OFFSET
1,2
COMMENTS
This sequence describes the unique solution for Pulsar puzzles.
A Pulsar puzzle is a special type of Latin square. An n x n grid is filled such that each integer 1 through n occurs exactly once in each row and exactly once in each column, while a fixed spiral of cells is circled. A spiral of circled cells is formed by successively wrapping around the grid of the puzzle: starting with the n cells of the first row, then the remaining n-1 cells of the n-th column, followed by n-2 cells of the n-th row, the n-3 cells of the second column, the n-4 cells of the third row, and so on.
This spiral of circled cells also leaves behind another spiral of uncircled cells, whose order (starting from the center) is called the "Pulsar Sequence". The restriction is each digit in a circled cell must equal the number of circled cells that contain that digit.
This rule ensures the Pulsar Sequence is unique. By knowing the Pulsar Sequence, one can immediately fill in the values for the circled cells, thus completing the puzzle.
Additionally, the Pulsar Sequence always has the same order regardless of the size of the puzzle. This means that the Pulsar Sequence solution for a puzzle of size n-1 will become initial terms for the Pulsar Sequence solution for a puzzle of size n.
Size 5:
+---+---+---+---+---+
|(5)|(2)|(3)|(4)|(1)|
+---+---+---+---+---+
| 4 | 1 | 2 | 3 |(5)|
+---+---+---+---+---+
| 2 |(4)|(5)| 1 |(3)|
+---+---+---+---+---+
| 3 |(5)| 1 | 2 |(4)|
+---+---+---+---+---+
| 1 |(3)|(4)|(5)|(2)|
+---+---+---+---+---+
Size 6:
+---+---+---+---+---+---+
|(6)|(2)|(4)|(3)|(5)|(1)|
+---+---+---+---+---+---+
| 5 | 1 | 3 | 2 | 4 |(6)|
+---+---+---+---+---+---+
| 2 |(4)|(6)|(5)| 1 |(3)|
+---+---+---+---+---+---+
| 3 |(5)| 1 |(6)| 2 |(4)|
+---+---+---+---+---+---+
| 4 |(6)| 2 | 1 | 3 |(5)|
+---+---+---+---+---+---+
| 1 |(3)|(5)|(4)|(6)|(2)|
+---+---+---+---+---+---+
"Pulsar" is the handle (a nickname) of an Internet user who allegedly created this puzzle. - Michal Paulovic, Oct 31 2025
LINKS
Ryan Liu and Vadim Ponomarenko, A Closed Form for the Pulsar Sequence, arXiv preprint arXiv:2510.17893 [math.HO], 2025.
Vadim Ponomarenko, The Pulsar Sequence, arXiv preprint arXiv:2507.14701 [math.HO], 2025.
YouTube, The Sudoku Discovery Of The Decade: The Sequel!!!, published Jul 17 2025.
EXAMPLE
The Pulsar Sequence is divided into multiple pieces for calculation. It is delimited by 1, so "1" appears at the end of each piece.
Since there is 1 element in the first piece, 2 elements in the second piece, 3 elements in the third piece, and so on, we can rearrange the Pulsar Sequence into a triangle shape as shown below.
1;
2, 1;
3, 2, 1;
4, 2, 3, 1;
5, 2, 3, 4, 1;
6, 2, 4, 3, 5, 1;
7, 2, 5, 4, 3, 6, 1;
8, 2, 6, 4, 5, 3, 7, 1;
...
Each element can be represented by T(n,k), where n is the n-th row in the triangle and k is the k-th column in the triangle.
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Ryan Z. Liu, Oct 08 2025
STATUS
approved