Design of Geometric Puzzles Marc van Kreveld Center for Geometry, Imaging and Virtual Environments...
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Transcript of Design of Geometric Puzzles Marc van Kreveld Center for Geometry, Imaging and Virtual Environments...
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