Twenty Questions (Of Maddening, Delicious Geometry)

By Ben Orlin

SURGEON GENERAL’S WARNING: BEWARE THESE PUZZLES. SIDE EFFECTS CAN INCLUDE LOST AFTERNOONS, HAIR PULLED OUT IN CLUMPS, AND EXCLAMATIONS OF “OHHHHHHH, THAT’S HOW YOU DO IT” SO LOUD THEY CAN DAMAGE WINDOWS.

I came across Catriona Shearer‘s math puzzles on Twitter a few months ago. I was immediately drawn in: they’re so tactile, so handcrafted, so ripe for solving. Each of her gorgeously tricksy problems can swallow an hour in a single bite.

She agreed to let me brainfish you folks by sharing 20 of her favorites. She even indulged my curiosity and admiration with an interview (see the bottom of the post).

Enjoy. And don’t say the Surgeon General didn’t warn you.

1.
The Garden of Clocks

“Unfortunately, my favourite one of the six is the only one I didn’t come up with myself,” says Catriona, “the dark blue one.”

2.
The Toppled Square

(This one feels like an instant classic to me.)

3.
It’s a Trap

“A ‘second attempt’ puzzle that was nicer than the first one I came up with.”

4.
Three Square Meals

“I quite like this one – I drew lots of pretty patterns based on it.”

5.
Shear Beauty

“Probably my all-time favourite. It just looks impossible! Apparently the solution method used here is called shearing (unfortunately, not in my honour).”

6.
All Men are Created Equilateral

“Another corollary that I much prefer to the original.”

7.
Semicircle Turducken

“I find angles puzzles much harder to write. My students told me this one was quite easy, but my parents found it really hard. I think you need to ‘know’ more to do this one, but the problem-solving aspect is easier.”
“I never learnt the intersecting chords theorem at school, so I love anything where I get to use it!”

9.
Tale of Two Circles

“This was a corollary to a different puzzle, but I like it more than the original!”

“I think this one’s quite neat, although it looks like a massive rip-off of Ed Southall’s puzzles.”

11.
All in the Square

“I like the fact that although you can work out all the dimensions of the orange triangle from the information here (and I did when I solved it), you don’t actually need to – using the area and the hypotenuse is enough.”

12.
Spike in the Hive

“This one’s quite neat – I like the fact that you don’t need to work out any of the actual side lengths, which are almost certainly horrible.”

13.
Isosceles I Saw

“I think the wording of this one is my favourite. Lots of people missed the important information and concluded there were infintely many solutions!

14.
Green vs. Blue

“Another one of my favourites.”

15.
Jewel Cutters

16.
Going, Going, ‘gon

“This one isn’t so neat, but the answer really surprised me. I think because it’s harder it didn’t get so much traction on twitter!

17.
Just One Fact

“This is one of my favourites, as it just doesn’t look like there’s enough information.”

18.
The Tumble Dryer

“I like the higgledy-piggledy squares here, like they’re rattling around in a tumble dryer. And the answer is surprisingly neat too.”

19.
Fly the Flags

“This one’s quite simple, once you see it – but I didn’t straight away so the simplicity of the answer surprised me.”

20.
The Tiger-gon

“This one I nearly didn’t post. But the picture reminded me of Tony the Tiger.”

BONUS:
Sunset Over Square City

“I like this one because it reminds me of a sunrise over a city of squares.”

In case you’ve made it this far down the post – in which case, it’s probably 6 months after you started, and your desk is surrounded by crumpled papers and empty Chinese food containers – then here are some questions I had for Catriona.

How did you get into designing these puzzles?

I went on holiday to the Scottish Highlands, but forgot to take a coat with me, so I ended up spending more time inside than my friends did! I kept doodling along the lines of “I wonder if I could work out…”

I wasn’t expecting it to turn into a hobby, but it gets a bit addictive – especially when people reply with their solutions, which I love. There’s almost always a neat shortcut that I’ve missed.

It just starts with doodling. I’ll end up with a whole page of overlapping squares at different angles, or regular(ish) pentagons with different parts shaded in, and then see if there’s any nice Maths hiding there – relationships between lengths or areas or angles.

Lots of your images are marker on paper. Why the low-tech approach?

I did try using Desmos and Geogebra, but I’m not very good. I found it way quicker to draw an inscribed circle by getting my compass out and doing a bit of trial and error than by constructing it nicely in geometry software.

Also, with felt tips you can fudge things because the lines are so thick. It’s a nice compromise between it looking ‘right’ but also knowing you can’t just get your ruler out and measure it.

One of the nice things about geometry is it’s very forgiving – I can show you a hopeless picture of a square or a circle, but it’s enough to communicate the concept because they’re so well defined.

Several of your puzzles provide just enough information. How do you find that boundary, where a diagram is just barely determined?

Sometimes giving the bare minimum is actually a giveaway, because it only leaves one avenue. My preference is for giving slightly too much information, so there are a couple of decoy routes. This also means I get to see more variety when people reply with their solutions!

I’ve posted a couple of puzzles that were impossible – luckily someone usually points it out quite quickly!

I’ve also posted puzzles that I’ve massively over-specified, because I didn’t see a nice shortcut that would only use half the information.