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
29 votes
Accepted

Social distancing in a 5x5 room

I'll get things started with:
RobPratt's user avatar
  • 13.7k
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
  • 77.4k
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
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
  • 13.7k
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
  • 146k
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
  • 1,137
19 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
18 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
17 votes

Smallest rectangle to put the 24 ABCD words combination

Pretty sure that the following is minimal.
Ed Pegg's user avatar
  • 271
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
15 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
  • 26.8k
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

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
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
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
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
  • 5,259
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
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
  • 13.7k
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
  • 29.8k
11 votes
Accepted

Multi-colored polyominoes inside a 7x7 grid

I think this would work as a possibility
hexomino's user avatar
  • 136k
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
  • 77.4k
10 votes

The farmer and the olive trees

I believe the answer is as shown in the flower kind figure below (This is the previous answer): The idea behind this is Here is the best answer: as you will see below:
Oray's user avatar
  • 30.3k
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
10 votes

Packing pentominoes in a circle

The radius of the smallest pizza that can accommodate all 12 cheeses is The cheeses can be arranged like this:
caPNCApn's user avatar
  • 19.2k

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