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Martin Gardner (1914-2010)
Scientific American – Mathematical Games column
• 1956-1981 (297 monthly columns)
Books:• Mathematical Games• Word puzzles• Annotated Alice• Books on
pseudoscience and skepticism
Presentation by Dennis Mancl, [email protected]
Magic squares
8 1 6
3 5 7
4 9 2
8 + 1 + 6 = 15
3 + 5 + 7 = 15
4 + 9 + 2 = 15
83
+ 415
15
+ 915
67
+ 215
• An array of numbers
• No duplicates• The sum of each row is the same
• The sum of each column is the same
Magic squares
? ? ?
? ? ?
? ? ?
?
?
?
? ? ? ?
• Use the numbers 1 through 16
• What will be the sum of each row?
(1+2+…+16) / 4
1+2+…+n = (n+1) n / 2
(1+2+…+16) / 4 = (17 16 / 2) / 4 = 136 / 4 = 34
Magic squares
1 2 3
5 6 7
9 10 11
4
8
12
13 14 15 16
16 15 14
12 11 10
8 7 6
13
9
5
4 3 2 1
• Start with 2 squares – numbers in reverse order
Magic squares
1 2 3
5 6 7
9 10 11
4
8
12
13 14 15 16
16 15 14
12 11 10
8 7 6
13
9
5
4 3 2 1
• Choose 8 cells from one square, 8 cells from the other
Magic squares
7
4 3
8
16 15 14
12 11 10
6
13
9
5
2 1
1 2 3
5 6 7
9 10
4
8
13 14
11 12
15 1616 2 3
5 11 10
9 7 6
13
8
12
4 14 15 1
Magic squares
16 2 3
5 11 10
9 7 6
13
8
12
4 14 15 1
16 + 2 + 3 + 13 = 34
5 + 11 + 10 + 8 = 34
9 + 7 + 6 + 12 = 34
4 + 14 + 15 + 1 = 34
1659
+ 434
2117
+ 1434
3106
+ 1534
138
12+ 134
16 2 3
5 11 10
9 7 6
13
8
12
4 14 15 1
Albrecht Dürer – Melencolia I (1514)
Puzzles
Spend 6 cents === guaranteed to have at least two red balls
Spend 8 cents === guaranteed to have at least two white balls
If there are 4 red balls and 6 white balls
2 cents is enough some of the time
3 cents is always enough
Puzzlesold
square tiles
newrectangular tiles
21 red squares19 white squares
• You can cover 38 of the 40 squares• But there will always be 2 red
squares left over• You need to cut one of the
rectangular tiles in half…
Hexaflexagons
A flexible hexagon made from a long strip of paper folded into triangles
You can “flex” the hexagon to show different faces
Discovered in 1939 by Arthur Stone (1916-2000) when he was a grad student at Princeton
• contributions by Bryant Tuckerman (1915-2002), John Tukey (1915-2000), Richard Feynman (1918-1988) [the “Flexagon Committee”]
B
A C
B
A
A
B
C
CA
B
C
z A C CA
BBExample: a tri-hexaflexagon
• 3 faces• Each face
has 6 triangles
CA
B
C
z A C CA
BB Step 1.Start with a strip of 10 equilateral triangles. Fold both ways on all of the lines
7 Steps to fold a tri-hexaflexagon
CA
B
C
z A C CABB Step 2.
Fold 3 triangles on the left towards the back
x
y
fold back
CA
B CA
BBB
y
x
Step 3.Fold over one triangle towards the front
fold to the front
A
CA
B CA
BB
yx
Step 4.Fold the 4 right triangles towards the front
Caution: Don’t fold towards the back… if you folded it the wrong way, your flexagon will look like this:
fold to the front
Step 5.Re-open the one triangle that was folded over in step 3
y
A
B
B
B
A
B
Ax
re-open one triangle
Step 6.Put glue on the top 2 triangles
A
B
B
BB
B
y
xglue
A
B
B
BB
B
B
y
xStep 7.Fold down the top triangle – done!
Front view Back view
Flexing a tri-hexaflexagon
HexaflexagonsThere is also a hexahexaflexagon: start with a strip of 19 equilateral triangles
Fold it into a coil
Then fold back the right-most 3 triangles; fold forward the left-most 4 triangles
Puzzles and mathematical games
It didn’t start with Martin Gardner…
• W. W. Rouse Ball (1850-1925)• Sam Loyd (1841-1911)
And the tradition goes on…• Ian Stewart (1945- )• A. K. Dewdney (1941- )• Dennis Shasha ()• Simon Singh (1964- )• Chris Maslanka (1956- )• Will Shortz (1952- )• Keith Devlin (1947- )• Jordan Ellenberg (1971- )
Tri-hexaflexagon template
See also:• https://www.youtube.com/watch?v=ngwuUqJZoxQ• https://www.youtube.com/watch?v=VIVIegSt81k• http://www.wikihow.com/Fold-a-Hexaflexagon