Paddy MacMahon

Trying to keep the question text to an absolute minimum. Ideally none, just the standing instruction to find the green area...

Does the tangent being so close to the curve in the first quadrant make this unclear...? Is this diagram clearer?

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Find the exact value of the green area.

...and tan(π/8)=sqrt2 - 1 ...

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Find the exact value of the green area.

No calculus needed today.

I'm assuming the problem is to find the area of the small circle (as a proportion of the square)?

Help me solve a beautiful geometric math problem!

I'm stuck and need a better approach!

Believe it or not, it's from 1898, I'm guessing from a temple in Japan (Sangaku).

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samjshah.com/2025/08/05/s...

I like that it boils down to a single integral. I started it in two parts, but realised I don't even need to find the extreme x-value, and just integrated ydx/dt between t=0 and t=1.

Excellent plan!

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Find the exact value of the green area.

That's staggering!

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Find the exact value of the green area.

I deliberately changed this as the y-intercept is 9, and the x-value we want is also 9, so it might have been a too-easy guess...

When normals are involved, I have to keep the scale the same in each direction.

But they provide a great service, right? No sewage in your rivers, no hosepipe bans because of massive amounts of water lost in infrastructure that hasn't seen a bean of investment since privatisation,...

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Find the exact value of the green area.

Yep!

a 💎 of a problem that came to mind after seeing Steve Phelps' great animations:

❓ Draw an equilateral triangle and shade the interior points closer to the center than any side; what is the resulting shape?

answers that are right, wrong, neither, or both: all welcome!

🚨 rot13.com to avoid spoilers

Lise Meitner (1878-1968) was an Austrian physicist whose pioneering work led to the discovery of nuclear fission. Find out all about her story in our latest blog post!

trinitycollegelibrarycambridge.wordpress.com/2025/08/01/s...

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Find the exact value of the green area.

In a bit of a turn up for the books...

...ChatGPT actually got this one correct. On its first attempt!

It wasn't very efficient/elegant in finding the value of a or using any symmetry to compute the integral, but it got there.

I was tempted to stretch one of the axes on desmos to make this less obvious...

Yep - recognising that the graphs represent inverse functions of one another makes life easier!

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Find the exact value of the green area.

Only around a month to go until TMiP 2025: a maths communication conference for anyone who communicates maths! Join us in Warwick/online from 28-30 Aug: talkingmathsinpublic.uk

Yes, sorry! Two parabolas, and 0<a<1.

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Here you go, @weduckworth.bsky.social. this one's certainly easier than yesterday's!

Find the minimum value of the green area as a varies between 0 and 1.

This is me with yoga. I will get back to it, I will...

A photo of Paddy MacMahon outside Paddy Mac's bar in Dublin.

Pilgrimage.

Yeah, the same. Find each equation in terms of a, use integration to find the area in terms of a, differentiate this and set to zero (the resulting equation being a quintic in sqrt(a), which happens to have 3 as a solution...)