Everyone loves an 'impossible maths exam' story

As I said above, I think it is more straightforward than many of the geometry questions DS faced in practice 9-1 GCSE papers last year, because the underpinning knowledge it requires is quite limited and of the same type (angles within a triangle / on a straight line), rather than requiring e.g. circle theorems.

If they ONLY accepted the elegant non-trig solution, then I think it would be trickier, but if they accepted the one basic trigonometry step as part of a successful answer, then i think it's OK as a 'later question' in the papers.

I agree that the junior UKMT challenge papers normally contain something like this, too - so accessible to 'reasonably able within their cohort' young mathematicians.

It's not a difficult problem. It's at the level of a junior UKMT challenge. The problem is that nowadays children are not trained to think. They rather learn by heart formulae to calculate things which is good enough for the accountant-like jobs they will be doing later. Geometry is for true mathematicians. Many great mathematicians started to love maths when they discovered geometry. This story is a great illustration of the fact that our kids are not really learning maths at school. Their mind is rather being shaped to run a system defined by others.

I've got post-grad maths. This is a terrible question for an exam.

Certainly hope not! I basically hid my head in my hands and didn't ask. He did get the right answer, and my guess is that whatever he did was probably correct but daft. He has form for that!

My DS apparently did it using the cos rule, sigh...

Seeline Imagine you have a compass, and you place its point at G. You draw your arc. You then take your compass away and draw 3 lines from G, to meet the arc.

You now have 3 lines, starting from G. Those 3 lines are GD, GB and GE.

You know they are all the same length (because your arc was the same distance from G all the way along).

Now you draw in your picture of the square, and the kite - and you rub out the GB line.

The distance from GB doesn't immediately look as though it's the same length as GD and GE but it is.

Does that make sense?

My dd drew that as a demonstration - which pushed towards the trig answer. We were really impressed by the equilateral triangle answer.

(And I was impressed by all of it. Dd found me pondering over a drawing and abandoned her Sociology homework to talk me through it. I have MN and noble to thank for a lovely half an hour of mother-daughter bonding. )

Think you're very right Rafa I asked a couple of Oxford science undergraduates last night - they made it very convoluted!

oh sorry ignore me I see it

DE = GB = GD

can anyone help me with this?

In order for the equilateral triangle answer to work, you have to know that DE = DG (=GB).

But how do you know that?

I think it might be slightly more than that. Purely because if your knowledge is limited to the angle properties of triangles and you are used to solving geometry problems by experimenting with drawing additional lines into diagrams that’s probably where you are going to expect the answer to be.

But if you are 15 and have a knowledge of trig, you are probably expecting the answer to be more difficult.

Doesn’t solve the issue of it being difficult to see the equal lines as being equal though.

Rafals,

I would be surprised and seriously impressed if many (more than 1%) of year 6 or 7 students could access that kind of problem.

It depends how much time you have in the exam. If the exam is purely based on reasoning they may be quite liberal with the time given for the paper because some questions need time spent on it.

Wouldn’t like to have met it in a GCSE though.

I wonder what would happen if you gave it to year 6/7 classes who heave had quite a lot of experience with reasoning activities? Might it be easier if you aren’t looking for a complex solution?

I would have enjoyed it in an exam aged 15.

Now, I enjoy it as an internet problem. If I see the answer I post; if not, I lurk and read someone else’s solution!

‘It trivially follows...’, always seemed to work for the lecturers at uni but I was never given credit for using it (because I could not link the two pieces of my proof....).

I would not have liked to have met that question in an exam as a 15 year old, even though the Geometry is basic. I would have found it a good homework question at that age.

Ah 'trivial' that mainstay of 'I don't know how to prove this but it must be true'

It's partly an optical illusion that GB doesn't look like GE and GD. Draw a sketch and then tip the diagram so that DB is horizontal, and you will be able to see that GB and GD do look equal.

That's in the question, see top of image.

I'm still confused by GE, GB and GD all being equal if the shape is a kite
Please explain

that teaches me to read properly, I had read Larry’s comment not noticing the speech marks and inserting an extra ‘is’ into ‘but ‘it is obvious’ is not a proof’.

Shouldn’t be allowed by ipad before 8:30 am!

ah never mind, further googling suggests that there was actually more information - like the equation!! I was trying to figure out how to solve it with no heights...

And how about that swing problem on the page linked with other exam questions?

It isn’t a proof Larry, but is it asking for a proof?