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36 votes
Accepted

Is this more than a packing puzzle?

They are:
phenomist's user avatar
  • 13.6k
35 votes

Social distancing in a 5x5 room

The problem is equivalent to Now,
Glorfindel's user avatar
  • 28.1k
29 votes
Accepted

Social distancing in a 5x5 room

I'll get things started with:
RobPratt's user avatar
  • 14.2k
28 votes
Accepted

12 piece cube packing puzzle

What a great puzzle! For me the key was to notice that you will quickly run out of corners. Since there is only one other way (plus a zillion symmetries) to place the hexacube, this means that we ...
Bass's user avatar
  • 77.7k
25 votes
Accepted

A COVID-19 puzzle: How large a class do you need to fit 30 pupils?

The solution that springs to (my) mind is to put them
Glorfindel's user avatar
  • 28.1k
24 votes

A COVID-19 puzzle: How large a class do you need to fit 30 pupils?

As in my answer to My Mother's Dish Collection, I used a nonlinear optimization solver, with variables $x_i$, $y_i$, $w$, $h$. The problem is to minimize $w\cdot h$ subject to: \begin{align} 0 \le ...
RobPratt's user avatar
  • 14.2k
21 votes
Accepted

Near-fill with 3x1 long triominos, how to do a different void square than the center square?

The trick to this puzzle is to: (And here are those tilings: the center was already given, and the rest are obtainable from these by rotation.)
Deusovi's user avatar
  • 147k
21 votes

How many distinct pentominoes are possible to place on an 8 x 8 board?

With integer programming, I managed to place like this. Here is my formulation. I happened to solve a similar model to solve a puzzle called One puzzle a day. Let $B$ be the set of cells in the 8x8 ...
xd y's user avatar
  • 525
19 votes

How many distinct pentominoes are possible to place on an 8 x 8 board?

I solved this completely by hand. Here is a clean proof of its optimality. No computer is needed. Mere pencil and paper suffice. Expand each pentomino by adding little right-angled isosceles ...
user21820's user avatar
  • 1,236
18 votes
Accepted

Packing pentominoes in a circle

UPDATE 2 A minor improvement. New best radius Arrangement /UPDATE 2 UPDATE New best radius: using arrangement /UPDATE I get a radius of about using the following scheme which is obviously ...
loopy walt's user avatar
  • 21.3k
17 votes
Accepted

Can you pack these tetracubes to form a rectangular block with at least one odd side length?

It is because Thanks to mousetail in the comments, here is a picture:
Jaap Scherphuis's user avatar
16 votes

Smallest rectangle to put the 24 ABCD words combination

Pretty sure that the following is minimal.
Ed Pegg's user avatar
  • 261
15 votes
Accepted

How many ways are there to solve the Mensa cube puzzle?

I used a computer to search for all solutions, and the number of solutions is Here is a picture of the solutions, with the top layer on the left, bottom layer on the right.
Jaap Scherphuis's user avatar
14 votes
Accepted

Can you stop the falling piano with 23 nets?

One straightforward way to arrange the nets for question 1 is as follows: Number the poles $0$ to $22$. Here are some thoughts on question 2:
Jaap Scherphuis's user avatar
14 votes
Accepted

Can you pack these pentacubes to form a rectangular block with at least one odd side length other the side whose length must be a multiple of 5

Yes, for instance we can make a 2x7x15 block. Put together two pentacubes to make a P pentomino two layers deep: Then, arrange 21 P pentominoes in 2D to make the 7x15 rectangle below. Image from ...
xnor's user avatar
  • 27.4k
14 votes
Accepted

PSE Advent Calendar 2023 (Day 8): A Quilt for Santa

I've been able to do it in You have to arrange the reindeers like this I found it by generating and evaluating all possible arrangements. It feels like a really good solution, but I cant proof if ...
Antikeks's user avatar
  • 206
13 votes

Packing pentominoes in a circle

I can get a radius of: Method: start with and then EDIT: I found a second solution with the slightly worse radius Method: start with and then I found both of these with the help of
Ravi Fernando's user avatar
13 votes

Can you pack these pentacubes to form a rectangular block with at least one odd side length other the side whose length must be a multiple of 5

5x5x5 block:
Magma's user avatar
  • 5,324
12 votes

Ten tetrominoes inside an 8x8 grid

what about the following arrangement of I found it manually by searching arrangement of same pieces, then had to change a bit strategy
franck vivien's user avatar
12 votes
Accepted

Cutting a square into integer triangles

An optimal 26-triangle solution: Previous manual construction of a solution with 25 triangles: GeoGebra construction to confirm validity:
Daniel Mathias's user avatar
12 votes
Accepted

Packing 25 three-dimensional N pentominoes into a 5x5x5 cube

You can solve the problem via integer linear programming as follows. For each of the $960$ placements $p$ of a piece, let $C_p \subset [5] \times [5] \times [5]$ be the set of cells covered by $p$, ...
RobPratt's user avatar
  • 14.2k
11 votes

Is this more than a packing puzzle?

A small addition to phenomist's excellent answer: Finally, here's a photo of all the pieces in the box. The tricky packing isn't visible.
Don Kirkby's user avatar
  • 2,602
11 votes
Accepted

Eighteen is not seventeen

Not a perfect circle, but it is clear that it works, and I didn't use a computer:
RobPratt's user avatar
  • 14.2k
11 votes

Dividing a piece of land

Alice can maximize her area by Why? Increasing the number of points will only decrease Alice's area because There are a few other things Alice can try: Reference:
I'm Nobody's user avatar
  • 1,334
11 votes
Accepted

Put three pieces of cake into a round box

An "improved" version of AxiomaticSystem's solution: PS: I realize the layout is actually the same as AxiomaticSystem, an optimal $\theta$ will put the more acute angle at the bottom as I ...
Florian F's user avatar
  • 30.6k
11 votes
Accepted

Multi-colored polyominoes inside a 7x7 grid

I think this would work as a possibility
hexomino's user avatar
  • 137k
11 votes

Multi-colored polyominoes inside a 7x7 grid

Here is a solution in which the red and green do not touch.
Jaap Scherphuis's user avatar
10 votes
Accepted

Scheduling based problem

Sure. Method: With some fiddling, it's also possible to get all the columns to add up to 25: And here's a magic square (with duplicates, unavoidably) followed by a row of fives: And finally:
Bass's user avatar
  • 77.7k
10 votes
Accepted

Ziggy - Make a square from 8 polyomino pieces

Here's one solution. The second solution can be achieved by: .
Riley's user avatar
  • 14.5k
10 votes
Accepted

The crossword packing puzzle

Note from /r/puzzles: The T in the top-right piece should be an I. Solved: Nice puzzle, thanks for posting!
Agargara's user avatar
  • 338

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