Ariadne thread sequence
Hello Math-Fun,
Our aim is to produce the lexicographically earliest infinite sequence S of distinct nonnegative integers such that S:
— forms a square spiral of square cells;
— square cells hosting a single digit d (from 0 to 8);
— d being the number of square «bricks» inside a cell;
— square brick having 3x3=9 possible positions inside a cell;
— walls design a kind of labyrinth;
— the labyrinth is crossed by a single path;
— this single path visits all the squares of the spiral (therefore d cannot be 9 – it would block the journey).
We are not interested in the labyrinth itself but in
the sequence S [a lot of different labyrinths
can be associated to a sequence — see for instance the three 1’s to the left of the starting zero; the upper one (being the first digit of 12) is inside a cell having only one brick; this brick could occupy any of the 9 possible positions, leaving S as it is].
Question:
Is S really the lexico-first such sequence? (We have a lot of doubts, as «sliding» some bricks from here to there changes everything…)
Best,
É.
Our aim is to produce the lexicographically earliest infinite sequence S of distinct nonnegative integers such that S:
— forms a square spiral of square cells;
— square cells hosting a single digit d (from 0 to 8);
— d being the number of square «bricks» inside a cell;
— square brick having 3x3=9 possible positions inside a cell;
4 is the number of bricks inside the cell; they are marked by X.
The thick blue lines are there to materialize the grid – they are not walls
— walls design a kind of labyrinth;
— the labyrinth is crossed by a single path;
— this single path visits all the squares of the spiral (therefore d cannot be 9 – it would block the journey).
Start on the central zero; move from cell to cell like a chess rook (no diagonal steps); unpassable bricks and walls are marked by X; you will visit every square of the grid.
(The reading direction of the integers is given by the thin yellow arrows)
can be associated to a sequence — see for instance the three 1’s to the left of the starting zero; the upper one (being the first digit of 12) is inside a cell having only one brick; this brick could occupy any of the 9 possible positions, leaving S as it is].
We have so far:
S = 0, 1, 2, 3, 4, 5, 6, 32, 7, 21, 8, 23, 10, 11, 12, 13, 30, 14, 33, 15, 16, 34, 31, 35, 36, 17, 22, 43, 44, 24, 25, 42, 18, 26, 45, 46, 20, 53, 54, 55, 27, 56, 52, 62, 41, 63, 51, 64, 65, 28, 61, 40, 66, 133, 130, 50, 60, 110, 70, 131, 303, 132, 71,...
Is S really the lexico-first such sequence? (We have a lot of doubts, as «sliding» some bricks from here to there changes everything…)
Best,
É.
____________________
Maximilian H. was quick to correct my sequence (after the yellow 25) — and compute more terms (note that Maximilian's square spiral turns counterclockwise) :
> Sequence =
[0, 1, 2, 3, 4, 5, 6, 32, 7, 21, 8, 23, 10, 11, 12, 13, 30, 14, 33, 15, 16, 34, 31, 35, 36, 17, 22, 43, 42, 24, 25, 44, 18, 26, 45, 46, 20, 53, 54, 55, 27, 56, 52, 62, 41, 63, 51, 64, 65, 28, 61, 40, 50, 66, 60, 303, 120, 110, 70, 101, 304, 230, 71, 203, 111, 112, 72, 130, 305, 100, 306, 333, 233, 334, 37, 132, 335, 323, ...]
Grid of size [-18..18] x [-18..18]
(Digits in [.] mean that the corresponding square is also filled with brick.)
Many thanks and bravo, Maximilian!
Hello Eric! I think instead of 44 you can put a 42 if you put the single X in the outside corner of the box with a 1 that makes up the (inner) corner where 42, then 24, go around.
RépondreSupprimerthen I get the 44 two steps later, where you have the 42, and then we get the same up to your 66 (after 40) where I get a 50.
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