1. Martin Gardner ( )
Scientific American – Mathematical Games column
(297 monthly columns)
Books:
Mathematical Games
Word puzzles
Annotated Alice
Books on pseudoscience and skepticism
Presentation by Dennis Mancl,

2. Magic squares 8 1 6 3 5 7 4 9 2 8 + 1 + 6 = 15 An array of numbers
No duplicates
The sum of each row is the same
The sum of each column is the same 3 5 7
= 15 4 9 2
= 15 8 3
+ 4 15 1 5
+ 9 15 6 7
+ 2 15

3. 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 ? ? ? ? ? ? ? ?

4. Magic squares 1 2 3 5 6 7 9 10 11 4 8 12 13 14 15 16 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
Start with 2 squares – numbers in reverse order

5. Magic squares 1 2 3 4 16 15 14 13 5 6 7 8 12 11 10 9 9 10 11 12 8 7 6 5 13 14 15 16 4 3 2 1
Choose 8 cells from one square, 8 cells from the other

6. Magic squares 1 2 3 5 6 7 9 10 4 8 13 14 11 12 15 16 7 4 3 8 16 15 14 12 11 10 6 13 9 5 2 1 16 2 3 5 11 10 9 7 6 13 8 12 4 14 15 1

7. Magic squares 16 2 3 5 11 10 9 7 6 13 8 12 4 14 15 1
= 34
= 34
= 34
= 34 16 5 9
+ 4 34 2 11 7
+ 14 34 3 10 6
+ 15 34 13 8 12
+ 1 34

8. Albrecht Dürer – Melencolia I (1514)16 2 3 5 11 10 9 7 6 13 8 12 4 14 15 1

9. Puzzles

10. If there are 4 red balls and 6 white balls
Spend 6 cents === guaranteed to have at least two red balls
Spend 8 cents === guaranteed to have at least two white balls

11. 2 cents is enough some of the time
3 cents is always enough

12. Puzzles
old
square tiles
new
rectangular tiles

15. 21 red squares
19 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…

16. Hexaflexagons B A C C A B z
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 ( ) when he was a grad student at Princeton
contributions by Bryant Tuckerman ( ), John Tukey ( ), Richard Feynman ( ) [the “Flexagon Committee”] B A C C A B z
Example: a tri- hexaflexagon
3 faces
Each face has 6 triangles

17. 7 Steps to fold a tri-hexaflexagonz B B
Step 1.
Start with a strip of 10 equilateral triangles. Fold both ways on all of the lines C A C B A C A C

18. C A B z y Step 2. Fold 3 triangles on the left towards the back x
fold back

19. C A B x Step 3. Fold over one triangle towards the front y
fold to the front C A B x
Step 3.
Fold over one triangle towards the front y

20. A C B Step 4. Fold the 4 right triangles towards the front y x
fold to the front
Step 4.
Fold the 4 right triangles towards the front y x
Caution: Don’t fold towards the back… if you folded it the wrong way, your flexagon will look like this:

21. A B y Step 5. Re-open the one triangle that was folded over in step 3
re-open one triangle x

22. y A B
Step 6.
Put glue on the top 2 triangles
glue x

23. A B x Step 7. Fold down the top triangle – done! y Front view
Back view

24. Flexing a tri-hexaflexagon

25. Hexaflexagons
There 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

26. Puzzles and mathematical games
It didn’t start with Martin Gardner…
W. W. Rouse Ball ( )
Sam Loyd ( )
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- )

27. Tri-hexaflexagon template
See also: