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Problem Space of the Tower of London

Summary by Distances / Lengths

explanations at bottom

N 0 1 2 3 4 5 6 7 8
number of tasks 36 108 168 216 174 168 186 174 66
generic tasks 6 18 28 36 29 28 31 29 11
                   
S_average 1.00 1.00 1.00 1.17 1.24 1.50 1.61 1.86 4.36
S_stdev 0.00 0.00 0.00 0.37 0.43 0.50 0.71 0.90 1.98
S_min 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00
S_max 1.00 1.00 1.00 2.00 2.00 2.00 3.00 3.00 8.00
                   
In_average 1.00 2.58 9.03 26.90 87.68 283.24 873.39 2862.22 6224.68
In_stdev 0.00 0.50 1.78 6.14 20.22 62.33 201.97 658.12 1062.70
In_min 1.00 2.00 6.00 15.00 49.00 183.00 586.00 1855.00 4254.00
In_max 1.00 3.00 11.00 37.00 119.00 379.00 1214.00 3896.00 7999.00
                   
Vn_average 1.00 3.00 4.67 6.00 4.83 4.67 5.17 4.83 1.83
Vn_stdev 0.00 0.83 0.96 1.17 0.70 0.76 1.36 0.70 0.91
Vn_min 1.00 2.00 3.00 4.00 4.00 4.00 4.00 4.00 1.00
Vn_max 1.00 4.00 6.00 8.00 6.00 6.00 7.00 6.00 3.00
                   
Vtn_average 1.00 4.00 8.67 14.67 19.50 24.17 29.33 34.17 36.00
Vtn_stdev 0.00 0.83 1.51 1.26 1.40 1.48 0.96 0.91 0.00
Vtn_min 1.00 3.00 6.00 12.00 17.00 21.00 28.00 33.00 36.00
Vtn_max 1.00 5.00 11.00 16.00 21.00 25.00 31.00 35.00 36.00
                   
En_average 3.00 5.67 8.33 7.83 7.17 7.33 7.50 6.67 0.50
En_stdev 0.83 0.96 1.39 1.59 0.70 1.82 1.52 0.96 0.51
En_min 2.00 4.00 7.00 5.00 6.00 4.00 6.00 5.00 0.00
En_max 4.00 7.00 11.00 10.00 8.00 9.00 10.00 8.00 1.00
                   
Etn_average 3.00 8.67 17.00 24.83 32.00 39.33 46.83 53.50 54.00
Etn_stdev 0.83 1.72 1.94 2.22 2.27 1.72 1.36 0.51 0.00
Etn_min 2.00 6.00 13.00 21.00 28.00 36.00 45.00 53.00 54.00
Etn_max 4.00 11.00 19.00 27.00 35.00 41.00 49.00 54.00 54.00
                   
Un_average 1.00 7.00 17.33 31.67 44.33 56.17 68.67 81.00 89.50
Un_stdev 0.00 1.66 3.19 3.14 3.59 3.63 2.32 2.19 0.51
Un_min 1.00 5.00 12.00 25.00 38.00 49.00 64.00 78.00 89.00
Un_max 1.00 9.00 22.00 34.00 48.00 60.00 71.00 84.00 90.00
                   
Pn_average 1.00 3.00 6.67 12.33 22.00 39.67 72.00 120.33 191.00
Pn_stdev 0.00 0.83 1.26 0.96 1.17 4.52 4.22 4.71 6.31
Pn_min 1.00 2.00 5.00 11.00 21.00 34.00 65.00 113.00 180.00
Pn_max 1.00 4.00 8.00 13.00 24.00 45.00 77.00 126.00 200.00
                   
Bn_average 1.00 3.00 9.67 30.67 97.67 310.67 988.67 3145.67 10009.67
Bn_stdev 0.00 0.83 2.08 7.04 22.01 70.49 223.64 712.59 2265.91
Bn_min 1.00 2.00 7.00 22.00 70.00 223.00 709.00 2257.00 7180.00
Bn_max 1.00 4.00 12.00 39.00 123.00 393.00 1248.00 3975.00 12642.00
                   

Explanations

number of tasks

generic tasks groups of tasks with same physical configuration (6 per group, permutations of colors)
moves authorized moves.
S number of solutions for each task.
I(task) average number of steps of random broad-first exploration algorithm for each task.
In average number of steps of random broad-first exploration algorithm  averaged by distance.
Vn number of nodes (vertices) at distance N.
Vtn number of nodes (vertices) at distance <= N.
En number of arcs (edges) at distance N. Distance counted from closest node
Etn number of arcs (edges) at distance <= N.
Un number of arcs + nodes at distance <= N (arcs at d <= N-1 + nodes at d <= N)
Pn number of direct paths (no loop) of length N.
Bn number of paths (with or without loop) of length N.





Copyright: (c) 2009 E.J. Fimbel, S. Lauzon, C. Rainville. This is open-access content distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.