[MATHS] The easiest problem nobody can solve 🤷‍♂️

The Collatz Conjecture (commonly known as 3x+1) is a long-standing problem in maths, which was popularised by the YouTuber Vertiasium. The premise of the problem is simple enough for children to understand, yet no mathematician has been able to prove/disprove the problem since it was originally posed in 1937.

The problem: Suppose you have a number x. If x is even, then divide it by 2, if x is odd, multiply it by 3, then add 1 to the result. The conjecture is the following: if you start at any positive whole number, and keep applying this operation to it, you will eventually get the value 1. For example, if I start at 12, I get the sequence: 12, 6, 3, 10, 5, 16, 8, 4, 2, 1. Easy, right?

The difficulty: The conjecture is asking us whether this will happen for ALL positive numbers.

1. We don’t know whether it is true. There is a possibility that there is some number out there, where if we keep applying the step, we will never reach the number 1. If such a counter example exists, but we have not found it yet, then no mathematician will be able to prove the Collatz conjecture to be true (since it isn’t).

2. If there is a counter-example, it’s unlikely that we can find it by trial and error. Believe me, it has been tried. Mathematicians have used a computer to manually try all numbers up to around 2^68 (300 quintillion), and so far all of them have reached 1 eventually. If, for example, 10 quattuorvigintillion (1 with 76 zeros) was a counter example, then we have absolutely no hope of stumbling across it by just testing the numbers.

💡 Things to Consider

• Is it true?: You may think that since it has worked for all numbers up to 300 quintillion, then it is probably true. However, probably true still isn’t good enough, as there have been other conjectures such as the Polya Conjecture, where a counter-example was found, and was approximately: 1845 followed by 358 zeros. With that in mind, do you think its true or not?

• Are there any patterns?: If you have access to a calculator or a computer, have a go at calculating the sequences of some numbers. Are there any sorts of patterns you can observe, or does the behaviour seem totally random?

• Will it ever be proved or disproved?: Maybe one day we will see, but I wouldn’t be surprised if within my lifetime this conjecture will remain a mystery. What do you think?

🔎 Find out more

• Veritasium’s video: https://www.youtube.com/watch?v=094y1Z2wpJg****

• Numberphile: https://www.youtube.com/watch?v=5mFpVDpKX70****

• Collatz Calculator: https://www.dcode.fr/collatz-conjecture

[PSYCHOLOGY] Can Psychology make revising easier? ✏️

Everyone has reached that point in their revision schedule where they’re too burnt out to focus, and questioning whether any information is even making it in, right? But did you know that there are many theories and tips that we can borrow from the field of Psychology to make revision easier? For this psychology scoop, we’re going to explore some key psychological ideas and the ways in which they can be applied to educational contexts!

💡 Things to Consider

• Basic Rest Activity Cycle: This refers to the biological rhythm our body operates on which consists of alternating periods of elevated and lowered alertness. On average, each cycle is 90 minutes long. Essentially, what this means is that our concentration levels reach their limit after around 90 minutes, making our brains more tired and decreasing the effectiveness of any work done after this point. A study of elite violinists conducted by Ericsson et al (2006) demonstrated that when practice sessions were scheduled in line with the basic rest activity cycle, this resulted in the best outcomes and improvements. This finding can also be used to increase the efficacy of your revision sessions! When creating revision schedules, instead of trying to power through hours-long sessions, limit each slot to no longer than 90 minutes in order to avoid fatigue and maximise performance. Following this, make sure to schedule in a break (or even a nap!) to allow your brain some recovery time and prepare it to be at an optimal level of alertness when you resume.

• Retrieval Failure Theory: This theory suggests that the reason why we seemingly forget information is due to a lack of sufficient cues to aid recall. In order to be able to ‘retrieve’ this information from our long-term memory store, we must ensure that the information is encoded alongside appropriate semantic cues which we can later use to trigger recall. In terms of revision, some ways that you can create cues whilst learning information include: 1.) creating meaningful links between information (e.g. using mnemonics to improve your recall of lists of things), & 2.) practising different types of recall when revising (e.g. using multiple choice questions to improve recognition of information, using flashcards with questions or prompts to stimulate cued recall etc…)

• Mindset Theory: Carol Dweck’s Mindset Theory (2006) states that those with a growth mindset have the belief that they can achieve their goals and be successful through their own efforts, whereas those with a fixed mindset believe that their intelligence and abilities are innate, and so often give up easily when they fail. Research by Blackwell et al (2007) showed that students who were taught to think with a growth mindset displayed improved motivation and academic performance. This suggests that adopting a more positive attitude towards revision and increasing your self-belief and self-motivation can be beneficial for your studies and overall grades!

🔎 Find out more

• https://www.tes.com/teaching-resource/how-to-structure-revision-and-how-long-to-revise-for-12104552

• https://www.tutor2u.net/psychology/reference/retrieval-failure-due-to-absence-of-cues

• https://psycnet.apa.org/record/2006-08575-000

• https://www.wiseinterventions.org/posters/learning-about-developing-your-intelligence-improved-math-grades-among-seventh-graders-over-the-next-semester-and-improved-classroom-motivation

🗳️ Poll

👀 Keep your eyes peeled for…

• 2nd October - UEA and Goldsmiths, University of London Open Lecture: insight4me Psychology – Psychopathology

• 3rd October - Sidney Sussex Webinar Series 2023-Finalising Applications and Admissions Assessments (Session 2)

• 4th October - English Literature Admissions Test (ELAT) Livestreams: Session 3

• 4th October - Oxford and Cambridge Foundation Years Information Event

• 4th October - Applying to Cambridge Webinars: Interviews & Assessments

• 4th October - University of York Open Lecture (History of Art) - Fakes and forgeries in the Islamic art market: A study of three problematic pieces

• 5th October - MAT Past Papers (2021) livestream

• 5th October - Sidney Sussex Webinar Series 2023-Finalising Applications and Admissions Assessments (Session 3)

• 5th October - UEA Open Lecture: Politics: Focus On Liberalism – The Many Critics of Liberalism

• 5th October - University of York Open Lecture: How electron cryo-microscopy aids the design of new medicines: Bringing proteins into focus

• 5th October - University of York Open Lecture (Physics): Challenges in the design and development of a combat UAV

• 6th October - St John's Admissions Clinic