Martin Gardner (21 October 1914 – 22 May 2010)

• Science and Mathematics: recreational, popularizing
• Literature: annotating problematic texts
• Skepticism: debunking pseudoscience and pseudoreligion
  
• Mathematical Games column in Scientific American (1956–1981)
• Notes of a Fringe-Watcher column in Skeptical Inquirer (1983–2002)
• More than 70 books

Gathering for Gardner (G4G)

• First two in Atlanta 1993, 1996 attended by MG himself
• Every even-numbered year after 2010
• The sort of stuff that interested MG
  
• After MG's death in 2010, annually on his birthday: 21 October
• Worldwide at dozens of venues
• In 2011 all continents are represented (also Antarctica)

MacMahon's coloured cubes

• Invented by Major Percy MacMahon in 1893.
  On the [number omitted] cubes that can be constructed with six differently coloured squares
• Featured by MG in Scientific American in 1961 and 1978
• Take six differently coloured squares
• Make them into a cube
• How many different cubes?
  
• Guesses please!

The reference cube, or standard die

• Sides ⚀ ⚁ ⚂ ⚃ ⚄ ⚅
• Pips on opposite sides add up to 7
• Has a corner ⚀ ⚁ ⚂ running clockwise
• $6!=720$ ways to paint it with 6 colours
• Not all different!

Counting different ways to paint it

• We can turn the die any way we like. It's hidden under the paint!
• Consider two given colours, say white and blue
• Case 1: they are adjacent
  • Put them at ⚀ and ⚁
  • $4!=24$ ways to paint the remaining four
• Case 2: they are opposite
  • Put them at ⚀ and ⚅
  • Put red at ⚁
  • $3!=6$ ways to paint the remaining three
• 24+6 = 30
• Does this convince you?

Counting ways of holding the cube

• Any of 6 colours may be in front
• Any of 4 sides may be on top
• $6\times 4$ ways to hold it
• Each cube is therefore counted 24 times when we say there are $6!$ ways to paint
• $720/24=30$
• Does this convince you?

A betting game

• You and I both take a cube from the set of 30.
• At random, with our eyes closed.
• You pick any two adjacent colours on your cube.
• If those colours are adjacent on my cube too, you win, otherwise I win.
• What odds should you give me?

My argument

• Say you choose red and blue.
• Put my cube on the table with the blue side up.
  
• You only lose if the red side is bottom.
• That's one case out of five.
• So you should offer me odds of 4:1.

Do you take that?

• If not, what should the odds be?
• Offers please!
• (Don't worry, we're not betting for real money!)

Edges

• There are $\left(\genfrac{}{}{0ex}{}{6}{2}\right)=15$ different edges.
• Each cube has $12$ edges.
• Giving $30\times 12=360$ edges in total on all the cubes.
• Each edge occurs on $360/15=24$ cubes.
• Each edge does not occur on $30-24=6$ cubes.

The correct odds

• 24 cubes contain your edge
• 6 don't
• You already have one of those 24
• Chances I also have one are 23:6
• If you give me 4:1, on the average every 29 rounds, I'll lose R1 23 times and win R4 6 times, so I expect to make R1 for every R29 I bet.

A notation for the cubes

• Number the colours, e.g.
  
• White, 2. Blue, 3. Red, 4. Green, 5. Yellow, 6. Orange.
• Choose ⚀ for the face on which colour 1 is.
• Choose ⚁ for the face on which the lowest-numbered neighbour of 1 is (this will be 2 most of the time, 3 when 2 is on ⚅)
• We call this the standard orientation of the cube.
• Write down the colour numbers under the faces in order ⚀ to ⚅ and group them in threes, e.g. 123-456. That's the notation.
• The 30 cubes are listed on the handout. This notation is used in the first column.

Corners

• The three colours around a corner are written clockwise. So 123, 231 and 312 all denote the same corner White-Blue-Red but not 321, 213 and 132 which denote White-Red-Blue.
• The two triplets in the notation 123-456 form a pair of opposite corners at ⚀ ⚁ ⚂ and ⚃ ⚄ ⚅.
• When we write a pair of corners as UVW-XYZ, U is opposite Z, V is opposite Y and W is opposite X.
  (Remember which side are opposite in ⚀ ⚁ ⚂ - ⚃ ⚄ ⚅!)

Find that cube!

• There are three other pairs of opposite corners:
  ⚀ ⚂ ⚄ - ⚁ ⚃ ⚅
  ⚀ ⚄ ⚃ - ⚂ ⚁ ⚅
  ⚀ ⚃ ⚁ - ⚄ ⚂ ⚅
• The other three opposite-corner pairs for each cube are also listed on the handout.
• Any opposite-corner pair fully specifies a cube.
• The same notation can be used to describe the cube, even when it is not in standard orientation.
• Which cube has the opposite-corner pair 451-632?
• Remember, 145 may also appear as 451 and 514, but not 541, 415 or 154.

Mirror images

• In the mirror, clockwise becomes anti-clockwise.
• So our standard cube
  ⚀ ⚁ ⚂ - ⚃ ⚄ ⚅
  becomes
  ⚀ ⚂ ⚁ - ⚄ ⚃ ⚅.
• Which cube on the handout is the mirror image of 451-632?

Complete corner sets

• There are $\left(\genfrac{}{}{0ex}{}{6}{3}\right)=20$ ways to choose the colours around a corner.
• Each of these may appear clockwise or anti-clockwise.
• That makes 40 different corners.
• One cube has 8 corners.
• Can you find a set of 5 cubes that between them contain all 40 corners?
• Remove those cubes. Can you do it again choosing among the others?
• Again, again, again and again?

Conway's notation

• Invented by John Conway, published by MG in Scientific American 1978.
• Independently discovered by DPL two months before MG published it.
• Both of us anticipated by Andrew Sobczyk, 1974
• We should really say "Sobczyk's classification"
• Take the six complete corner sets, call them $A,B,C,D,E,F.$
• Take their mirror images, call them $a,b,c,d,e,f.$
• Each cube appears twice, once in an uppercase set, once in a lowercase set. Those two letters are the notation for the cube.
• Those are different letters: no $Aa,Bb$ etc. Why?

Puzzles for which Conway's notation helps

• The first four puzzles involve building a $2\times 2\times 2$ cube with single coloured faces.
• Pick any cube as "prototype". Select eight others from which you can build a replica of that cube, with touching faces in the inside being the same colour.
• (The Mayblox puzzle): like the above, but you are given the set of eight cubes, and are not told which cube is the prototype. Always has two solutions.
  
• (The Madness puzzle): like the Mayblox puzzle, but the inside faces need not match.
• Build a cube with only four colours showing, two pairs of opposite faces having the same colour. You may choose your eight cubes any way you like.
  
• $3\times 3\times 3$ versions of the first three puzzles.

The MAYBLOX puzzle, patented by P.A. MacMahon and J.R.J. Jocelyn

Further reading

Jürgen Köller, MacMahon's Coloured Cubes, http://www.mathematische-basteleien.de/macmahon.htm
(8 references)

Torsten Sillke, MacMahon's cubes,
http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/macmahon-cubes
(21 references)