# Tag Info

Accepted

### Is this duplo train track under too much tension?

First, we can check that there is no angular misalignment. Since 12 curved pieces are needed to make a full circle, the number of left pieces minus the number of right pieces must be a multiple of 12. ...
• 18.9k
Accepted

### A colorful dodecahedron

Partial Answer: Solution: Other Solutions: Fun Stuff:
• 8,136

### Capture a laser beam

Assuming that you can create curved mirrors with infinite precision, then you could catch a ray ... About that footnote: If curved mirrors are more expensive then you could simplify the room: A ...
• 1,445
Accepted

• 13.1k
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 ...
• 77.3k

• 3,178
Accepted

### Mishustin's circle problem

Here's my go (click to embiggen) Steps: Connect A to P and pick an arbitrary point Q between them, near-ish to P. Then, draw lines as shown, constructing the points in alphabetical order, which ...
• 77.3k
Accepted

### World Tour of Planet Rhombicosidodecahedria

There are exactly Proof:
• 14.7k
Accepted

### Connect dots on a grid with one continuous line 2.0

I mean, as long as the number of segments doesn't matter..
• 77.3k
Accepted

• 697

### What is the total area of the two quarter circles?

Thus from Pythagorean theorem, Total area of the quarter circles is
• 6,301

### Join six cities with roads

This was not as easy as it first appeared. Here is a solution with a bit of geometric flair:
• 14.7k
Accepted

### Wizard of subsets

Unless I'm missing something
• 6,593

### What is the total area of the two quarter circles?

Similar to the Napkin ring problem, there is a trick solution: In a comment Sneftel pointed out that it is even easier to
• 52.2k

### Covering a 15x15 grid with rectangles

Here is a proof of the lower bound of 13: Why it works (and how I found it):
• 957
Accepted

### Rearrange words to make a sentence

A confusing string of negatives. To put it another way,
• 6,599
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:
• 52.2k

### Wizard of subsets

Move those marked l to the left, then the rs right and then the us up.
• 20.9k
Accepted

### An immortal ant on a gridded, beveled cube divided into 3458 regions

The answer is that it because
• 52.2k
Accepted

### 9 trees in 7 rows with 3 trees in each row

A different solution is possible:
• 6,524

### Join six cities with roads

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• 281
Accepted

### Drowning Squirrel

The optimal solution has length So, can the squirrel escape when T=30 seconds? This problem is called Bellman's lost in a forest problem, and for an equilateral triangle, the optimal solution was ...
• 5,837

### Wizard of subsets

Very similar to @loopy-walt solution, but still OC :)
• 779
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 ...
• 26.6k

### Rearrange words to make a sentence

Another take: This is true since
Accepted

### Colouring a rug

Here is a 19: UPDATED: Colours now roughly follow OP's. Patches are numbered A-S in order of appearance. In each panel the latest patch is also highlighted by double density boldface lettering. ...
• 20.9k
Accepted

### Walking in a random direction

Start at the origin $(0,0)$. If the two directions are $t$ and $u$, the ending location is $\pi(\cos t+\cos u, \sin t+\sin u)$ and the distance from the origin is \pi\sqrt{(\cos t+\cos u)^2+(\sin t+...
• 13.2k
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:
• 14.7k
Accepted

### Smallest polyomino adjacent to 3 copies

Assuming (like Retudin's answer) that smallest P means smallest polyomino, and that placements of polyominoes have to be perfectly grid-aligned: by With monominoes The reason why a domino solution ...
• 2,468