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