A353234 Maximum length of 2 finite snakes in the Snake Number Problem with n-periodic instructions in an infinite square grid (see Comments).

32, 44, 138, 226, 326, 310, 409, 1138, 5265, 10499

Offset: 4

Author

Rodolfo Kurchan, May 01 2022

Keywords

• nonn
• more

Comments

We start with 2 infinite snakes, and 4 possible directions: up, right, down, left.

If on its turn one of the snakes cannot execute an order because that square is occupied, it goes to the next order, and so on.

The snakes can be blocked and finish there or can continue infinitely.

Which are the longest finite snakes using n instructions for 2 snakes that start in the same square (with n >= 4 because with 3 or fewer instructions are infinite)?

Examples

			4 INSTRUCTIONS
URDL: 32
   --  30  31  --  --  --
   25  26  32   7   8  --
   24  20   2   3   9  13
   17  18   1   4  15  14
   16  12   6   5  19  21
   --  11  10  29  23  22
   --  --  --  28  27  --
   1) Both snakes start in square 1.
   2) First snake go U = 2, second snake can't go Up.
   3) First snake goes R = 3.
   4) Second snake goes R = 4.
   5) First snake cannot go D, so second snake D = 5.
   6) First snake cannot go L, so second snake goes L = 6
   7) First snake can go U = 7, and second cannot go U.
   8) First snake can go R = 8, second snake cannot go R.
   9) First snake can go D = 9, second snake can go D = 10.
  10) First snake cannot go L, second snake can go L = 11.
  11) First snake cannot go U, second snake can go U = 12.
  12) First snake can go R = 13, second snake cannot go R.
  13) First snake can go D = 14, second snake cannot go D.
  14) First snake can go L = 15, second snake can go L = 16.
And so on.
                      | Instructions that give
   n | Maximum length |   the maximal length
  --------------------------------------------
   4          32              URDL
   5          44              URDLU
   6         138              UURDLL
   7         226              UURDUUL
   8         326              UURDLUUL
   9         310              UUUURDUUL
  10         409              URUDLLDURR
  11        1138              UDDRDDRDUDL
  12        5265              URUDRDURDDDL
  13       10499              UUDRUDDLUULDD
Computer solutions found by Giorgio Vecchi.
		

Links

• Ariel Futoransky, Snake Program, Snake Program to try the snakes, April 2022 (see bottom animation and label for different options).
• Rodolfo Kurchan, Puzzle Fun, Snake Number Problem, March 2022.

Crossrefs

Cf. A353259, A353060, A352388.