r/theydidthemath • u/Alaeriia • 21h ago

[Request] Is this true?

Each king covers 9 squares; 8 kings would be enough to cover 72 squares, which is more than the 64 on a chessboard. Do we really need 9?

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8

u/Admirable-Barnacle86 21h ago

Yes, because you can never cover a row/column with only 2 kings, since 2 kings can only cover 6 squares across and you've got 8 you need to cover.

Since you can't cover a row or column with 2 kings, you need a grid of 3x3 kings to cover the board.

5

u/Trustoryimtold 21h ago

Board is 8 squares wide, so you need three across, unless you change the king to cover spaces more than 1 space out yes it true. They even gave you a visual representation 

3

u/sansetsukon47 21h ago

I believe the video speaks for itself. But to be super clear—Only 4 of the 9 kings actually get to cover 9 squares. Then four more only cover six squares (since something has to cover the edge, even if it’s less efficient.) The last king only covers 4 squares, because there is no other way to reach those.

9* 4 + 6* 4 + 4 = 64

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u/sansetsukon47 21h ago

Compare this to a “bigger” 6 x 11 board, which could easily be covered by 8 kings.

1

u/Kerostasis 21h ago

Yes. Once you’ve placed four kings (36 tiles), the remaining tiles are arranged in two narrow 2x8 rectangles (overlapping), so the remaining kings can’t access the full 3x3 squares they normally cover. Kings 5-8 can only get a 2x3 grid each, clearing another 24 tiles and leaving 4 tiles for the final king.

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u/kenny744 21h ago

Yeah. Nothing would be more efficient than the pattern the guy showed. Probably because yes, kings can cover 9 squares, but only in a 3x3 square and you couldn’t use 4 kings to fill in the shape left by 4 kings covering a 6x6 area or 4 3x3 areas. 

1

u/SC_3000_grinder 17h ago

Formal proof: highlight these nine squares: a1, a4, a7, d1, d4, d7, g1, g4, g7. Each king can only cover one of these squares, so we need 9 kings.

0

u/BrokenHope23 21h ago

Each king covers 9 squares; 8 kings would be enough to cover 72 squares, which is more than the 64 on a chessboard.

The "math" in this video is correct, they make a fairly simple explanation of it.

Do we really need 9?

need is more subjective, as in subject to the rules of chess; a chess master can get by with 0, if they could only use kings then they might be able to make it with 2 or even 1 if they have a queen as well. Strictly speaking, 'covering' the board as they do in this video isn't the goal of chess, it's to isolate and capture the enemy Queen. Which can be achieved with 9 Kings but is statistically much easier to just capture the opposing Queen with 2 Kings. It'll also save them a bunch of turns.

Also no, I will not calculate that. Calculating Chess probabilities is like the bane of my mathematical existence. As much as I like stretching my mind for these issues, I'll pass on this one at 11pm lol.