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Using only kings, and as few rooks as possible, set up a position where 30 of black's kings are checkmated.

Checkmate:

  1. The king is in check
  2. The king has no moves out of check

Black only has kings, exactly 30 of them. White has rooks and kings, and the goal is to use as few rooks as possible.

  • White must have at least one king, but can have more than one.
  • White kings and black kings can not stand next to each other.
  • All black kings must be checkmated, stalemate is not enough.

Examples.

Left: NOT valid. The king on g8 is checkmated, but the king on h8 is merely stalemated.
Right: A valid solution for 2 kings with 1 rook.

enter image description here enter image description here

Left: NOT valid. Either king can capture the rook.
Right: A valid solution for 3 kings with 2 rooks.

enter image description here enter image description here

Online board editor

SE - stop firing the good guys
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    In that case, White King on c4 should be on b4 , then it becomes a Valid Solution – Prem Jun 22 '22 at 19:50
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    Even though this is a generalization of chess (as known by common rules) it shows wonderful imagination to even think of this problem! – JosephDoggie Jun 22 '22 at 20:49
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    Is it better to have 3 kings and 2 rooks, or 2 kings and 3 rooks? – Joe DiNottra Jun 23 '22 at 03:11
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    @d-b Then what is the point of any question on this site, let alone its existence? – Rewan Demontay Jun 24 '22 at 05:46
  • what about kb2 to a1 on the right board? – sidgate Jun 24 '22 at 11:20
  • @sidgate then ka1 is threatened by rb1. – RobPratt Jun 24 '22 at 12:05
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    As @Prem said, the last board has more white kings than necessary. Moving the one from c4 to b4 would still get the job done with 2 total, you didn't need to add another, unless you're trying to demonstrate that sub-optimal solutions can be valid. Not sure if it's intentional, since Prem's comment pointing that out came 10 minutes before your last edit. – Peter Cordes Jun 25 '22 at 11:18
  • @d-b Literally EVERY question in this site is arbitrary. Why do we waste our time on puzzles hmmm? Your question is useless for this site. I also see you have no questions or answers for a year old account, so I believe you have some puzzle soul searching to do. – Rewan Demontay Jun 27 '22 at 15:29

6 Answers6

30

Here's a solution with 11 rooks and 1 king:

enter image description here enter image description here

kkkRkkRk/2kRkkRk/R1kkkkkk/2k2Rk1/k1k1R1k1/k4Rk1/kRkk3R/kR1kkk1K b - - 0 1 https://lichess.org/editor/kkkRkkRk/2kRkkRk/R1kkkkkk/2k2Rk1/k1k1R1k1/k4Rk1/kRkk3R/kR1kkk1K_b_-_-_0_1

RobPratt
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  • Impressive. Two neat rook positions where distance takes care of rook not being attacked. Is this the unique solution or are there more? I presume this is the optimal and that there is no solution with 10, right? – Zizy Archer Jun 23 '22 at 07:56
  • There's at least one other solution, if the G5 king moves to H5 – user7868 Jun 23 '22 at 08:34
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    @user7868 nope, h6 doesn't get checkmated then. Similar issue with e1 king if you move c2 to e2. And various other moves. I don't see a move that would leave all kings checkmated with this solution. – Zizy Archer Jun 23 '22 at 08:51
  • That's a good point – user7868 Jun 23 '22 at 08:53
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    I have encoded this problem as a SAT problem, and if my encoding isn't wrong, then 10 rooks is in fact impossible, and there are tons of 11 rook solutions. – Mateon1 Jun 23 '22 at 09:14
  • what stops black to take a6 rook? What threatens h4, d4 and f2? Is this a valid solution? – akostadinov Jun 23 '22 at 15:04
  • @akostadinov There is no black king threatening a6. The h2 rook threatens h4. The e4 rook threatens d4. The f3 and h2 rooks both threaten f2. The black cells are kings, and the gray cells are unoccupied. I added a lichess board for clarity. – RobPratt Jun 23 '22 at 15:19
  • Ah, got it, I read it the other way around. – akostadinov Jun 23 '22 at 15:51
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    I left my SAT encoding version run in the background today to figure out what's the maximum number of kings you can pack into an 11-rook solution. 5 white kings is impossible, but here's one with 4 white kings: KK1kR2k/R2k1K1k/kkkk3R/kRRkk1K1/kkkkk3/1kkRRkkk/RRkkkkRk/kkk3Rk w - - 0 1 – Mateon1 Jun 23 '22 at 18:05
16

To kick things off, here is a solution with 1 white king and 14 white rooks.

enter image description here

I have a feeling this could be tweaked to remove a rook by adding some more kings, but haven't quite managed yet.

Jaap Scherphuis
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14

This just gives a lower bound on what the best solution could be.

For a black king to be in mate, all neighboring fields must be either threatened or blocked and the field where the king is must be threatened. The blocking of neighboring fields can be done in a variety of ways but threatening the king itself can only be done by a rook. A rook can move in any of the 4 directions but it can only threaten one king in every direction. Hence you need at the very least one rook per 4 kings which gives a lower bound of 8 rooks to mate all 30 kings.

At the time of writing the best solution has 11 rooks, I don't know where in the range of 8 to 11 the ideal solution is.

quarague
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12

Solution with 12 rooks and 3 kings: https://lichess.org/editor/K1k2Rkk/2kkkRk1/1kRRkkk1/2kkkkRR/K1kkRkk1/3kRkkk/1KRk1kRk/1R1kkkRk_w_-_-_0_1 12 rook solution

There are some trivial modifications to add 2 checkmated black kings (move Rb1 to c1, remove Kb2, put black kings on a1 and a2), but I don't see a way to remove a rook.

Zizy Archer
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5

Here is a solution with 1 king, 13 rooks.

It feels like it might be possible to remove one rook with a few adjustments, but I haven't achieved this after hours of shuffling pieces around.

Mate 30 kings with 13 rooks

For a slightly different challenge, here is a way to mate 41 black kings:

mate 41 kings with king and rooks

Evargalo
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    I guess this possible alternate challenge could be defined as maximizing kings instead of minimizing rooks. Extending @Jaap Scherphuis's pattern, I found one with 42 kings – SE - stop firing the good guys Jun 22 '22 at 08:29
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    @SE-stopfiringthegoodguys oups, thank you. I have edited a correct version in. I had shuffled to much trying to save one rook and posted a bad version ! – Evargalo Jun 23 '22 at 09:36
4

Another solution with 12 rooks and 3 kings:

enter image description here

FEN: k1RR1k1K/kkkkkk2/RRkkRRk1/kkkkkk2/k1RRk2K/kkkkk1K1/RRkkR3/kkk1Rk2 w - - 0 1

JLee
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