## Introduction: Square Peg in a Round Hole

This is a simple but fun two-piece plaything that you can 3D print. It allows you to put a square peg in a round hole. And in a triangular one. I saw it on an episode of QI (series H, episode 1) and thought "hey, that's cool, I should make that" then forgot about it for years. You could make it with conventional tools but it is not trivial to cut perfect square or triangular holes in things nor to safely and precisely cut funny-shaped small objects.

## Step 1: Design

I always like to use the simplest tool that will do the job, and in this case it was pretty clear that Tinkercad would be ideal. I made the cylinder into a wedge by drawing a cylinder 30 mm in diameter and 30 mm high, then added 30×30×15 mm wedge-shaped holes on either side. I then drew a square prism, a cylinder and a rotated roof of the matching dimensions of each side, and placed them in a row. I followed that up with a flattish oval cylinder big enough to safely encompass the 3 shapes (43×127 mm), then turned the shapes into holes.

## Step 2: Print

I exported it as an .STL file and printed it on a Makerbot Replicator. I think it looks more interesting if you print the two pieces in different colors, but it hardly matters. I print them together then mix-and-match prints of different colors.

If there is any binding preventing a perfect fit, shave or sand off the offending pieces. The file below generates a pretty tight fit, so if you want it to be looser, scale down the wedge a percent or two.

The file is attached below, if you'd like to print it yourself.

## Step 3: Play

It's enjoyable to watch people figure it out; it's slightly more thinky than the block toy. It's a good inexpensive stocking-stuffer for kids.

Post a picture of your print (or make it conventionally!) and I'll send you a 3 month pro membership.

I really like this, I'm going to try to make one out of wood and give it a good finish. I think that if I make the square into a rectangle I can make it fit one way in each hole. Either way, this old dog of a toy just learned some new tricks. Thank you!

I'd love to see a good quality one made of wood. Good luck!

Interesting! Great idea, however what bugs me about it is that the square hole is unnecessarily large in this version. Meaning: if the circle is diameter X, and the square is X per side, and the triangle is also X per side, then the triangle and circle can also fit through the square hole.

I want to see if this could be modified so that the block can only go through each hole ONE way. (i.e. along ONE axis, either direction)

Seems as if the square could be reduced to just a square of side Square Root 2X. (approx 0.71 X), and this would work. However the problem is that the triangle is not equilateral.

See what I mean in the sketchup image attached.

Anyhow, I'll try printing this out and see how well it works!

Cool. But does the square now not fit through the round hole?

Aaargh you're right! I can make it work with a Triangle, Square and Pentagon (triangle is again not equilateral), as attached. Or it could work with an equilateral triangle, square and hexagon. But not a circle.

Nice. But it looks like the square would still fit through the pentagon, no? It's an interesting challenge, for sure. The 6-shape cube in my earlier comment has plenty of flaws - all of the shapes fit through the square, the triangle fits through the arch, and that's not a regular hexagon. But the challenge then becomes to fit each shape through the matching hole. It becomes a minimalist shape-sorter.

@makendo: The square has diagonal length of X, and the pentagon also has longest diagonal X, so they don't fit through each other! (See yellow pic attached). Like I said earlier, though - either the triangle OR the pentagon will not be a perfect regular polygon, as the ratio width-to-height does not match.

I've also found that it could work with an equilateral triangle, square and hexagon (See second pic in white), and none of the 3 shapes will fit through any others. (And most importantly the Equi triangle and hexagon have the same width to height ratio, so that this 3-sided puzzle works to the ridiculous mathematical constraints I set up.)

Ok enough math geeking. Time to print!!

And thanks to Dave for the cube idea!

Makendo - Thanks for pointing out that the square fits through the round hole. I attempted to 'rotate' the square but the resulting block always ended up with an imperfect corner somewhere. I wonder if anyone else can solve the Circle-Triangle-Square puzzle!

If we don't use a circle then the solution is easy. Just have 3 different N-sided polygons NOT including a circle, which trumps them all. Eg. A triangle, square and pentagon. (See image). However the triangle is not equilateral. Again - any solutions to designing this using pure regular polygons?

It would be cool instead of a flat plate to make a cube with a circular hole through in one dimension, a square hole in another and triangular hole in the third.

Great job. Voting for you.

Dave - it worked well. And it occurred to me that you could make a cube with six different shapes and 2 blocks. Here, the second block is a hexagon/pentagon/arch. Not sure when I'll get to print it, but once it is done I will send you a print. Stay tuned. Thanks again for the suggestion.

Dave... that's a *great* suggestion. I will try that for sure, and send you one if it works!