11 Geometry Puzzles That Drive Mathematicians to Madness

“There’s a meta-problem,” mathematician Timothy Gowers recently mused, “that it’s vaguely on my to-do list to think about.”

Gowers, a 1998 Fields medalist, has done breakthrough work in combinatorics. Dude’s a Royal Society Research Professor at University of Cambridge. His to-do list is no doubt a catalog of deep and important mathematical questions. So what is this meta-problem nagging at him?

“HOW DOES CATRIONA SHEARER DO IT???”

Catriona Shearer is a math teacher whose Twitter account features homemade geometry puzzles. But “puzzles” perhaps undersells them. These are puzzles that entice and entrance mathematicians of every stripe.

Puzzles that elicit caps-lock, triple-punctuated expressions of wonder.

“I don’t think I’ve ever seen anyone in my entire life,” says the mathematician Mike Lawler, “who has an eye for neat geometry problems like Catriona Shearer does.”

“These problems can’t just pop into her head,” insists Gowers. “Does she have a general theory? Or a nice bag of tricks? Or what?”

“Yesterday I read a tweet of hers,” chimed in John Carlos Baez, a leading category theorist, “where she said she’s not as creative as some people seem to think: she keeps using the same tricks over and over again.

“This,” Baez noted, “is also what Feynman said.”

In reply, Catriona shared a video snippet of her notebook. It was another tantalizing glimpse of her fertile thought processes:

Anyway, Catriona continues to decline my offers to put her in touch with publishers, but kindly picks out favorite puzzles to share here.

So, for your pleasure: eleven delightful excursions into geometry.

 

1.
Transit Across a Purple Sun

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What’s the total shaded area?

“Easily my most popular tweet ever, this one,” says Catriona. “There are lots of very nice replies, but I particularly like this animation where the various possibilities seem to flow around the one fixed value.”

 

2.
Shearer’s Emerald

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Four squares. What’s the shaded area?

“This is a bit of a trick question,” says Catriona. Spoiler alert: “One of the lengths is a red herring – you only need [redacted] to be able to answer. It was actually based on an earlier puzzle that you featured in your first collection.”

 

3.
The Pyramid with Two Tombs

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Two squares inside an equilateral triangle. What’s the angle?

There are lots of ways to approach this,” says Catriona. “One of my favorites – which I would never have come up with myself – is this one, where the top part of puzzle is tessellated to create a [spoiler redacted].”

Also, please note: my solutions to Catriona’s puzzles are uniformly plodding, and usually devolve into calculation at the end. The ones on Twitter are always glorious, Olympic-gymnastic-level feats of symmetry. So it is.

 

4.
Setting Sun, Rising Moon

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What fraction of the rectangle is shaded?

This one is a bit of a hangover from all those semicircles-within-rectangles puzzles I made back in December!” says Catriona.

 

5.
Hex Hex Six

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Both hexagons are regular. How long is the pink line?

Maths is a unique(?) international language,” commented one fan on this one. “See the word replies not in English, but the mathematical solutions are totally understandable.”

“My favourite thing about twitter,” Catriona agreed, “is being able to do maths with people all over the world.”

 

6.
Four, Three, Two

c6
What’s the area of this triangle?

“I sat down next to somebody at a training event in Birmingham,” says Catriona, “who recognised my name from twitter and proceeded to tell me his favourite geometry fact: the inscribed circle in a right-angled triangle with integer sides has an integer radius. We spent some of the quieter moments of the day trying to figure out all the patterns, and I made this puzzle on the train home.”

Funny coincidence! When I taught in Birmingham, one of our spectacular students taught this fact to me and other faculty.

 

7.
The Trinity Quartet

c7
All four triangles are equilateral. What fraction of the rectangle do they cover?

This one,” Catriona says, “is actually featured in Alex Bellos’s latest book, which I think is very cool. The design reminds me a lot of a stained-glass window in a chapel. My favourite solution was this one from Mike Lawler’s son, just because (unusual for social media) we get to see the entire thought process, including the bits where he gets stuck.”

(That is, in general, one of the delightful things about Mike’s blog; it’s a unique document of mathematical learning and teaching.)

8.
The Falling Domino 

c8
This design is made of three 2×1 rectangles. What fraction of it is shaded?

This one inspired a nice spin-off puzzle from Vincent Pantaloni of Geometry Snack fame,” says Catriona. “I mainly like it because it’s not often that you spot out in the wild.”

 

9.
Slices in a Sector

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The three colored sections here have the same area. What’s the total area of the square?
The 13 isn’t a coincidence!” says Catriona, adding: “I liked this approach to the solution.”

 

10.
Disorientation

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The right-angled triangle covers ¼ of the square. What fraction does the isosceles triangle cover?

“Less than a month after I posted this,” says Catriona, “I accidentally re-derived it. I thought I’d made a new puzzle until I noticed that the numbers were familiar, and realized it was exactly the same set-up, just in a different orientation!”

11.
Sunny Smile Up

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What fraction of the circle is shaded?
“Because,” says Catriona, “everyone needs a smile at the moment.”

11 thoughts on “11 Geometry Puzzles That Drive Mathematicians to Madness

  1. I’ve always been surprised over the years to talk to math-types who found geometry boring or uninteresting or uninspiring… when I thought it was an exercise-of-unending-beauty! 😉 So great to see Catriona luring in old & new fans alike to such puzzles and simple visualizations!

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