UKMT Olympiad + STEP/TMUA/MAT General Advice Needed

Experienced Mathematicians (fingers crossed @Muttley79 sees this) - how do you learn to start chipping into really hard problems that you just cannot seem to crack, even after an hour of contemplation, even after a night or 2 of sleeping on a problem. Even when you are tenacious by nature?
I'm trying to support a maths student who wants to progress but becomes demotivated by hours of little to no progress on tough Olympiad style questions .
Can you teach yourself techniques to problem solve on your own?
Student was gifted George Polya's How to Solve it but it's too dry for their age.
I've seen Paul Zeitz's 'Art & Craft of Problem Solving' mentioned on Reddit but am worried the examples may be too advanced for a Y10?
Any general advice would be most welcome - either resources or approaches.
I am sure this skill comes with age but I don't want demotivation to kill the fun in the meantime.

This is a fantastic reply @mqb2766 and the one I was looking for - thank you so much! I'm going to get both books on the back of your recommendation.
May I ask about AoPS....is it the US equivalent of UKMT? I'm trying to navigate the site as an ignoramus and it looks great but I'm struggling with two things:
1) it all seems geared at selling online classes. Am I missing some static content area of past papers or questions banks that correlate with the 'range of different solutions online' that you mention.
2) I'm not sure which level to direct the Y10 student to. They score fullish marks in IMC, obtained JMO silver & Cayley gold but found Hamilton hard this year - what level would that be? Very 'naturally' good at UKMT, hasn't been hot-housed, obviously hasn't studied A level maths yet but often asks teacher to explain higher level maths that they've not yet come across in lessons

Can I just check that the 'Frost slides' are the ones listed under Riemann Zeta Club?

your observations are the seeds of personal statement material for down the line & not something I'd pick up on but will be pointing out for possible future use...nice!

These suggestions are brilliant & so helpful.
Just purchased the Gardner books as they'll totally hit the mark. A love of recreational maths has always been there though I tread slightly carefully at this stage where constant suggestion of resources on my part makes it less fun. I know that if I left those two 'lying around' they'll be lingered over for a good deal of time.
I'm going to struggle to get my hands on Proofs without Words but a preview reminded me slightly of some of the visuals I saw on a trial of Brilliant.com. Do you happen to know whether the content on there is any good? It's quite an expensive subscription just for fun
Also keen to hear of any good logic based app games - Simon Tatham's Puzzle collection is the go to time passer on tube journeys for both of us. The Maths.org link is perfect. The Plus 'Magazine' puzzles are great (wish they were on an app) and the articles are perfect. Student is more included to read those at the moment than get stuck into a book because is currently in a phase of reading immersive & emotionally moving modern novels - fair enough. Likes to watch Three Gods-type TedEd Riddles, Josephus problem, The 100 Prisoners Riddle etc on YouTube to wind down at night. Soaks it all up like a sponge. Wish I'd had access to all these resources as a teenager. In an attempt to better understand why maths is so loved I read an untouched copy of Fermat's Last Theorem last summer - completely brilliant & I now get it even though I can't do it.

I keep looking at the Roger Nelson book...it looks fantastic (but so £££)

@mqb2766 I have a new question if that is okay?
Obviously Hamilton went pretty well BUT her layout to Q2 was HORRID. I wouldn't know but she knew it was horrid & her maths teacher had a look too and also commented that it was horrid- the script came back (with the sole comment of 'well done' on it) so they had a look at it.
She'd actually been in a panic when attempting Q2 as had been somewhat thrown by both Q1 & Q2 but I suspect her layout is not the best even when things are going well.
My question :
How do mathematicians learn the discipline of a neat layout with all the rights statements etc ?
I'm guessing it is not worthy of a dedicated book but do you know of an essay or webpage on it? I think from memory I saw one on AoPS when you directed me there.
Or is it something you pick up just doing questions and looking at model answers and having your scrappy layout pointed out to you 😀

In the words of a poster on that linked thread Thank you SOO MUCH! This is EXCEPTIONALLY helpful.
She's a massive scribbler/rough worker so I just don't know what happened on Q2. There was definitely an element of panic which is unusual as she normally does not break a sweat in a crisis. It's all good though as I think the marker report acknowledged that it was a very very slightly off paper this year.
She's going to attempt Q6 this weekend even though she said she didn't understand the answer. I've cut & paste a few approach hints of yours from the other thread for her to look at so fingers crossed.

In the words of a poster on that linked thread Thank you SOO MUCH! This is EXCEPTIONALLY helpful.
She's a massive scribbler/rough worker so I just don't know what happened on Q2. There was definitely an element of panic which is unusual as she normally does not break a sweat in a crisis. It's all good though as I think the marker report acknowledged that it was a very very slightly off paper this year.
She's going to attempt Q6 this weekend even though she said she didn't understand the answer. I've cut & paste a few approach hints of yours from the other thread for her to look at so fingers crossed.

Was mulling this over a bit yesterday and another way to get started on the problem would be to think about the logical reasoning (another problem solving section) associated with the general "n" problelm. Its also differs from the previous way in that youre less interested thinking about the sequence of cells dug and more interested in the properties of the final solution. The usual thing is that when you hit a brick wall with one approach, try another thing.
So to approach the problem from the other direction (and it would be partially related to how they describe the solution). So roughly there are two main rulesand were looking for a minimal solution.
Taking the first rule, that means there must be at least one dug cell in each row and column (n rows and n columns so 2n in total). The reasoning is very simple as if there isnt, the first rule means you can dig a cell on the first empty row/column next to the dug area and keep iterating out over the undug rows/columns. The smallest number of dug cells which would satisfy this is n and youd arrange those along the diagronal and each dug cell would be double counted for a row and a column. A dug diagonal also passes through the centre cell which is where moley starts, so it seems a reasonable arrangement to consider. So M>=n.
Now think about the second rule, how can a minimal number of additional cells be introduced so the diagonal forms a connected set of cells? ....
The above is to get her started and there is a reasonable amount of extra reasoning in the ... But when you come up with the answer theres a few interesting insights about such paths.