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So my question is: was this set up so that the golden ratio ϕ=(1+√5)/2 would turn up, or could you cook up a similar puzzle without it?

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\[\left(\frac{5}{3}\right)^x = \frac{1 \pm \sqrt{5}}{2}\] (quadratic formula)
\[\left(\frac{5}{3}\right)^x = \frac{1 + \sqrt{5}}{2}\] (only one root is real)
\[x \ln \left(\frac{5}{3}\right) = \ln\left(\frac{1 + \sqrt{5}}{2}\right)\] (take logs)
\[x \left(\ln5 - \ln3\right) = \ln\left(1 + \sqrt{5}\right) - \ln2\]
\[x = \frac{\ln\left(1 + \sqrt{5}\right) - \ln2}{\ln5 - \ln3}\]

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Here's my working:

Want to solve for \(x\):

\[ 9^x + 15^x = 25^x \]

First guess: it's a little bit less than \(1\):

9^0 + 15^0 &=& 2 &>& 1 = 25^0 \\
9^1 + 15^1 &=& 24 &<& 25 = 25^1

Do some rearranging:

\[1+ \left(\frac{5}{3}\right)^x = \left(\frac{25}{9}\right)^x\] (divide by \(9^x\))
\[\left(\frac{25}{9}\right)^x - \left(\frac{5}{3}\right)^x - 1 = 0\]
\[\left(\frac{5}{3}\right)^{2x} - \left(\frac{5}{3}\right)^x - 1 = 0\]

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Someone at work sent me this algebra puzzle, which I think came from The Times newspaper:

Solve 9ˣ+15ˣ=25ˣ.

(Put a CW on solutions in replies, please!)

Have you seen it before? Did you see it in The Times?

Attention everyone, I have reached my breaking point. Thanks to the internet, I have a new internet-legal name.

@esoterica Multiplicative/product calculus, as described by this preprint, is also described nicely by some of the books hosted on this site: and this other preprint: .

In Python, you can filter a list comprehension by adding an if statement to the end, e.g.:

[x for x in list if x<y]

I wish you could do the same in a for loop, e.g.:

for x in list if x<y:

Gang, I think I'm going to start putting a little bit of space between things that are multiplied together.

\[ \frac{- b \pm \sqrt{b^2 - 4 \, a \, c}}{2 \, a} \]

instead of

\[ \frac{- b \pm \sqrt{b^2 - 4 a c}}{2 a} \]

Making this document look more like serious academic writing by switching from sans-serif to serif.

A notation question:
The polar form of a complex number is \(r e^{iθ}\).

Which, if any, of these are in polar form?

A) \(5e^{2i}\)
B) \(2e^{i\pi}\)
C) \((4+\sqrt{2})e^i\)
D) \(e^{(4+\sqrt{2})i}\)
E) \(e^{0i}\)
F) \(e^0\)
G) \(1\)

I've been playing with Cuttle a bit more. Here I made a nice tiling using the Rotational Repeat and Tile Repeat modifiers, then I wrote a custom modifier to fade out the colours on the outer pieces.
Here's a link:


An amazing discovery today during my random stroll (someplace in Greece, in a secret location)! As I took a shortcut through a tiny street, I was thrilled to see that someone had filled the place with beautifully drawn Geometry theorems! [1/9]


To take time off, I have to fill in an "annual leave request form". It's a web form hand-coded by local IT, because nobody else does this, so we need a bespoke tool, of course?

Anyway, the form asks for "hours required". Turns out the label is rewritten to "days required" in the email the office staff get, but the number stays the same.

I only discovered this because the person who normally handles it is on leave herself. So she'd just been ignoring this for years!

What's my pattern?


(not a , or at least I haven't done the maths to make it easy to chant without stopping to think yet)

Does anyone disagree that the cube is the least prismy prism?

*clicking the "regenerate values" button until I get the ones I want*

"hmm, there are 100 combinations, I'd have to be very lucky to get the one I want"

*three clicks later*


@christianp Calibre calls it "Library Order". I think I've also seen it referred to as "title sort".

It's weird being asked to make a certificate of participation for someone who gave a talk at a conference.
Like, their employer won't accept seeing a recording of them giving the talk on the conference website, but will accept a PDF I knocked up in a few minutes?

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