40
votes
Accepted
22
votes
Accepted
What is the least number of colours Peter could use to color the 3x3 square?
The minimum is
because
- 28.2k
21
votes
Accepted
Fitting 10 pieces of pizza in a box
Thanks to 2012rcampion for crunching the numbers to find a minimum side length of
For this arrangement of slices:
- 16.7k
18
votes
Fitting 10 pieces of pizza in a box
Here is a simple but rather effective packing:
The sides of the square box are
Explanation:
- 7,735
16
votes
What is the least number of colours Peter could use to color the 3x3 square?
Basically a beginner here.
Start with a diagonal. All three cells must have unique colours:
Then, the two unshaded corners must be given unique colours because both of them have a diagonal with the ...
- 524
15
votes
Use all eight of the given polygons to tile a parallelogram
The total area of all tiles together is
With that in mind it doesn't take much time to find:
- 7,735
9
votes
Accepted
Cutting a 27×27 square into incomparable rectangles
The widths and heights of the rectangles are:
They tile the square:
- 16.7k
8
votes
Packing cubes into spheres
(partial answer)
Simplification
Let's start with an intuitive simplification. The cubes must be stacked in independent "layers", where each layer "sits" on $y=k$ for some integer $...
- 2,143
7
votes
What is the least number of colours Peter could use to color the 3x3 square?
As described in many answers, five colors is the minimum. Here we bring in the theory of pandiagonal Latin squares to show some hidden features of the solution and allow a generalization to $n×n$ ...
- 1,080
7
votes
Accepted
Packing cubes into spheres
Some empirical evidence that n=6
The best axis-aligned configuration I found uses a radius of $1.630998544...$.
Here's configuration with $6$ cubes that fits in a sphere with radius ~$1.627$. (The ...
- 2,143
7
votes
Fitting 10 pieces of pizza in a box
Here's the best setup I could come up with, using unit slices:
Achieving a length of
Here, the highlighted slice is shifted by $b \approx 0.01626$ in both directions, and the rest of the pieces fit.
...
- 1,086
7
votes
Accepted
7
votes
Accepted
Discrete cops and robbers
The cop wins up to and including diameter
which is the diameter of the circumcircle of a
If the diameter is larger the robber wins by
Conversely, if the diameter is equal or smaller
A recipe for ...
- 7,735
6
votes
4
votes
What is the least number of colours Peter could use to color the 3x3 square?
I got
by "coloring" with numbers:
- 221
3
votes
Accepted
3
votes
Fitting the 9th piece into the pizza box
The first approach to this problem is to see if it is possible at all, and so the first, most trivial idea is to inscribe a circle within the 2x2 square (b/c you want to keep the shapes as close as ...
2
votes
2
votes
2
votes
1
vote
Only top scored, non community-wiki answers of a minimum length are eligible