a1
a2 b2
a3 b3 c3
a4 b4 c4 d4
a5 b5 c5 d5 e5
Square lattice representation and board notation.
(0,0) hole in bold.
Normal representation with board notation.
Green holes are starting/finishing locations along the center line. Red holes are starting/finishing locations off the center line. We can vacate any green or red hole, and (potentially) finish in any other green or red hole. All such problems constitute every possible single vacancy to single survivor problem (solvable by considering the fundamental classes). Problems are grouped by the color of the finishing hole.
15 Hole Triangular Board      
Single Vacancy to Single Survivor Problems
# Vacate Finish at Length of Shortest Solution Number of Solutions Longest Sweep Longest Finishing Sweep Shortest Longest Sweep Number of Final Moves #(Longest, Second longest, Final) [Comment]
1 (0,0) a1 (0,0) a1 10 (S) 15 3 3 3 4 15(3,2,3)
x (1,-2) b3 (0,0) a1 Impossible 0          
2 (0,-3) a4 (0,0) a1 10 15 3 3 3 4 15(3,2,3) [Same as #1]
3 (2,-4) c5 (0,0) a1 9 (S) 6 3 3 3 4 2(3,3,3), 4(3,2,3)
x (0,0) a1 (1,-2) b3 Impossible 0          
x (1,-2) b3 (1,-2) b3 Impossible 0          
x (0,-3) a4 (1,-2) b3 Impossible 0          
4 (2,-4) c5 (1,-2) b3 10 (S) 5 2 2 2 1 5(2,2,2)
5 (0,0) a1 (2,-4) c5 10 (S) 16 4 4 2 4 1(4,1,4), 1(3,2,1), 10(2,2,2), 4(2,2,1)
6 (1,-2) b3 (2,-4) c5 11 (S) 5 2 1 2 1 5(2,2,1)
7 (0,-3) a4 (2,-4) c5 10 16 4 4 2 7 1(4,1,4), 1(3,2,1), 10(2,2,2), 4(2,2,1) [Same as #5]
8 (2,-4) c5 (2,-4) c5 9 (S) 15 5 5 2 8 3(5,1,5), 3(4,2,4), 1(3,3,3), 1(3,2,3), 2(3,2,2), 5(2,2,2)
9 (0,0) a1 (0,-3) a4 11 16 2 1 2 1 16(2,2,1)
x (1,-2) b3 (0,-3) a4 Impossible 0          
10 (0,-3) a4 (0,-3) a4 11 8 2 1 2 1 8(2,2,1)
11 (3,-3) d4 (0,-3) a4 11 8 2 1 2 1 8(2,2,1)
12 (2,-4) c5 (0,-3) a4 10 8 3 2 2 2 2(3,2,1), 1(2,2,2), 5(2,2,1)
          Total: 133          
Column Definitions:
Length of Shortest Solution This is the length of the shortest solution to this problem, minimizing total moves
Number of Solutions This is the number of unique solution sequences, irregardless of move order and symmetry
Longest Sweep This is the longest sweep possible in any minimal length solution [link to solution]
Longest Finishing Sweep This is the longest sweep in the final move of any minimal length solution [link]
Shortest Longest Sweep There is no minimal length solution where all sweeps are shorter than this number [link]
Number of Final Moves This is the number of different finishing moves (up to symmetry)
#(Longest, Second Longest, Eg. 12(8,7,2) indicates there are 12 solutions with different move sequences, where
, Final) the longest sweep is 8, the second longest sweep is 7, and the final sweep is 2
(S) Problem is symmetric, multiple solutions counted as one
Solution differences can be very subtle.
Download a text file with all 133 solutions

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