Martin Gardner (1914-2010)

1. 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, dmancl@acm.org

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

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

8. Albrecht Dürer – Melencolia I (1514) 16 2 3 13 5 11 10 8 9 7 6 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 A C A A flexible hexagon made from a long strip of paper folded into triangles You can “flex” the hexagon to show different faces B B z B C A C B A C A C • 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”] C B B • Example: a tri-hexaflexagon • 3 faces • Each face has 6 triangles A

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

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

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

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

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

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

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

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 (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- )

27. 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