Q: Henry Goldman's Square of Digits

Jaap Scherphuis introduced The Arithmachinist by Henry Goldman in his post Goldman's Transformation Puzzle.

In the book I found a curious figure on p.52. I wonder what it means. Could you let me know?

I have posted this question on Jaap Scherphuis's post as an answer. Some users commented as follows:

I have pondered that too, but have not figured out what it means, apart from it being a latin square. – Jaap Scherphuis 9 hours ago

Also, a sudoku square. – P.-S. Park 8 hours ago

Nice. Also the main diagonal is in order 1 to 9, though the other diagonal and the broken diagonals don't seem to have anything special. – Jaap Scherphuis 8 hours ago

The other diagonal still has the 1-9 digits, just not in order – El-Guest 8 hours ago

it seems a solved sudoku problem to me with diagonal is 1 to 9. – Oray 7 hours ago

To keep everything in one package, I'm also adding the notes from earlier.

The numbers 1-9 can be found on

• every row, column and diagonal

• in every 3x3 square whose top row is on the 1st, 4th or 7th row

• in every 3x3 square whose left column is on the 1st, 4th or 7th column

• If the 3x3 square fits both of the above categories (so it's one of the 9 "sudoku squares"), then each row and column will add up to 15. The diagonals also add up to 15 in the middlemost sudoku square.

• The numbers 1-9 can also be found in the corners and midpoints of the square

• and in here

• and also if you pick the same point in all the sudoku squares. Any point will do, so you can move this pattern one step in any of the eight directions. If you do, you'll always get a square with the sum of 15 on each row and column. The pattern consisting of the sudoku square centre points is the only one that gives sum 15 on both diagonals too, though:

Maybe there's more? That would seem to be quite enough for me to print it in a book, though :-)

EDIT: Well, turns out there's more, as lined out in the other answer. There's also one more really significant feature that user @Magma already pointed out in the comments:

The grid has an antisymmetry through the center and the number 5: If you make a copy of the grid, rotate it 180 degrees, and add it to the original, you get a 10 in every spot.

This is probably a necessary condition for getting all the other magical properties to work, but that doesn't make it any less neat.

• I agree. Thank you. – P.-S. Park Apr 12 at 5:22
• As I know, the earliest sudoku-like puzzles were printed on French newspapers, Le Sièle (1892) and La France (1895). en.wikipedia.org/wiki/Sudoku#History_of_sudoku Goldman's book was published in 1898, so it's not so far from the French sources. Can we say he is one of the inventors of sudoku puzzle? – P.-S. Park Apr 12 at 5:31

The diagonals form a 10's-complement:

as do the anti-diagonals, and the main diagonal and anti-diagonal are self-complementing.

And another $$123456789$$ block:

A skew grid with wrapround (adjacent grids work too):

• Looks like the entire square becomes its own complement when rotated by 180°. – Magma Apr 12 at 12:21