cp's OEIS Frontend

A072174 Maximum path length of a crippled knight on an n X n board.

%I A072174 #54 Aug 18 2025 00:10:14
%S A072174 1,1,5,9,16,27,38,51,66
%N A072174 Maximum path length of a crippled knight on an n X n board.
%C A072174 A crippled knight moves one square vertically and two horizontally (or vice versa) and can't land on or pass over any square on which it is previously rested. The initial placement counts as the first move.
%C A072174 a(9) >= 63. - _Jud McCranie_, May 25 2021
%C A072174 a(9) >= 66. - _Giovanni Acerbi_, May 20 2024
%C A072174 a(10) >= 79. - _Jud McCranie_, Aug 17 2025
%D A072174 A crippled knight is defined by Dario Uri in the Journal of Recreational Mathematics, problem 2465, Vol. 29 #4.
%D A072174 Vol. 30 #4 has an example for 8 X 8 with 48 moves found by Henry Ibstedt.
%e A072174 For 3 X 3, the longest path is:
%e A072174   1 . 3
%e A072174   4 . .
%e A072174   . 2 5
%e A072174 The knight cannot move from #5 because it would have to cross over 2 or 3, so a(3)=5.
%e A072174 For 8 X 8, a(8)=51 has a unique solution:
%e A072174    .  1  8 19 22 25 28 31
%e A072174    7 20 23 26 29 32  .  .
%e A072174    2  9 18 21 24 27 30 33
%e A072174    .  6  3 10 17 34 37 40
%e A072174    4 11 16 35 38 41  .  .
%e A072174   49 46  5 12 15 36 39 42
%e A072174    .  . 50 47 44 13  .  .
%e A072174   51 48 45 14  .  . 43  .
%e A072174 Best known solution for 9 X 9 (66 moves):
%e A072174    . 56 53 50 47 44 27  .  .
%e A072174    .  .  . 55 52 49 46 43 28
%e A072174   57 54 51 48 45 42 29 26  .
%e A072174   64 61 58 41 38 35 32  . 30
%e A072174    .  . 65 62 59 40 37 34 25
%e A072174   66 63 60 39 36 33 24 31  .
%e A072174    .  2  5  8 11 14 17 20 23
%e A072174    4  7 10 13 16 19 22  .  .
%e A072174    1  .  3  6  9 12 15 18 21
%Y A072174 Cf. A003192, A157416, A323131.
%K A072174 nonn,walk,hard,more
%O A072174 1,3
%A A072174 _Jud McCranie_, Jun 29 2002
%E A072174 a(8) by _Jud McCranie_, Mar 18 2021
%E A072174 a(9) by _Jud McCranie_, Aug 12 2025