Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly...

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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 sentation by Dennis Mancl, [email protected]

Transcript of Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly...

Page 1: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.

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]

Page 2: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.

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

Page 3: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.

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

Page 4: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.

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

Page 5: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.

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

Page 6: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.

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

Page 7: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.

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

Page 8: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.

16 2 3

5 11 10

9 7 6

13

8

12

4 14 15 1

Albrecht Dürer – Melencolia I (1514)

Page 9: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.

Puzzles

Page 10: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.

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

Page 11: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.

2 cents is enough some of the time

3 cents is always enough

Page 12: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.

Puzzlesold

square tiles

newrectangular tiles

Page 13: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.
Page 14: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.
Page 15: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.

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…

Page 16: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.

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

Page 17: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.

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

Page 18: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.

CA

B

C

z A C CABB Step 2.

Fold 3 triangles on the left towards the back

x

y

fold back

Page 19: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.

CA

B CA

BBB

y

x

Step 3.Fold over one triangle towards the front

fold to the front

Page 20: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.

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

Page 21: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.

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

Page 22: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.

Step 6.Put glue on the top 2 triangles

A

B

B

BB

B

y

xglue

Page 23: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.

A

B

B

BB

B

B

y

xStep 7.Fold down the top triangle – done!

Front view Back view

Page 24: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.

Flexing a tri-hexaflexagon

Page 25: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.

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

Page 26: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.

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- )

Page 27: Martin Gardner (1914-2010) Scientific American – Mathematical Games column 1956-1981 (297 monthly columns) Books: Mathematical Games Word puzzles Annotated.

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