I translated it all so that they were distances from the origin, thought about the equation they satisfied, and did an expansion to find the product of its roots. I think you can avoid the point case as there is a factor of z everywhere.

That sounds like we did it in very different ways!

I can't remember where I adapted it from - it certainly isn't wholly original.

A very difficult maths question.

I have been saving this one for a while.

I should have added #ALevelMaths (Further).

A series of six differential equations for which the left hand side first five is the total derivative of a product. The sixth, not so.

In #MathsToday, my Y13 class really enjoyed the first five of these questions (and then the sixth shortly thereafter).

We teach OCR, and never have to do the correction*, I think for more or less this exact reason.

*Except in Further Maths.

It is worth adding that this does not happen for the mean.

Averaging lots of sample means tends to the true mean (which we know from Normal Distribution Hypothesis tests).

That is mean!

'These data' in particular, even though we are told it is a sample.

I agree!

The real answer is along the lines of:

If we repeatedly do this, and average our answers, the average tends to a predictable amount below the real answer, relative to sample size.

In order to 'unbias' this, we need to multiply by a factor of n/(n-1), and the n terms cancel out.

Hence, by analogy, sample statistics will almost always underestimate population ones.

Dividing by n-1 makes th result slightly bigger than dividing by n does, which is good enough.

Someone once explained it to me by thinking about the range.

If you calculate the range of a sample, you are very unlikely to randomly pick the largest and smallest values, so the range of a sample is almost always going to be too small.

n-1 when you are using sample data to estimate the population parameter.

n when you have all the data.

(But whether you have to do this varies by exam board and Single/Further Maths.)

An exam question on proof with the name crossed out and replaced with 'Santa'. Also, a snowman.

I did something similar?

Could it be along the lines of the numerator having to be always larger than the denominator when added, so there must be an intersection, hence they can't be mutually exclusive?

The Cogwheel Brain by Doron Swade with an empty pint glass.

One of the most interesting and exciting Maths books I have read for some time. Very balanced on its view of Babbage and his contemporaries, and stuffed full of fascinating detail.

An excellent way to do some Maths (sort of) over half term.

After looking at trig derivatives and the chain rule, I differentiate sin^2 x for my class, ask them to do cos^2 x, and then we look at cos^2 x + sin^2 x, which should obviously be more complicated...

Anamorphosis - Wikipedia

en.m.wikipedia.org/wiki/Anamorp...

Is it this?

I remember reading something recently about how in some sports they are now added digitally to look 'wrong' from a particular camera angle, and therefore more realistic.

This is also the shape that an orange makes if you peel it in one piece starting at the top. (Presumably for similar reasons?)

The red feels like the odd one out to me.

Which gives me back your original graph.

Maybe it was right all along?

Here is my suggestion then:

Convert it to something parametric (not sure what).

Stretch the two x and y equations separately.

Make it cartesian again.

I shall continue to think.

I think one needs to be 0.5, and the other 2, not both the same.

I can reason that out, but it could definitely confuse a pupil.

It is neater, but this version doesn't feel complete to me without, for example, a justification that k(sqrt(2)-1) must be an integer. It is tacitly assumed above.

A wordless solution to this lovely puzzle.

Here is my attempt.

What I like about the Quadratic Formula is not the Quadratic Formula.

Proof is not required; it's a bonus.

You don't need all these big numbers, like ten.

A class did one for me an unspecified number of years ago. Such a good gift.

Clopen Mic Night Clopen Mic Night

Do you like maths? Do you like entertainment? Want to be entertained in a variety show about maths? Come along to the latest Clopen Mic Night and see these cool people (also me and my excellent bandmates): clopenmicnight.com/events/2025-...

And there was another page of them somewhere else in the magazine as well.