A392177 - OEIS

A392177

Consider the square spiral with its cells numbered starting at 0, as in A308884 and A274641. Two players, Black and Red, take turns. When it is Black's turn, he places a knight at the smallest unoccupied cell not attacked by an existing Red knight, and when it is Red's turn, she places a knight at the smallest unoccupied cell not attacked by an existing Black knight. Sequence lists squares occupied by a Black knight.

0, 2, 5, 9, 11, 15, 20, 21, 30, 31, 36, 40, 42, 47, 48, 50, 56, 61, 65, 67, 69, 70, 71, 75, 76, 81, 83, 85, 87, 89, 93, 99, 109, 110, 111, 112, 116, 117, 126, 132, 133, 138, 144, 148, 150, 152, 154, 156, 161, 162, 176, 180, 182, 187, 193, 197, 199, 201, 203, 205, 207, 208, 209, 211, 213, 214, 219, 229, 231, 233, 235, 237, 238, 239, 243

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

There is a subtle point about the definition. Because the players take turns in placing a knight, the spiral is not filled in consecutively. To obtain the sequence, one must first fill in the whole spiral, and then return to the start, and record the Black knights that are encountered as one walks along the spiral.

To view the knights in the order in which they are placed on the spiral, see A395355. - N. J. A. Sloane, Apr 25 2026

This is a problem, not a game. The (unsolved) problem is to explain the extraordinary patterns formed by the Black and Red knights (see the links).

By considering the evolution of the spiral as the number of terms grows, it seems clear that, apart from narrow bands along the x- and y-axes, the upper half-plane will be occupied by red knights, and the lower half-plane by black nights.

The reason for this is unclear.

REFERENCES

Michael S. Branicky, Jonas Karlsson, and N. J. A. Sloane, Proof of convergence of the Red and Black Knights structure, MS in preparation, May 14 2026.

Jonas Karlsson, Letter to N. J. A. Sloane, Jan 23 2026, and emails from Feb 05-06 2026.

LINKS

Jonas Karlsson, Table of n, a(n) for n = 1..45763 [More than the usual number of terms are shown, in view of the complexity of the sequence.]

Michael S. Branicky, The first 10^6 cells of the spiral.

Michael S. Branicky, The first 4*10^6 cells of the spiral

Michael S. Branicky, The first 16*10^6 cells of the spiral

Michael S. Branicky, The first 64*10^6 cells of the spiral

Michael S. Branicky, The first 256*10^6 cells of the spiral

Michael S. Branicky, Python program for OEIS A392177 and A392178

Brady Haran, Jonas Karlsson, and N. J. A. Sloane, Red & Black Knights (extraordinary result); Amazing Chessboard Patterns (extra), YouTube Numberphile videos, May 2026

Jonas Karlsson, The first 8 shells (288 cells) [The shells are outlined by black lines, and the cells are both numbered and colored.]

Jonas Karlsson, The first 1000 cells of the spiral.

Jonas Karlsson, The first 100000 cells of the spiral.

Jonas Karlsson, Table of a(n) for n = 1..10^7 [95 MB file]

Jonas Karlsson, Python program for these sequence and similar sequences involving other chess pieces of two or more colors

Rémy Sigrist, Illustration of variants where players use kings, queens, rooks, bishops, knights, pawns (heading north, east, west or south), wazirs or ferzes

Rémy Sigrist, C# program for generating various sequences (including A392177, A392178, A392179, A392180, A395486, A395506) and drawing the spiral, with knights or other pieces.

Rémy Sigrist, The first 16*10^6 cells of the spiral (solid colors correspond to pieces with record indexes, light colors to other pieces)

N. J. A. Sloane, The first five layers of the spiral. [Black circle = Black knight, red circle = Red knight, black slash = attacked by a Black knight, red slash = attacked by a Red knight.] Note that these links are deliberately arranged in logical order, not alphabetical order (see Comments)

PROG

(Python) # see linked program

(C#) // See Links section.

CROSSREFS

Cf. A274641, A308884, A308885 (if all the knights are of the same color), A392178-A392180.