parent configurations can be partitioned into  5 by 5  squares.  The key remark is that two parents clearly have the same children if one of those small squares is either empty or has only its central cell occupied.  So, the number of distinct children of parents contained in a 5n by 5n square is  no greater than :

( 225 - 1 ) n2

If that number is less than the number of 5n-2 by 5n-2 configurations, some of those must have no parent !  Let's simplify the relevant inequality:

( 2 25 - 1 ) n2   <   2 (5n-2)2

With  k  =  lg ( 2 25 - 1 )  =  24.999999957004336643612528...  this inequality becomes  (taking the binary logarithm of both sides):

k n2   <   25 n2 - 20 n + 4

The leading term of the polynomial   (25-k) n2 - 20 n + 4   being positive, it is itself positive for sufficiently large values of  n.  Numerically, the inequality holds when  n  is beyond  465163191.59...  So, there must be  Gardens of Eden  among the populations contained in a square  2325815956  cells on a side!   QED

The above can be used to show that a (very) large configuration is most likely to be a  Garden of Eden  (the probability that it isn't vanishes exponentially as a function of its size).  It's still a challenge to find  small  Gardens of Eden, though.

The first explicit  Garden of Eden  to be discovered was the following pattern, inscribed in a  9 by 33  rectangle.  It was found by Roger Banks, Mike Beeler, Rich Schroeppel et al. at MIT in 1971.  Curiously,  Achim Flammenkamp  noticed many years later  (on June 16, 2004)  that the  5  rightmost columns of this historical example are essentially not needed  (yielding a  9 by 28  Orphan ).

 Banks Garden of Eden

At this writing, the smallest known  Garden of Eden  is a pattern of  72  live cells in an  11 by 12  rectangle.  It was discovered by  Achim Flammenkamp  on June 23, 2004.
   Ralph William Gosper, Jr.
R. William Gosper

The Gosper Glider Gun :

Early on, Conway had conjectured that there were finite  life forms  which would grow indefinitely but he could not find one...  So, he put up a  $50  reward for an example.

Bill Gosper (1943-)  claimed the prize with the following  grand  thing, obtained by studying the interaction of two  queen bee  shuttles  (stabilized by  blocks ).

 Gosper Gun

This was the first example of what's called a  glider gun.  The  Gosper gun  emits a steady stream of gliders but its core returns to its former self after  30  steps.

Glider guns  have since been devised for any arbitrary period above 14.  They are a key ingredient in the so-called  universalization  of the game of life performed independently by Gosper and by Conway  (using the same approach).  As described in the next section, this establishes, essentially, that  anything  boils down to a question about Conway's game!

 Alan M. Turing

Conway's  Game of Life  is a universal computer :

In the last chapter of the first edition of  Winning Ways  (1982)  John Conway proves that his automaton is just as powerful as a Turing Machine  (or any other type of computer with an unbounded amount of read/write memory).

Remarkably, an engineering approach is used to show how all the components of modern computer circuitry can be simulated within the  Game of Life  (program and input data being encoded in the starting configuration).

Basically, Conway uses  clocked  streams of rarefied  gliders  as the basic digital signals  (the presence or absence of a glider in a stream at a scheduled time is interpreted as a specific bit being  1  or  0).  Such streams are produced by  guns  and absorbed by  eaters  (guns  with arbitrarily low output rate exist, so that synchronized wires will not interact as they cross each other).

Conway uses a large zoo of special configurations and a bunch of clever techniques to simulate logic gates and all the circuitry of a  finite  computer endowed with an  unbounded  external memory which it can read and write...

As a beautiful final touch, he shows how such a simulation can  completely self-destruct  to indicate that the corresponding computer program has halted  (otherwise, something remains on the board).

The engineering details are quite intricate but the guiding principles are simple and the glorious conclusion is inescapable:  The  Game of Life  is an automaton which is just as powerful as a Turing machine.  Any problem which  (like most interesting logical questions)  is equivalent to the ultimate halting of a computing machine  (with unlimited storage capabilities)  can actually be  rephrased  in terms of the ultimate vanishing of a specific starting configuration in Conway's  Game of Life .  In other words:  Life  is  hard.    Isn't it?  

October 1970 (Martin Gardner)   |   Conway's Game of Life (Wikipedia)
What is the Game of Life?  by Paul Callahan and Alan Hensel
Golly  by Andrew Trevorrow and Tomas Rokicki  (with Dave Greene, Jason Summers & Tim Hutton)
Logicell 1.0  by  Jean-Philippe Rennard, Ph.D.  (implementing logic functions in Conway's game).
26-Cell Pattern with Quadratic Growth  (Bill Gosper and Nick Gotts, March 2006)
Eric Weisstein's Treasure Trove of the Life Cellular Automaton  by Eric Weisstein  (2000-2005)
Open Directory Life Index  (initially maintained by Mirek Wojtowicz)
 
John Conway Talks About the Game of Life   Part 1   |   Part 2


 David Singmaster (2009-01-14)   Rubik's Cube:  The Craze
Martin Gardner's cover story  (March 1981).

The notation which is now standard to describe sequences of moves in Rubik's cube was invented by David Singmaster in 1979.

A capital letter indicates a clockwise rotation of a quarter turn.  The same letter  primed  denotes a counterclockwise rotation.  The following six letters are used, which refer to the location of the center of rotation, irrespective of its color:

F (front), R (right), L (left), U (up), D (down) and B (back).

In practice, B is rarely used.

 Come back later, we're
 still working on this one...

Rubik's Cube (Wikipedia)   |   Solving the Rubik's Cube Systematically  by Alex Fung Ho-San
Beginner Solution to the Rubik's Cube  by Jasmine Lee
A simple trick to crack all Rubik puzzles  by  Burkard Polster   (Mathologer, 2016-01-15).


(2010-06-06)   Impossible Knot
Misleading parsing.

You are allowed to lie a little, but you should never mislead.
Paul R. Halmos  (1916-2006)

 John Conway 2 minutes and 53 seconds into the aforementioned video presented by David Suzuki  (CBC, 1996)  John Conway says:

This, I'm sure, was in Martin's column sometime.  You know, it's impossible to tie a knot without leaving go of the ends of the strings the way I just did... 

Up to this point, Conway did not lie but he did mislead...  Indeed, the last thing he said could be parsed as applying only to the locution "leaving go of the ends of the strings"  (which is precisely what Conway did, secretly).

This perfect example of a misleading true statement is followed by Conway's concluding remark which either turns the whole thing into a straight lie  (expected of an illusionist, amateur or not)  or can be forcefully reparsed into a true statement.  Your pick:

... but Martin will tell you many different ways of doing it.

Conway's performance is flawless and well photographed.  Even if you know what to look for and play the video frame by frame, you simply won't detect the fallacy.


(2010-06-11)   The knowledge of Man

Martin Gardner loved the following Indian legend.  He first run across it in  The People, Yes (1936)  by  Carl Sandburg (1878-1967):

The white man drew a small circle in the sand and told the red man,  "This is what the Indian knows,"  and drawing a big circle around the small one,  "This is what the white man knows."  The Indian took the stick and swept an immense ring around both circles:  "This is where the white man and the red man know nothing."

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