A379393 - OEIS

%I #13 Dec 23 2024 01:58:15

%S 1,6,72,277,1910,8657,27442,97132,295752,967914,2922814

%N Maximum number of connections for a 3 X n rectangle.

%C In a 3 X n board (with n > 1) with numbers 1, 2 and 3, at least 2 of each, find the arrangement with more solutions connecting a pair of numbers 1 and a pair of number 2 and a pair of number 3, covering the entire board and without passing through the same square twice.

%C Terms a(5) and a(7)-a(12) from Giorgio Vecchi.

%H Rodolfo Kurchan and Claudio Meller, <a href="https://www.puzzlefun.online/problems">Number Connections</a>, Puzzle Fun, Problems (2024).

%e For n = 2 with the board

%e   +---+---+

%e   | 1 | 1 |

%e   +---+---+

%e   | 2 | 2 |

%e   +---+---+

%e   | 3 | 3 |

%e   +---+---+

%e There is only 1 solution being the squares with these letters:

%e   +---+---+

%e   | A | B |

%e   +---+---+

%e   | C | D |

%e   +---+---+

%e   | E | F |

%e   +---+---+

%e Solution:

%e 1) AB - CD - EF

%e There is one solution so a(2) = 1.

%e .

%e For n = 3 with the board

%e   +---+---+---+

%e   | 1 | 3 | 3 |

%e   +---+---+---+

%e   | 1 | 2 | 2 |

%e   +---+---+---+

%e   | 1 | 2 | 2 |

%e   +---+---+---+

%e the maximum number of solutions is 6 being the squares with this letters:

%e   +---+---+---+

%e   | A | B | C |

%e   +---+---+---+

%e   | D | E | F |

%e   +---+---+---+

%e   | G | H | I |

%e   +---+---+---+

%e Solutions:

%e   1) ADG - BC - HEFI

%e   2) ADG - BC - FEHI

%e   3) ADG - BC - EFIH

%e   4) ADG - BC - EHIF

%e   5) ADG - BEFC - HI

%e   6) ADEHG - BC - FI

%e There are six solutions so a(3) = 6.

%Y Cf. A379241.

%K nonn,more

%O 2,2

%A _Rodolfo Kurchan_, Dec 22 2024