Six new geometry puzzles. How many can you solve?

Another month, another outpouring of brilliant puzzles from Catriona Shearer.

Apparently she’s going through a semicircles phase, which I’m sure will be remembered with the same fervent enthusiasm as Picasso’s blue period, or the Era of Peak TV, or the year the Beatles got really into acid.

Speaking of which: when will Catriona’s blue period arrive? More urgently, what would tripping acid do for one’s geometric imagination?

Without further ado, six puzzles. Feel free to discuss and solve below.


The Three Amigos

Three congruent rectangles. What’s the angle?

See also Catriona’s original tweet (and the ensuing discussion).


The Broken Purple Moon

The two shaded semicircles are the same size. What fraction of the larger semicircle do they cover?

When it comes to this puzzle, Catriona explains:

I spent a week thinking about how to pack two semicircles into a larger one, with very little progress. I only managed to get anywhere when I made a scale drawing. I hoped someone would show me why the solution is obvious; I learned lots from reading the solutions, but it seems it is genuinely tricky.


Double Decker

The red square has double the area of the yellow square. What’s the angle?

Catriona’s preferred solution involves a hidden insight, but she also gives props to this “more physical” solution. See also her original tweet.


The Box of Tangents

What’s the angle?

I’m very fond of this one. More discussion at the original tweet.


Sizing the Aquarium

What’s the area of the dark blue rectangle?

Check out this tweet for a beautiful animated hint.


The Trisected Corner

The three angles are equal. What fraction of the rectangle is shaded?

Original here. Catriona explains:

Most people I showed it to (including my students) managed the correct answer in their first guess but then got into all sorts of a muddle trying to explain why.

I did it with trigonometry, but there are nice ways without – such as this.

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