A379241 -id:A379241 - cp's OEIS frontend

A379393 Maximum number of connections for a 3 X n rectangle.

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

Offset: 2

Rodolfo Kurchan, Dec 22 2024

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.

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

			For n = 2 with the board
  +---+---+
  | 1 | 1 |
  +---+---+
  | 2 | 2 |
  +---+---+
  | 3 | 3 |
  +---+---+
There is only 1 solution being the squares with these letters:
  +---+---+
  | A | B |
  +---+---+
  | C | D |
  +---+---+
  | E | F |
  +---+---+
Solution:
1) AB - CD - EF
There is one solution so a(2) = 1.
.
For n = 3 with the board
  +---+---+---+
  | 1 | 3 | 3 |
  +---+---+---+
  | 1 | 2 | 2 |
  +---+---+---+
  | 1 | 2 | 2 |
  +---+---+---+
the maximum number of solutions is 6 being the squares with this letters:
  +---+---+---+
  | A | B | C |
  +---+---+---+
  | D | E | F |
  +---+---+---+
  | G | H | I |
  +---+---+---+
Solutions:
  1) ADG - BC - HEFI
  2) ADG - BC - FEHI
  3) ADG - BC - EFIH
  4) ADG - BC - EHIF
  5) ADG - BEFC - HI
  6) ADEHG - BC - FI
There are six solutions so a(3) = 6.

Rodolfo Kurchan and Claudio Meller, Number Connections, Puzzle Fun, Problems (2024).

Cf. A379241.

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

Original entry on oeis.org

1, 31, 800, 6466, 60778, 441492, 3216584, 18693320

Offset: 2

Rodolfo Kurchan, Dec 26 2024

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.

Terms a(5)-a(9) from Giorgio Vecchi.

			For n = 2 with the board
+---+---+
| 1 | 1 |
+---+---+
| 2 | 2 |
+---+---+
| 3 | 3 |
+---+---+
| 4 | 4 |
+---+---+
There is only 1 solution being the squares with these letters:
+---+---+
| A | B |
+---+---+
| C | D |
+---+---+
| E | F |
+---+---+
| G | H |
+---+---+
Solution:
1) AB - CD - EF - GH
There is one solution so a(2) = 1.
.
For n = 3 with the board
+---+---+---+
| 1 | 2 | 2 |
+---+---+---+
| 1 | 2 | 2 |
+---+---+---+
| 3 | 4 | 4 |
+---+---+---+
| 3 | 4 | 4 |
+---+---+---+
the maximum number of solutions is 31 being the squares with these letters:
+---+---+---+
| A | B | C |
+---+---+---+
| D | E | F |
+---+---+---+
| G | H | I |
+---+---+---+
| J | K | L |
+---+---+---+
Solutions:
  1)	AD - GJ - BC - HEFILK
  2)	AD - GJ - BC - IFEHKL
  3)	AD - GJ - BC - KHEFIL
  4)	AD - GJ - KL - EHIFCB
  5)	AD - GJ - KL - FIHEBC
  6)	AD - GJ - KL - BEHIFC
  7)	AD - GJ - BCFE - HILK
  8)	AD - GJ - BCFE - ILKH
  9)	AD - GJ - BCFE - LKHI
  10)	AD - GJ - BCFE - KHIL
  11)	AD - GJ - CFEB - HILK
  12)	AD - GJ - CFEB - ILKH
  13)	AD - GJ - CFEB - LKHI
  14)	AD - GJ - CFEB - KHIL
  15)	AD - GJ - FEBC - HILK
  16)	AD - GJ - FEBC - ILKH
  17)	AD - GJ - FEBC - LKHI
  18)	AD - GJ - FEBC - KHIL
  19)	AD - GJ - EBCF - HILK
  20)	AD - GJ - EBCF - ILKH
  21)	AD - GJ - EBCF - LKHI
  22)	AD - GJ - EBCF - KHIL
  23)	ABED - GJ - CF - HILK
  24)	ABED - GJ - CF - ILKH
  25)	ABED - GJ - CF - LKHI
  26)	ABED - GJ - CF - KHIL
  27)	GHKJ - AD - IL - BCFE
  28)	GHKJ - AD - IL - CFEB
  29)	GHKJ - AD - IL - FEBC
  30)	GHKJ - AD - IL - EBCF
  31)	ABED - GHKJ - CF - IL
There are 31 solutions so a(3) = 31.

Rodolfo Kurchan and Claudio Meller, Number Connections, Puzzle Fun, Problems (2024).

Cf. A379241, A379393.

cp's OEIS Frontend

A379393 Maximum number of connections for a 3 X n rectangle.

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