Gothic Window Puzzle – Mind Your Decisions (2024)

Gothic Window Puzzle – Mind Your Decisions (1)

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Posted October 1, 2024 By Presh Talwalkar. Read about me, or email me.

This problem is from @bilalsarimeseli on Twitter.

The gothic window design below has arc BC centered at A and arc AC centered at B. Semicircles D and E have a radius of 1 and the circle F is tangent to the arcs and the semicircles. What is the radius of circle F?

Gothic Window Puzzle – Mind Your Decisions (2)

As usual, watch the video for a solution.

Gothic Window Puzzle

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"All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon.

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Answer To Gothic Window Puzzle

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

Thanks Stephan for alerting me of a typo!

The solution is by @ilarrosac on Twitter.

If two circles are tangent, their centers and the tangent point are collinear. Thus we can construct a radius from A to the arc BC through the tangent point and center of circle F, and we can similarly construct a line between the centers and tangent point of circles D and F. The center of F is equidistant from points A and B so it is along the perpendicular bisector of AB and passes through O. Let h be the distance from F to O. So we have the diagram:

Gothic Window Puzzle – Mind Your Decisions (3)

From the right triangles DOF and AFO we have:

h2 + 22 = (4 – r)2
h2 + 12 = (1 + r)2

Subtracting the second equation from the first gives:

22 – 12 = (4 – r)2 – (1 + r)2
4 – 1 = 16 – 8r + r2 – 1 – 2rr2
3 = 15 – 10r
10r = 12
r = 1.2 = 6/5

And that’s the answer!

Special thanks this month to:

Lee Redden
Mike Robertson
Daniel Lewis
Kyle

Thanks to all supporters on Patreon!

References

problem
https://twitter.com/bilalsarimeseli/status/1410621140775542786
solution
https://twitter.com/ilarrosac/status/1410641346445127682

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PRESH TALWALKAR

I run the MindYourDecisions channel on YouTube, which has over 1 million subscribers and 200 million views. I am also the author of The Joy of Game Theory: An Introduction to Strategic Thinking, and several other books which are available on Amazon.

(As you might expect, the links for my books go to their listings on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay.)

By way of history, I started the Mind Your Decisions blog back in 2007 to share a bit of math, personal finance, personal thoughts, and game theory. It's been quite a journey! I thank everyone that has shared my work, and I am very grateful for coverage in the press, including the Shorty Awards, The Telegraph, Freakonomics, and many other popular outlets.

I studied Economics and Mathematics at Stanford University.

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Gothic Window Puzzle – Mind Your Decisions (4)

Mind Your Decisions is a compilation of 5 books:

(1) The Joy of Game Theory: An Introduction to Strategic Thinking
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(4) The Best Mental Math Tricks
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Gothic Window Puzzle – Mind Your Decisions (5)

The Joy of Game Theory shows how you can use math to out-think your competition. (rated 4.3/5 stars on 290 reviews)


Gothic Window Puzzle – Mind Your Decisions (6)

40 Paradoxes in Logic, Probability, and Game Theory contains thought-provoking and counter-intuitive results. (rated 4.2/5 stars on 54 reviews)


Gothic Window Puzzle – Mind Your Decisions (7)

The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias is a handbook that explains the many ways we are biased about decision-making and offers techniques to make smart decisions. (rated 4.1/5 stars on 33 reviews)


Gothic Window Puzzle – Mind Your Decisions (8)

The Best Mental Math Tricks teaches how you can look like a math genius by solving problems in your head (rated 4.3/5 stars on 116 reviews)


Gothic Window Puzzle – Mind Your Decisions (9)

Multiply Numbers By Drawing Lines This book is a reference guide for my video that has over 1 million views on a geometric method to multiply numbers. (rated 4.4/5 stars on 37 reviews)


Gothic Window Puzzle – Mind Your Decisions (10)

Mind Your Puzzles is a collection of the three "Math Puzzles" books, volumes 1, 2, and 3. The puzzles topics include the mathematical subjects including geometry, probability, logic, and game theory.

Math Puzzles Volume 1 features classic brain teasers and riddles with complete solutions for problems in counting, geometry, probability, and game theory. Volume 1 is rated 4.4/5 stars on 112 reviews.

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Math Puzzles Volume 3 is the third in the series. (rated 4.2/5 stars on 29 reviews)

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Gothic Window Puzzle – Mind Your Decisions (2024)

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