Felt-Tip Geometry

I am pleased as punch (which is, as I understand it, a very pleased drink?) to offer you four new puzzles from geometric visionary Catriona Shearer.

They are arranged in roughly ascending order of challenge, but can be consumed in any order (unlike most four-course meals).


Three Little Squares Went Out to Play

catriona 1
The three small squares each have an area of 4. What is the total shaded area?

People kept asking Catriona if this one was a “trick question.” It’s not; just a tasty little hors d’oeuvre, which you may wish to nibble at for a while, or to eat in a single bite. (See the original here.)


The Parallelogram of Eternal Balance

catriona 2
What fraction is shaded?

I love this construction. Nothing but midpoints, connected with the elegance and economy of a calligrapher. “I’m sure this is a classic that’s been done many times,” says Catriona, “but it was new to me.”

(See the original here, including a beautiful animated solution, as well as a lovely solution by dissection.)


Sunny Dome

catriona 3
The two red arcs are the same length. What fraction of the semicircle is shaded?

Catriona describes this as a riff on her very first puzzle (which I dubbed Garden of Clocks). I’ll take this moment to make the vapid point that, in addition to impeccable geometric taste, Catriona always chooses the perfect colors. (See the original here.)


Three Squares and a Slash

Catriona 4
The areas of the squares are given. How long is the line?

Some of Catriona’s puzzles yield to a single geometric insight. This one offers a different kind of pleasure. Catriona explains: “I doubt many people could do this one in their head – I couldn’t – but it makes the highlight reel on account of an unreasonably nice answer.” (See the original here.)

15 thoughts on “Felt-Tip Geometry

  1. What’s the best way to see all of Catriona’s puzzles? I see that you have posted collections 4 (?) times. Is that the complete set so far? Do I need to wade into twitter to find one-off puzzles?

    1. In partial answer to my own question, I think there are at least 5 sets of CS puzzles. Four are tagged “Catriona Shearer, Puzzle Wizard” but this one is not: https://mathwithbaddrawings.com/2018/10/03/twenty-questions-of-maddening-delicious-geometry/
      By title, the list is:
      Twenty Questions of Maddening Delicious Geometry
      Tilted Twins (and other delights)
      Geometry Puzzles for a Winter’s Day
      Six New Geometry Puzzles
      Felt Tip Geometry (this page)

  2. Where is this going to ever be relevant in one’s life if one doesn’t work in a stem field? That stuff is a worthless to me now as it was when I had it shoved down my throat 40 years ago in high school.

      1. Unfortunately, it reminds me of my horrendous time in high school when I had that stuff rammed down my throat. The worst of it was that no one would ever take the time to explain how and why it would be helpful in real life. Thus, in my future careers in accounting and law, I say again that it is just as useless to me today as it was 40 years ago.

        1. There are some arguments to be made that a good course in geometric proof is great preparation for law (Lincoln and Jefferson, for example, were both huge fans of Euclid, and Lincoln made this case explicitly, saying that geometric proofs are what taught him how to construct an argument). But a bad course in geometric proof is not very good preparation for anything!

          In any case, geometry is like most subjects – whatever the benefits of learning it, most people will be pretty much fine without it.

          These puzzles, for what it’s worth, are just that: puzzles. They belong to the same family as crosswords, or sudoku, or jigsaw puzzles. Everyone will have different preferences on these – e.g., I enjoy crosswords and these geometry puzzles, am neutral-positive on sudoku, and don’t like jigsaws – but it comes down to how much you enjoy the activity itself, rather than any promise of usefulness.

        2. I don’t have anything against the puzzles. My only point is that one would be a little disinclined to see the utility of an academic topic when one is made to learn it with a gun at his back…again without ever being told of it’s real life relevance.

        3. Oh, for sure! I think way too many people, through no fault of their own, have a terrible experience with school math, and that understandably shapes their outlook on the subject for decades thereafter.

        4. Sad but true. Unfortunately for me, my outhouse of a high school had a mentality of LEARN THIS OR ELSE. Consequently, I have both a lifetime hatred of my old high school and and a triggered aversion to all things STEM, I’m afraid. And yet, college and law school was so much fun

    1. Sorry for jumping into the middle of a conversation… but
      Your main argument is, these puzzles (and geometry in general) don’t have any real-life application. Well, I think as long as they challenge someone’s mind they are much more useful than many other things such as pointless games and surfing social media etc. Why do we care about real life applications only when it comes to math and puzzles and not other pointless activities that might occupy most of our spare time?

      By the way, it’s a scientifically proven fact that puzzles and mind games help preventing or postponing brain deterioration and diseases such as dementia, Alzheimer’s and such.

  3. This website was useful because there were a lot of different examples, so when I have to narrow down my topic this will be a great website to refer back to.

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