Stage 5 Cipher Challenge

Exams begin in 4 weeks but I seem to want to do everything BUT revise for them. Today I started a cipher challenge I found on NRICH and was so excited to get the first clue. I made a frequency chart using Excel and found the letter E pretty quick and then, with some deduction, found a few more letters. It got to a certain point where it was easy to guess the entire thing. This is it:

Well done, you've cracked this code! Have you worked out how this message has been enciphered? The letter 'a' was passed to 'd', 'e' to 'e' etc. This is called a Caesar shift, with a shift of three letters in this case. We also made things a bit easier by leaving punctuation and the spaces between the words in. How did you decipher this? You may have tried looking for repeated three letter words such as 'the', or counted how many of each letter appeared in the ciphertext and guessed that the most common letter corresponds to 'e'. This second method is the basis of a method called frequency analysis and is very useful for monoalphabetic ciphers. If you know how to program, you can save yourself a lot of time by writing some code to do it for you! Don't worry if you don't though, there are lots of ways you can do it. The 'find and replace' tool in a word processing program can be very useful, must make sure you don't change again the letters you've already replaced! One way around this is to turn the whole message into lower case, and then use capitals for the decrypted message. The neat message will be slightly harder, good luck!

Now I need to figure out how to force myself to revise and get A LOT of schoolwork done. Wish me luck !

UPDATE Making the Equilateral Triangle on the Parabola y=x^2 a 3, 4, 5 Triangle

I failed. It's almost as if Dr Munro had predicted it as I (thankfully) found this solution:
https://www.maths.ox.ac.uk/system/files/attachments/Interview_Question_2019_JM.pdf

Key mistakes I was making:

1. As it had to be similar to a 3, 4, 5 triangle I was equating equations to a certain 3λ, 4λ and 5λ. Introducing another variable into the mix was not brilliant so the idea of representing each side as a proportion of the other was a lot cleverer.
2. I somehow managed to forget that two perpendicular lines have gradients that multiply to make -1. That would've been a good piece of information to know (!)

Anyway, I have a series of questions from IWantToStudyEngineering that I did on Sunday that shall be written up (and possibly typed up) soon. For now I will be reconsidering my application to Oxford because that wasn't even as hard as I made it out to be. 

Three Equal Circles in a Semicircle

If the small circles each have a radius of 1, what is the radius of the semi-circle?

Papa showed me this really cool problem courtesy of STEPMaths. At the time I was doing a really frustrating chemistry paper and by the time I got to starting to work on it Papa showed me the solution... So I am going to present it here to (hopefully) learn it.

SOLUTION:

Join up the midpoint of two of the circles as well as the midpoint of semi-circle. Extend the hypotenuse of the right-angled triangle formed to extend to touch another point on the semi-circle.

By definition, this extended line is the radius of the semi-circle. Let this be R
Therefore the hypotenuse of the triangle is (R - 1)
Using Pythagoras' Theorem on the right-angled triangle we can show that:

(R-1)² = 1² + 2²

(R-1)² = 5

R - 1 = ±√5

R = 1 ±√5

As R ⟩ 0, R = 1 + √5

Now let's pretend we have no given values and look at the relationship between the radius of the semi-circle and that of the small circles.

We will keep the radius of the semi-circle as R and let the radius of the smaller circles be r

(R-r)² = r² + (2r)²

(R-r)² = 5r²

R - r = ± r√5

R = r ± r√5

As R ⟩ 0, R = r + r√5 = r(1 + √5)

This is interesting because we can see a link between the radius of the semi-circle and the Golden Ratio. The radius of the semi-circle is equal to radius of the small circle times twice the Golden Ratio.

In other words: the ratio of the radius of the semi-circle to the diameter of the small circles is the golden ratio.

I need to do some more research on the Golden Ratio because right now all I know is what Emily tried discussing with some very confused year sevens. 

Oxford Maths Interview Question

On Sunday I spent a great deal of time thinking about a question. A question presented in the Open Day I talked about yesterday that was used in Interviews for the 2019 Applicants. And it just kept whizzing around my head. I was watching TV with Mummy and Kunnu when I had to grab my notepad and have a go at it again. I annoyingly missed a big chunk of the show but it's okay because I now have two solutions to it!

This is a screenshot of the question. The diagram isn't perfect but that's not really the problem here. The question is: you are given the curve y = x2 and you have three points: A, B and C, where B is the origin (0, 0). Find A and C such that the triangle is equilateral.

Although it took me a while (due to a idiotic mistake) my two methods are (1) using the length of the sides and (2) using trigonometry

Solution for (1)

Solution for (2)

Now this is where it gets interesting. Dr Munro then suggested that from here he would ask students if this triangle could be similar to a 3, 4, 5 triangle (obviously no, due to the proportions being completely different). Then he wants us to MAKE IT INTO ONE!

I haven't got around to it yet because I have been annoyingly busy today (which, of course, means I was yet again unsuccessful at planning my time) but I definitely have some ideas. I'm going to have a go at it and update ASAP.

Announcement to Charlotte and Charlotte

Today I finally did it. I mustered up the courage to tell Charlotte and C Mitch.

I don't really know why I was worried to be perfectly honest. I guess I thought they would virtually spit in my face. No well, that's not true; I know I was worried because from now on every time I am awful at Maths it will be even more embarrassing considering I'm practically dedicating my life to it. But I guess online school has taken away that fear as now they won't be able to see all the ridiculous blunders I make, as my blunders and I will remain in the comfort of my own home. Of course, they gave the perfect responses.

Let's start with C Mitch:

Me: hey guys i have something to tell you🙈
C Mitch: Ffs
C Mitch: Spit it out
Me: Umm
Me: guys
Me: i think I'm going to
Me: 🙈🙈🙈so shy rn
C Mitch: wtf is going on
Me: um apply to study mathematics at university🙈🙈
C Mitch: wait r u being serious
C Mitch: this is so exciting
Me: .. yes
Me: IVE BEEN SO SCARRD TO TELL U GUYS
C Mitch: stfu

beautiful <3

and when Charlotte finally saw she hit me with a
omg...omg
honestly go for it i can see it now... 4 years of studying.. maybe a phd... and then you will get a job as an accountant😂😂😂
thats so crazy u will be the next [cat bird bird]

I went along with the whole accountant thing which definitely disgusted C Mitch. Evidence: "They're rly boring appaz r u sure"

Then I called myself the worst at maths out of us three (which I still think was fair) which sparked a fire in C Mitch to desperately defended her place as last. I surrendered and was hit with the "Yh ur stupid; And I always win; U shld know by now". Needless to say today has been a whirlwind of emotions. Also - since when does she always win?

Finally Charlotte unionised us with a beautiful image. We then laughed considering our assigned subjects are all the ones we performed worst on in the mocks.


That is all for today. I have something else to post on a maths question but I've decided to postpone until I've figured out the whole thing. Also - still convincing C Mitch to hop on the Maths journey. Who knows what'll happen in six months time?

Oxford Maths (Online) Open Day

Its 2pm and around two hours ago I got off the virtual Oxford Maths Open Day headed by Dr James Munro. A hilarious man who I have a really deepfound respect for. A man who studied Mathematics for eight years at the best institution for the subject in the world, before becoming the Admissions Officer for  the second-best one. He has this real childlike excitement for maths that makes me just adore the way he speaks.

To be honest, Kunnu was arguably more excited for it than me; her screams were the ones to wake me up at 10:20 this morning, 10 minutes before the start of the talk. Although I didn't learn too many new things (I have been scouring TheStudentRoom for months about all things Oxford Maths), I found the talk really motivating and reassuring in a way. The undertone of the whole event was pretty much "do well in the MAT and the interview, and you're in". Obviously the statement is easier said than done, but it gives me a goal. And that goal is to get really good at maths. Really good.

Watching it made me want to go to Oxford so painfully bad. Talking to Sally makes me want to go to Oxford so painfully bad. I want to go so badly but I feel weirdly hindered in a way, I have this overwhelming feeling of definitely not being good enough. I am terrified by the thought that I don't have that spark of "natural talent". I wasn't even good at maths until maybe year 5, and I put hard work towards that every night just so that I could show up to secondary school and breeze through every Maths lesson. As tedious as I found it at the time, I can't say it didn't work. If I still have that energy inside of me to work towards the MAT, then hopefully I can reach my goal of getting an okay score too (we're talking 60s-70s, if I got in the 80s I'd probably combust).

I sometimes feel as though I spend more time looking for resources than I do actually using them. So I am going to list what I have right now, all of which I want to work through ideally before summer. Then we can see if that goal is in any way attainable:

• Finish all the puzzles in Algorithmic Puzzles by Levitin & Levitin
• Complete all the questions on I Want to Study Engineering 
• Complete Dr Frost Maths Maths Challenge sheets
• Complete Dr Frost MAT topic-by-topic worksheets
• Complete edX Imperial Maths Extension of A-Level course
• Stanford Mathematical Thinking course
• STEP Integral Maths resources
• Do NRICH Questions
• Do UKMT Maths Challenge questions

Okay it really is a lot and some of these might not be of top priority. Perhaps the two courses aren't going to be the most useful things to do right now in this short period of time. Especially considering I have several exams coming up testing my skills on all my A-Level subjects (the part I always forget - I need A-Levels to get into uni!) On top of all that I have to read a bunch of wider reading books to prove my "passion". By the sounds of it that might be the easiest step. 

What I am saying here is I need to put a pause on the laziness for the better half of the next year. Right now all I want is to work my hardest to get into [insert good university name here because I am too scared to put Oxford] and I think I will. 

The scariest part of this whole ordeal is that I need to email my teachers and tell them of my intentions to pursue their subject for the rest of my life. Horrible thought so let's put that off for as long as possible. Or possibly never tell them? They don't need to ever find out, do they?

First Post (and probably The Last)

I feel a bit stupid making this right now and I honestly don't know why I am. Simply logging onto this website and looking at my profile picture, user name (no one has called me "Meeka" since year 8) and bits about me is honestly a bit scary.

I am becoming weirdly nostalgic of being in year 7 and living the idyllic lifestyle of being a child in a big school trying to discover her personality. I'm 17 now and I fluctuate between absolutely loving and absolutely hating who I am. I feel that for the past three years my life has been a conquest to suddenly transform myself into alpha-Vamika. An ultra-productive, healthy and balanced teenager who people respect. Right now, I'm afraid, I'm pretty much the opposite. Just yesterday, I stayed up the whole night before (online) school due to the sudden urge to down a mug of coffee, did copious amounts of physics work whilst my heart sped like a train thumping against the tracks. And I loved it. I now understand the photoelectric effect and made it through ten minutes of my next physics lesson at 9am before feeling completely lost again. But still, ten minutes! I call that a win. I then proceeded to knock out on my mattress at lunchtime, wake up in shock at 3pm and join my fifth period physics class parading "WiFi issues". I have no doubt in my mind that the teacher saw right through the whole facade. WiFi issues for half an hour and I didn't think once to use the hotspot from my phone? But, hey, we're in the midst of a global pandemic; he doesn't care.

The coffee came right after a series of unfortunate events in an afternoon that made me positively break down. It was Wednesday A week: a day of relaxation. Someone in administration obviously gave up decided to give me four free periods accompanied by an afternoon of PE which, to me, translates into: Inset Day! I woke up at 11pm and spent the day peacefully doing nothing productive until. at 4:15pm, I was greeted to copious messages on my phone and the realisation that I had missed a group Teams chat with my friends. After joining I was asked what I had done that day. I am a relaxed girl, I keep it real. I monologued my affairs which involved only reading the obituary of a very eccentric mathematician. In came flooding in the lectures about productivity, the cries of "how do you do four A-Levels?!", the daily routines of people who seem to want to shove their perfect lifestyles down my throat. All done in a friendly, joking way of course because that makes it better (!) Flustered, I waited for the whole ordeal to come to an end and joined in with messing with our video backgrounds. I was really frustrated with myself for wasting a whole day, but also frustrated with my friends who ridiculed me for something that honestly seems quite natural.

Then I went downstairs and was greeted by my sister showing me the FaceBook profile of a girl aspiring to study Mathematics at university too. Or should I call it a resume? If perfection was a person, they would look up to her. All 9s in her GCSEs, music and academic scholarship to private school, Grade 8 Distinction in Piano (in year 9, as opposed to my year 10) and a searing passion for maths that led her to creating a blog for aspiring female mathematicians. Her first post? An award-winning essay she wrote on types of maths I've never even heard of. Note to self - find out what topology is. And of course a gold award on the Senior Maths Challenge accompanies with Best in Year and Best in School.

Both of these factors culminated in me realising I am not good enough to study Mathematics at Oxford University. There are undoubtedly 100 versions of her hidden around the country ready to snatch up the places. And I am angry because? Because she worked harder than me? Because she has a stronger passion than me? I am afraid to say the answer is probably yes, as unreasonable as it is. I am also greatly annoyed at myself for choosing this subject. Why couldn't I have been given a deep-rooted passion for Chemistry instead? After a lot of pacing and silent crying and TheStudentRoom browsing I downed the coffee and experienced the most productivity I have in a long time. But I also barely paid attention in my lessons the following day and today (a real Inset Day) I can honestly say I have achieved nothing.

I have several courses and books and puzzles and worksheets to get through over the next few months to make myself feel a bit better about the whole process. I also have a real addiction. to YouTube, Twitter and Snapchat that I need to get over ASAP. Will I ever accomplish either of these things? I cannot be certain but I know I need to try. Harder. But tonight I am setting an alarm to wake up a lot earlier than I am probably capable of. Let's see if I can at least do that.