"Recreational Math: What's That?" presentated at fall MichMATYC meeting October, 2010

1. Recreational Math: What's That? Presented byNancy J. Sattler nsattler@terra.edu Terra Community College

2. When is the impossible possible?

3. How can Mathematics be Recreational? Think Big! How can you cut a 3x5 card so that you can walk through it?

4. Why Recreational Mathematics? Historically many important mathematical concepts arose from problems mathematical in origin. Ideal for introducing topics covered in liberal arts mathematics courses. Something a bit different to do on a day when students need something a bit different to do!

5. Types of Recreational Mathematics Puzzles Games Others

6. Puzzles Puzzles require mathematics in order to solve them. They have specific rules, but mathematical puzzles don't usually involve competition between two or more players. Instead, in order to solve the puzzle, the solver must find a solution that satisfies the given conditions. Logic puzzles fall into this category.

7. Games Rules, strategies, and outcomes can be studied and explained by mathematics although players may not use mathematics in order to play the game.

8. Others Curiosities and pastimes of non-trivial mathematical interest Optical Illusions Videos Brain Teasers Cat's cradle and other string figures Origami (many mathematical results, some deep)

9. Puzzles

10. Tower of Hanoi http://www.mazeworks.com/hanoi/index.htm Game consists of three rods, and a number of disks of different sizes which can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape. The objective of the puzzle is to move the entire stack to another rod, obeying the following rules: Only one disk may be moved at a time. Each move consists of taking the upper disk from one of the rods and sliding it onto another rod, on top of the other disks that may already be present on that rod. No disk may be placed on top of a smaller disk.

11. Tangrams The aim of the puzzle is to seamlessly arrange all the geometric pieces to form problem figures. It is said that the Pythagorean theorem was discovered in the Orient with help of Tangram pieces... The 7 polygons or 'tans' that form the Tangram are: • 5 right triangles: 2 small (hypotenuse of n/2 and sides of n/2√2); 1 medium (hypotenuse of n/√2 and sides of n/2); 2 large (hypotenuse of n and sides of n/√2). The large triangle is 4 times the size of the small triangle, but curiously its perimeter is only 2 times as big! • 1 square (side of n/2√2). • 1 parallelogram/rhomboid (sides of n/2 and n/2√2). Of these 7 pieces, the parallelogram (or rhomboid) is the only piece that may need to be flipped when forming certain shapes; in fact, it has no reflection symmetry but only rotational symmetry, and so its mirror image can only be obtained by flipping it over.

12. An Even Hundred Can you insert addition or subtraction signs among the following digits to get an expression equal to 100? 123456789

13. An Even Hundred Solution There are many are possible solutions 123 – 4 – 5 – 6 – 7 + 8 – 9 = 100 123 – 45 – 67 + 89 = 100

14. Make a Thousand Try making 1000 with Eight 8’s Seven 7’s Six 6’s

15. Make a Thousand Solution 888 + 88 + 8 + 8 + 8 = 1000 777/.7 – 77/.7 = 1000 666/.666 = 1000

16. 7 pairs; 7 sums Using the numbers 1-6, create a three by two grid where the seven pairs of adjacent squares have seven different sums

17. Solution

18. Congruency How many different ways can you cut a grid 4x4 in half along the grid lines? Six different polygonal shapes are possible from the halves.

19. The Simple Solution . . .

20. Another solution . . .

21. Another solution . . .

22. Another solution . . .

23. Another solution . . .

24. Another solution . . .

25. Logic Puzzles Printable logic puzzles in pdf format found at http://www.logic-puzzles.org/ Monthly Logic Puzzles that to not need to be printed found at http://www.puzzlersparadise.com/page1034.html

26. Games

27. Strategy Games A strategy is a rule or decision making formula that tells the player which choice to make at each turn. Winning strategy – Strategy enables the player to win no matter what moves his or her opponent makes. Drawing strategy – Strategy does not guarantee a win for a particular player but does guarantee that he or she does not lose. The game can end in a draw.

28. Strategy Games What are the rules? What constitutes a win or a loss? What is a move? Is it advantageous to go first? What should be the opening move?

29. Types of Games Chance – The player’s fortune depends on roll of the dice or the deal of the cards. Chance-Free – Each player at each turn is free to choose any legal moves. Decisions are not made by chance. Perfect Information- Each player is aware at all times of all aspects of the structure of the game. Finite- The game must necessarily end or terminate in a finite number of moves. Bounded – If there is a number “n” such that the game cannot last for more than “n” moves.

30. Nim Game of Nim, said to have originated in China http://education.jlab.org/nim/index.html The "classical" Nim game is a game by two players. It consists of 16 matches. Two players alternately pick a certain number of matches and the one, who takes the last match, loses.

31. 9 Cell Tic Tac Toe Directions: Each player has three markers. Alternate turns putting own marker on any cell. When all six markers have been placed on the board, players alternate moving their own marker into an empty adjacent cell either horizontally or vertically. The first player to get three in a row is a winner.

32. Split, Strategy Game Game for two players Play begins with a pile of 32 matchsticks or checkers. Each player must separate any existing pile into two unequal piles. For example, 4 chips must be split into a pile of 3 and 1, not two and two. A pile of two can never be separated. The first player who cannot make a move is the loser.

33. Other Games Jeopardy http://www.jmu.edu/madison/teacher/jeopardy/ jeopardy.htm Other Power Point Games (Who Wants to be a Millionaire, etc.) http://teach.fcps.net/trt10/PowerPoint.htm

34. Other

35. Optical Illusions . . . Escher

36. YouTube as a resource . . . Pa & Pa Kettle due mathematics? http://www.youtube.com/watch?v=OettgNpe4n4 Where does the extra man come from? http://www.youtube.com/watch?v=4XbrUHMDTg0&feature=related

37. Brain Teasers Puzzles, games (speed games, memory games, problem solving games), illusions, and logic games. You will need to create a free account found at http://www.brainbashers.com/ Make your own puzzles (Pencil puzzle, Letter puzzles & Others) found at http://www.creatievepuzzels.com/spel/speel1/sites.htm Collection of brain teasers, and puzzles found at http://www.internet4classrooms.com/brain_teasers.htm

38. String Games, Cats Cradlehttp://www.ifyoulovetoread.com/book/chten_cats1105.htm

39. Origami Wolfram Mathworld Origami found at http://mathworld.wolfram.com/Origami.html Using Origami to Teach Mathematics – Math on the Street found at http://math.serenevy.net/?page=Origami-TeachingLinks

40. Opportunity to Share What have are you doing for FUN in your classroom?