maths summer work

Look online. You could also try to see what local community colleges offer, most do run a variety of english and math qualifications and courses. See details that are found on their website.

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As far as I know, there isn't a great deal over the summer. Doesn't mean that theres nothing,for instance
https://oxfordsummercourses.com/prices-and-dates/
seems a tad expensive, but activities (competitions, ...) are typically during term time for fairly obvious reasons and theres the ukmt camp, but thats very limited and you have to be invited. This
https://debatechamber.com/project/mathematics-summer-school/
doesnt look too bad but Ive no personal knowledge of any of these.

If youre motivated (forgetting about the social side of things) there are plenty of things you could work through at that level, such as Gardiners https://www.openbookpublishers.com/product/979 or many others, but you lose the social side of things.

i don't mind if its not social tbh i would prefer that lol, but it doesn't really have to be over the summer i was just looking for something that i can say i have completed so i can like learn new things and so i am able to like put it on applications for jobs/ university. thank you for all the links though and i will definitely have a look

What sort of thing are you looking to do at univerity / what level? Most university personal statements don't make a lot of difference as long as you show some interest. For a maths degree, doing the ukmt challenges... would be more than enough as well as a bit of reading. If youre into coding (and maths), there is the project Euler which you can work through and is well structured. In term time there are a variety of maths (team and personal) competitions: ukmt, ritangle, mathsbombe, lincoln, liverpool, ... as well as coding and cypher ones. Veritasium has good pop-maths videos and there are many pop-maths books which give an entertaining read to "harder" maths topics and applciations. Theres the oxford maths club, mat/step/... courses, our local university runs a monthly maths club ... and that is just focussing on the maths side of stem stuff.

Whatever you do, pick something that is doable / interesting and don't think of it in terms of getting yourself job/university ready. Most of the time they want you to be interested/enthusiastic and doing stuff, whatever that is, is generally more than enough.

You could even go back 2,300 years and look at Euclid/greek maths. Assuming you did the Edexcel gcse exam recently, does proposition 1 look familar?
https://themathpage.com/aBookI/propI-1.htm
So all the chatter about it being "hard" ....

I don't know what I'm looking to do at university but I know I want to do something that is heavily maths based? Maybe maths and physics or something like that but I'm still not sure. At a level I'm planning on doing maths, further maths, computer science, and physics. The project Euler thing looks very interesting to me so thank you for the recommendation
I have taken part in the recent ukmt and got silver but I regret not preparing for it because I feel like I would like to get gold and do the kangaroo but I guess there is always next year!
The only maths book i have read is '50 Mathematical Ideas You Really Need to Know' and whilst I understood most of it, there was some of it that confused me a bit.
I see myself as a very enthusiastic person when it comes to maths, sometimes too enthusiastic, however I know that currently there is a bigger competition for stem subjects so i thought having some other things to add to applications might make me stand out?
I actually found that question on the exam very interesting, and I did some working out but ran out of time to answer it fully. I tried the question when I got home and got an answer, but I feel like learning more about it would be fun.
Thank you for your help, I really appreciate it

The 50 Things ... book is a good overview of different ideas and silver on the ukmt is decent and wouldn't stop you getting into anywhere, but also maybe doing some preperation (over the summer?) would give an improvement. For me, its less about aiming to max your score, rather its more about understanding the math ideas behind the questions as well as learning about problem solving. Gardiners book isnt bad to work though at that level (its heavily question based), drfrost has some slides for his smc club, there are books about general problem solving etc.

Also, Im presuing you know about isaac physics and the physics challenges if you're thinking of doing it?

The main things for maths/stem subjects applications is your A level grades/entrance exams/interviews and an interest. I'd repeat, do your own learning because you're interested in it. However, you should find that it will help you with A levels/entrance exams/interviews as well.

I didn't know dr frost had ukmt stuff so thank you for that!
I actually use isaac physics to revise for gcse and I knew about the physics challenges but I kind of forgot about them so I will look into the type of questions over the summer.
Personally, I know maths is not for everyone but I have a genuine interest in it and I actually find doing questions quite theraputic in a way? I don't know if that sounds weird but I find it very satisfying to get answers to the questions.
Thank you, you have been extremely helpful and I have started the project euler thing - it does make you think lol