177 votes
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. ...
2012rcampion's user avatar
  • 18.9k
44 votes
Accepted

A colorful dodecahedron

Partial Answer: Solution: Other Solutions: Fun Stuff:
DqwertyC's user avatar
  • 8,136
32 votes

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 ...
Kris Van Bael's user avatar
31 votes
Accepted

Join six cities with roads

msh210's user avatar
  • 13.1k
28 votes
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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
  • 77.3k
24 votes

Mishustin's circle problem

daw's user avatar
  • 3,178
24 votes
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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 ...
Bass's user avatar
  • 77.3k
23 votes
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World Tour of Planet Rhombicosidodecahedria

There are exactly Proof:
Daniel Mathias's user avatar
22 votes
Accepted

Connect dots on a grid with one continuous line 2.0

I mean, as long as the number of segments doesn't matter..
Bass's user avatar
  • 77.3k
21 votes
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Most polyominoes in an 8x8 grid

user1502040's user avatar
21 votes

What is the total area of the two quarter circles?

Thus from Pythagorean theorem, Total area of the quarter circles is
ACB's user avatar
  • 6,301
21 votes

Join six cities with roads

This was not as easy as it first appeared. Here is a solution with a bit of geometric flair:
Daniel Mathias's user avatar
20 votes
Accepted

Wizard of subsets

Unless I'm missing something
Daniel S's user avatar
  • 6,593
20 votes

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
Jaap Scherphuis's user avatar
19 votes

Covering a 15x15 grid with rectangles

Here is a proof of the lower bound of 13: Why it works (and how I found it):
mathlander's user avatar
19 votes
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Rearrange words to make a sentence

A confusing string of negatives. To put it another way,
codewarrior0's user avatar
  • 6,599
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

Wizard of subsets

Move those marked l to the left, then the rs right and then the us up.
loopy walt's user avatar
  • 20.9k
16 votes
Accepted

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

The answer is that it because
Jaap Scherphuis's user avatar
16 votes
Accepted

9 trees in 7 rows with 3 trees in each row

A different solution is possible:
Nautilus's user avatar
  • 6,524
16 votes

Join six cities with roads

${}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$
Pumpkin_Star's user avatar
15 votes
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 ...
isaacg's user avatar
  • 5,837
15 votes

Wizard of subsets

Very similar to @loopy-walt solution, but still OC :)
DL33's user avatar
  • 779
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
  • 26.6k
15 votes

Rearrange words to make a sentence

Another take: This is true since
EphraimRuttenberg's user avatar
14 votes
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. ...
loopy walt's user avatar
  • 20.9k
14 votes
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+...
RobPratt's user avatar
  • 13.2k
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
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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 ...
new Q Open Wid's user avatar
13 votes
Accepted

Fewest polyominoes adjacent to 3 copies

The minimum number of polyominoes needed is Proof:
Carmeister's user avatar
  • 1,875

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