Design of Geometric Puzzles
Marc van KreveldCenter for Geometry, Imaging and
Virtual Environments
Utrecht University
http://www.cs.uu.nl/~marc/composable-art/
Two warnings
• This is not computational geometry• This talk involves user participation
Overview
• Classical puzzles: cube dissections • New cube dissections• Design of a ‘most difficult’ puzzle• Some more puzzles• The present• The future
Two famous cube dissections
Puzzles and blocks
Naef - cubicus
New cube dissection
• 6 pieces: 2 of 3 types• 2 types are
mirrored
Variation: 8 pieces
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
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?
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
Analysis of the pieces
• The type of a piece (BDD):
• Choose the alphabetically smallest type(not DDB or DBD, but BDD)
Exercise
• Which pieces (types) are these two?
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
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
Which types fit?
• A and D always fit; B and C always fit
• Nothing else will fit
Additional requirement
• Every type of half-edge - A, B, C and D - appears exactly 6 times in the puzzle
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?
208
A puzzle solver?
• For all 8 pieces: Place the first piece– 2nd piece: 7 positions, 3 orientations– 3rd piece: 6 positions, 3 orientations– …
• So: 7! · 37 = 11,022,480 ways to fit• All 125.970 candidate puzzles:
1,388,501,805,600 ways to test
Different approach
• Take a cube a split all 12 edges in the 4 possible ways
Different approach
• When we know how the 12 edges are split, then we know the 8 pieces; this gives the 412 = 16,777,216 solutions ofall 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
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
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
The easiest puzzle
• With 1656 69 solutions
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
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
… then I made one of these puzzles …
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 CC B
+ another 31 puzzles
Results
• There are 5 equivalence classes in the 34 uniquely solvable puzzles
But: is there any difference in difficulty?
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
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
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
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
Computation of difficulty
• With a program: the 5 non-equivalent puzzles have 107, 116, 116, 118, and 122 loops
• Easiest puzzles & maximum: 230 loops
Difficultymost difficult puzzle =
No. of good loops
Total no. of loops=
6
122
… I made one of the easiest of the uniquely solvable puzzles !
How about 6 types?
• To be named A, B, C, D, E, and F:
E and F havediagonal pinsand fit only oneach other
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?
More puzzles
A personal puzzle
Hinged puzzle
Gate puzzle
The present
36 squares 12 pieces needed
The future
• Ideas for new puzzles
24 different pieces
More future
• Based on the composable painting
The end
some puzzle jugs