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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

Henry Goldman's Square of Digits

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2 Answers 2

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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
    enter image description here

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

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

  • 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.

enter image description here

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

enter image description here

  • and in here

enter image description 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:

enter image description here

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.

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    $\begingroup$ 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? $\endgroup$
    – P.-S. Park
    Apr 12, 2020 at 5:31
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The diagonals form a 10's-complement:

diagonals

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

And another $123456789$ block:

3x3block

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

skew grid

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    $\begingroup$ Looks like the entire square becomes its own complement when rotated by 180°. $\endgroup$
    – Magma
    Apr 12, 2020 at 12:21

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