1. Design of Geometric Puzzles Marc van Kreveld Center for Geometry, Imaging and Virtual Environments Utrecht University http://www.cs.uu.nl/~marc/composable-art/

2. Two warnings This is not computational geometry This talk involves user participation

3. Overview Classical puzzles: cube dissections New cube dissections Design of a ‘most difficult’ puzzle Some more puzzles The present The future

4. Two famous cube dissections

5. Puzzles and blocks

6. Naef - cubicus

7. New cube dissection 6 pieces: 2 of 3 types 2 types are mirrored

10. Variation: 8 pieces

12. Idea for a puzzle 8 pieces, 1 for each corner of a cube Adjacent pieces must fit in their shared edge Every piece has 1 corner and 3 half- edges

15. Requirements of the puzzle All 8 pieces different No piece should be rotationally symmetric As difficult as possible (unique solution) Does such a puzzle exist? And how do we find it?

16. Analysis of the pieces How many different pieces? –There are 4 possibilities for half-edges  call them types A, B, C, D A C B D A

17. Analysis of the pieces The type of a piece (BDD): Choose the alphabetically smallest type (not DDB or DBD, but BDD)

18. Exercise Which pieces (types) are these two?

19. Assignment (2 minutes) How many different pieces exist? At most 4 x 4 x 4 = 64, but exactly? Hint: –How many with 3 letters the same? –How many with 2 letters the same? –How many with 3 letters different? + AAA, AAB, AAC, AAD, ABA, … the same

20. Answer 3 letters the same: 4 2 letters the same: 4 choices for double letter, another 3 for single letter: 12 3 letters: 4 choices which letter not used, for each choice two mirrored versions (e.g. ABC and ACB): 8 + 24

21. Which types fit? A and D always fit; B and C always fit Nothing else will fit

22. Additional requirement Every type of half-edge - A, B, C and D - appears exactly 6 times in the puzzle

23. The pieces There are 24 different pieces, but 4 of these we don’t want There are ( ) = 124,970 sets of 8 different pieces. Which set fits in one unique way? 20 8

24. A puzzle solver? For all 8 pieces: Place the first piece –2 nd piece: 7 positions, 3 orientations –3 rd piece: 6 positions, 3 orientations –… So: 7! · 3 7 = 11,022,480 ways to fit All 125.970 candidate puzzles: 1,388,501,805,600 ways to test

25. Different approach Take a cube a split all 12 edges in the 4 possible ways

26. Different approach When we know how the 12 edges are split, then we know the 8 pieces; this gives the 4 12 = 16,777,216 solutions of all cube puzzles! –Test every piece for: not AAA, BBB, CCC, DDD –Test every pair for being different –Test whether A, B, C and D appear 6 x each

27. Different approach There are 1,023,360 solutions of puzzles, according to the computer program Final requirement: Unique solution  Find different solutions that use the same 8 pieces; such puzzles are not uniquely solvable

28. Results The 1,023,360 solutions are of 2290 puzzles that fit 3 requirements The minimum is 24 solutions (34 puzzles) The maximum is 1656 solutions (4 puzzles) 24 solutions  1 solution

29. The easiest puzzle With 1656  69 solutions

30. Question (1 minute) All 34 most difficult puzzles use the pieces AAD, ADD, BBC and BCC Is this logical? Explain Note: All 4 easiest puzzles use the pieces AAB, ABB, CCD and CDD, or AAC, ACC, BBD and BDD

31. Results 34 different puzzles are uniquely solvable: AAB, AAD, ABC, ADD, BBC, BCC, BDC, CDD AAC, AAD, ACB, ADD, BBC, BCC, BCD, BDD AAD, ACB, ACD, ADB, ADD, BBC, BCC, BDC + another 31 puzzles

32. … then I made one of these puzzles …

33. Results 34 different puzzles are uniquely solvable: AAB, AAD, ABC, ADD, BBC, BCC, BDC, CDD AAC, AAD, ACB, ADD, BBC, BCC, BCD, BDD AAD, ACB, ACD, ADB, ADD, BBC, BCC, BDC B  C C  B + another 31 puzzles

34. Results There are 5 equivalence classes in the 34 uniquely solvable puzzles But: is there any difference in difficulty?

35. Towards a definition of difficulty How does a puzzler solve such a puzzle? Probably: start with the bottom 4 pieces = 1 loop / lower face of the cube

37. Towards a definition of difficulty After making the bottom loop, it is only a puzzle with 4 pieces Difficulty puzzle = No. of good loops Total no. of loops

38. Assignment (5 minutes) Make a (crude) estimate of the difficulty of the most difficult puzzle Hint: For the total no. of loops, consider a ‘random’ puzzle instead. Recall: There are 6 each of A, B, C and D

39. Answer No. of good loops: 6 Estimate total no. of loops ‘random’ puzzle: –Place a piece, say, with AB on the table –About 5 - 6 half-edges will fit the A, say, 5.25 –About 4 - 5 half-edges will fit the B, say, 4.5 –4th piece of the loop must fit on 2 sides: probability 1/16; the 5 remaining pieces have 5 x 3 = 15 ordered pairs –This gives an estimate of 5.25 x 4.5 x 15/16 = 22 loops –There are 8 x 3 = 24 choices for the first pair (AB) –We over-count by a factor 4 –So estimated 22 x 24/4 = 132 loops in a puzzle Difficulty puzzle  132/6  22

40. Computation of difficulty With a program: the 5 non-equivalent puzzles have 107, 116, 116, 118, and 122 loops Easiest puzzles & maximum: 230 loops Difficulty most difficult puzzle = No. of good loops Total no. of loops = 6 122

41. … I made one of the easiest of the uniquely solvable puzzles !

42. How about 6 types? To be named A, B, C, D, E, and F: E and F have diagonal pins and fit only on each other

43. Question What happens: still puzzles that fit all requirements (now equal usage of A, B, C, D, E and F)? Is the new most difficult puzzle more difficult or easier?

44. More puzzles

45. A personal puzzle

49. Hinged puzzle

50. Gate puzzle

52. The present 36 squares  12 pieces needed

53. The future Ideas for new puzzles 24 different pieces

54. More future Based on the composable painting

55. The end some puzzle jugs