cp's OEIS Frontend

A379589 Maximum number of connections for a 4 X n rectangle.

%I A379589 #7 Dec 26 2024 19:58:56
%S A379589 1,31,800,6466,60778,441492,3216584,18693320
%N A379589 Maximum number of connections for a 4 X n rectangle.
%C A379589 In a 4 X n board (with n > 1) with numbers 1, 2 3 and 4, at least 2 of each, find the arrangement with more solutions connecting a pair of numbers 1, a pair of number 2, a pair of number 3 and a pair of number 4, covering the entire board and without passing through the same square twice.
%C A379589 Terms a(5)-a(9) from Giorgio Vecchi.
%H A379589 Rodolfo Kurchan and Claudio Meller, Number Connections, Puzzle Fun, Problems (2024).
%e A379589 For n = 2 with the board
%e A379589 +---+---+
%e A379589 | 1 | 1 |
%e A379589 +---+---+
%e A379589 | 2 | 2 |
%e A379589 +---+---+
%e A379589 | 3 | 3 |
%e A379589 +---+---+
%e A379589 | 4 | 4 |
%e A379589 +---+---+
%e A379589 There is only 1 solution being the squares with these letters:
%e A379589 +---+---+
%e A379589 | A | B |
%e A379589 +---+---+
%e A379589 | C | D |
%e A379589 +---+---+
%e A379589 | E | F |
%e A379589 +---+---+
%e A379589 | G | H |
%e A379589 +---+---+
%e A379589 Solution:
%e A379589 1) AB - CD - EF - GH
%e A379589 There is one solution so a(2) = 1.
%e A379589 .
%e A379589 For n = 3 with the board
%e A379589 +---+---+---+
%e A379589 | 1 | 2 | 2 |
%e A379589 +---+---+---+
%e A379589 | 1 | 2 | 2 |
%e A379589 +---+---+---+
%e A379589 | 3 | 4 | 4 |
%e A379589 +---+---+---+
%e A379589 | 3 | 4 | 4 |
%e A379589 +---+---+---+
%e A379589 the maximum number of solutions is 31 being the squares with these letters:
%e A379589 +---+---+---+
%e A379589 | A | B | C |
%e A379589 +---+---+---+
%e A379589 | D | E | F |
%e A379589 +---+---+---+
%e A379589 | G | H | I |
%e A379589 +---+---+---+
%e A379589 | J | K | L |
%e A379589 +---+---+---+
%e A379589 Solutions:
%e A379589   1)	AD - GJ - BC - HEFILK
%e A379589   2)	AD - GJ - BC - IFEHKL
%e A379589   3)	AD - GJ - BC - KHEFIL
%e A379589   4)	AD - GJ - KL - EHIFCB
%e A379589   5)	AD - GJ - KL - FIHEBC
%e A379589   6)	AD - GJ - KL - BEHIFC
%e A379589   7)	AD - GJ - BCFE - HILK
%e A379589   8)	AD - GJ - BCFE - ILKH
%e A379589   9)	AD - GJ - BCFE - LKHI
%e A379589   10)	AD - GJ - BCFE - KHIL
%e A379589   11)	AD - GJ - CFEB - HILK
%e A379589   12)	AD - GJ - CFEB - ILKH
%e A379589   13)	AD - GJ - CFEB - LKHI
%e A379589   14)	AD - GJ - CFEB - KHIL
%e A379589   15)	AD - GJ - FEBC - HILK
%e A379589   16)	AD - GJ - FEBC - ILKH
%e A379589   17)	AD - GJ - FEBC - LKHI
%e A379589   18)	AD - GJ - FEBC - KHIL
%e A379589   19)	AD - GJ - EBCF - HILK
%e A379589   20)	AD - GJ - EBCF - ILKH
%e A379589   21)	AD - GJ - EBCF - LKHI
%e A379589   22)	AD - GJ - EBCF - KHIL
%e A379589   23)	ABED - GJ - CF - HILK
%e A379589   24)	ABED - GJ - CF - ILKH
%e A379589   25)	ABED - GJ - CF - LKHI
%e A379589   26)	ABED - GJ - CF - KHIL
%e A379589   27)	GHKJ - AD - IL - BCFE
%e A379589   28)	GHKJ - AD - IL - CFEB
%e A379589   29)	GHKJ - AD - IL - FEBC
%e A379589   30)	GHKJ - AD - IL - EBCF
%e A379589   31)	ABED - GHKJ - CF - IL
%e A379589 There are 31 solutions so a(3) = 31.
%Y A379589 Cf. A379241, A379393.
%K A379589 nonn,more
%O A379589 2,2
%A A379589 _Rodolfo Kurchan_, Dec 26 2024