Enumeration of Hidato puzzles

Gyora Benedek

solutions

Hidato solution = Hamiltonian path on a grid

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How many are th

ere?

Let’s try something easier:2 by n grids

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Enumerating 2 by n LR Hamiltonian paths

• LR paths: start at BL and end TR

2 by 6• H(2,6) = ?• H(2,n) = ?

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Enumerating 2 by n LR Hamiltonian paths

• Note that for any 2 by n path we can exchange any two values in the same column and get a new path

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Enumerating 2 by n LR Hamiltonian paths

• Note that for any 2 by n path we can exchange any two values in the same column and get a new path – both have the same neighbors

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Ordered LR 2 by n path

Def. Ordered LR 2 by n path:A 2 by n path where 1 is in the left col., 2n is in

the right col. and in every column the number below is smaller than the number above.

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How many ordered LR 2 by n paths?

Def. K(n) = #(ordered LR 2 by n paths)

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2 by 1

9

K(1) = 1

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2 by 2

10

K(2) = 2

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2 by 6

11

K(6) =?

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2 by 6

12

K(6) =?

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2 by 6

13

K(6) =K(5)+?

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2 by 6

14

K(6) =K(5)+?

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2 by 6

15

K(6) =K(5)+?

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2 by 6

16

K(6) =K(5)+K(4)

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How many ordered LR 2 by n paths

K(1) = 1K(2) = 2K(n) = K(n-1) + K(n-2)

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How many ordered LR 2 by n paths?

K(1) = 1K(2) = 2K(n) = K(n-1) + K(n-2)

Fibonacci numbers shifted – as any biologist would recognizeF(1) = 1; F(2) = 1; F(3) = 2; F(n)=F(n-1)+F(n-2)

K(n) = F(n+1)

Interesting to note that this is also the number of ways for 2 by 1 dominoes to cover a 2 by n board.

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Enumerating 2 by n LR Hamiltonian paths

Reminder LR: start at BL and end at TR

19

)1(2),2( 2 nFnH n

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Divide and conquer

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Lower bound by brute force

• 5x5 {?,?} => {A,B}: 83,781,512• 5x4 {C,D} => {E,F}: 202,374• 4x4 {G,H} => {I,J}: 9,868• 4x5 {K,L} => {?,?}: 1,306,425• Symmetries: 16• Total: 3,497,329,213,274,509,542,835,200

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Big numbers

• Rubik’s 3×3×3 cube ≈ 4.3 × 10^19• Sudoku 9×9 ≈ 6.6 × 10^21• Hidato King 9×9 >> 3.49×10^24

× 500

• Still, the lower bound is extremely pessimistic. Can we do better?

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Project proposal: Enumerate all Hamiltonian paths in a 9×9 grid

with diagonals

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A Walk in the shadows

Shadows

2,000,000 774 544(c) Copyright, Gyora Benedek 2014 24

Shadows

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Shadows

• 2,000,000*774*544 = 842,112,000,000 paths(c) Copyright, Gyora Benedek 2014 26

Feasibility

• #shadows – Comb(27,81) ≈ 2.3 × 10^21– Trim non-connected, with holes (? 10^9 ?)– Entry, exit points– If too large try 4 shadows of 20 + 1– Or smaller boards (8×8 even 7×7)

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Challenges

• Efficient algorithms and code• Efficient representation of the shadows

– Shadow: 81bit, entry: 7bit, exit: 7bit, counter: 32?

• Efficient search: hash of the shadow• Parallel/Distributed processing

– Cloud

• Efficient disk access (SSD?) and memory use• Testing: we have data up to 5×5

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Summary

• Numbers are men’s friends – get to know them!– Help the young know them

• Hidato is great for your brain – play and enjoy• Enumeration of Hidato solutions is hard

– Will you pick up the challenge and walk in the shadows?

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Thank You!

Questions?

Ideas?

Visit hidato.com

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