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/...
- 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 ...
- 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:
...
- 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.
- 44.1k
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, ...
- 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 ...
- 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
- 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 ...
- 1,695
10
votes
Multi-colored polyominoes inside a 7x7 grid
Here is a solution in which the red and green do not touch.
- 44.1k
10
votes
Accepted
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.
- 44.1k
9
votes
Factorials whose product is a perfect cube
How small can the largest of these be?
Proof positive:
- 9,990
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 ...
- 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:
- 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
- 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 ...
- 111k
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 ...
- 7,964
6
votes
Accepted
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 ...
- 44.1k
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...
- 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 ...
- 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 ...
- 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
- 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...
- 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.
- 44.1k
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 ...
- 9,990
1
vote
1
vote
Only top scored, non community-wiki answers of a minimum length are eligible