30 votes
Accepted

Checkmate 30 kings with rooks

Here's a solution with 11 rooks and 1 king: kkkRkkRk/2kRkkRk/R1kkkkkk/2k2Rk1/k1k1R1k1/k4Rk1/kRkk3R/kR1kkk1K b - - 0 1 https://lichess.org/editor/kkkRkkRk/2kRkkRk/R1kkkkkk/2k2Rk1/k1k1R1k1/k4Rk1/kRkk3R/...
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  • 7,964
20 votes
Accepted

Checkmate N Kings with M Knights

50 Kings,14 Knights: This is optimal but not unique, see bottom of this answer. Reasoning: I think the problem is equivalent to covering every square on the board with as few knights as possible and ...
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  • 12.2k
16 votes
Accepted

The Game of Golden Squares

I've achieved tiles, and can prove that this is the optimal solution. Reasoning: Golly 4.0+ pastable RLE of this solution: ...
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  • 4,454
15 votes

Checkmate 30 kings with rooks

To kick things off, here is a solution with 1 white king and 14 white rooks. I have a feeling this could be tweaked to remove a rook by adding some more kings, but haven't quite managed yet.
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15 votes

The universal ticket

Very unlikely to be optimal, but got to 120 on my first go: Approach: mess around with the problem until it becomes clear that connectivity of the squares will be the main problem. invent glue, ...
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  • 68.9k
14 votes

Checkmate 30 kings with rooks

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 ...
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  • 419
12 votes

The universal ticket

The previously best-known solution has score of 165, with the following grid: From a clever brute-force search, one can learn that However, you can do better! The ticket achieves a score of
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  • 759
12 votes

Checkmate 30 kings with rooks

Solution with 12 rooks and 3 kings: https://lichess.org/editor/K1k2Rkk/2kkkRk1/1kRRkkk1/2kkkkRR/K1kkRkk1/3kRkkk/1KRk1kRk/1R1kkkRk_w_-_-_0_1 There are some trivial modifications to add 2 checkmated ...
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  • 1,695
10 votes

Multi-colored polyominoes inside a 7x7 grid

Here is a solution in which the red and green do not touch.
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10 votes
Accepted

Multi-colored polyominoes inside a 7x7 grid

I think this would work as a possibility
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  • 123k
10 votes

Checkmate N Kings with M Knights

This seems like it could be optimal [Edit: it is not - see loopy walt's answer]: 16 knights, and 48 kings.
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9 votes

Factorials whose product is a perfect cube

How small can the largest of these be? Proof positive:
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8 votes

The Game of Golden Squares

Update: Up to by introducing a small asymmetry Update ends. Original answer: I can do using this setup. This is essentially a big Over time (1275 turns) this will fill up to a First few groups ...
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  • 12.2k
7 votes
Accepted

Four pipes on a 8x8 grid

Assuming you pick all 8 start/end points and then all the pipes get laid: I think I've got but not sure if I can prove it's the shortest possible route, with these starting points:
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  • 186
7 votes
Accepted

The longest path of edges on a 3x3 grid

I think the highest number of edges you can visit is Reasoning Example
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  • 123k
6 votes

Factorials whose product is a perfect cube

Possibly-partial answer (if "how few can they be?" means how few absolutely, rather than how few conditional on making the largest as small as possible, I have not answered that): [There is ...
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6 votes
Accepted

Fitting pentominoes inside a 10x10 grid

Bonus: I used integer linear programming as follows. Introduce binary decision variable $x_p$ for each possible placement of a pentomino in the grid. Let binary decision variable $y_{ij}$ indicate ...
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  • 7,964
6 votes
Accepted

Tetromino in a Pentomino Lair

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  • 7,964
5 votes

Fitting pentominoes inside a 10x10 grid

Rob Pratt beat me to it, but I'll post anyway because my computer found a couple of other solutions to the bonus question. I used my own program to solve it. I ran it overnight, and after 15 hours it ...
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5 votes

Fitting pentominoes inside a 10x10 grid

As a quick baseline solution: Bonus: (Edit) I also found a nice symmetric solution for the first question: I wonder what Hexomino's solution is...
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  • 10.6k
5 votes

Checkmate 30 kings with rooks

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. For a ...
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  • 5,088
4 votes

The universal ticket

Update: Honing in the parameters allowed for a score of 153. This is much closer than I expected to get to the 165 mentioned on the website. original: I decided to go for a brute force approach and ...
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  • 41
4 votes

Checkmate 30 kings with rooks

Another solution with 12 rooks and 3 kings: FEN: k1RR1k1K/kkkkkk2/RRkkRRk1/kkkkkk2/k1RRk2K/kkkkk1K1/RRkkR3/kkk1Rk2 w - - 0 1
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  • 15k
2 votes

Multi-colored polyominoes inside a 7x7 grid

An obvious upper bound for the maximum number of distinct shapes is $2+4+4=10$, and...
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  • 7,964
2 votes

The Game of Golden Squares

While I cannot beat loopywalt's answer, my construction got just over halfway there. I'm posting it in case it sparks ideas for better constructions.
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2 votes

Checkmate N Kings with M Knights Perfectly

I'll stand by my initial comment: All kings should be attacked by at least two knights. If any king is attacked by only one knight, the attacking knight must not be threatened by another king. This ...
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1 vote

Checkmate N Kings with M Knights

The answer I found is: Explanation:
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  • 1,022
1 vote

The Game of Golden Squares

I can get up to How do:
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  • 5,426

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