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 




























