BAFFLING British Olympiad Proof By Induction

I hope this has been interesting! Haven't done any proof based videos for a while but if you enjoy olympiad related content please let me know. Comment with any questions or other suggestions for new topics and subscribe to stay updated.
~ Thanks for watching all!
Another huge thanks to ‪@maths_505‬ for inspiring me from the start and reposting about my channel - he is the GOAT!
#maths #mathematics #proofs #olympiad #bmo #impossible #hiddeninplainsight #imo #euler #funproblems #proofs #functions #physics #sums #series #limits #whiteboard #math505

[@zunaidparker]

Found you from Maths505. If I can make a small suggestion to be more UA-cam friendly: please find a camera angle that's more square-on and makes the whiteboard easier to read. With it at an angle it becomes real work to focus on the far end.
Have a look at Michael Penn and bprp for inspiration.
I like your style and the way you explain your thinking (contrast to Michael who absolutely jumps to solution methods without motivating why or how to come up with it). Keep it up and hope your channel blows up!

[@OscgrMaths]

Okay thank you! The reason I'm stuck at this angle is because of the way the room is laid out but I'm going to try and find a better one in my next video. Really appreciate the feedback and glad you liked the style!

[@maths_505]

This was awesome. Keep it up mate.

[@OscgrMaths]

Thanks a lot! Glad people are liking the olympiad content too. Means so much that you're giving so much support bro.

[@dimitrikatsouris7114]

Great question and really well explained👍

[@OscgrMaths]

Thanks so much! Really glad you enjoyed it.

[@randommaheshwari9667]

nice bro keep it up..loved the way you explained

[@OscgrMaths]

Thanks so much! I'm very glad you enjoyed it.

[@W.RonanJohannesPaulDiii]

Great video!

[@OscgrMaths]

Thanks so much! Glad you enjoyed.

[@BreezeTalk]

(I like that you use a whiteboard and marker because it means I can copy you more easily. The visuals are more realistic).

[@OscgrMaths]

Thanks that's so good to hear! I quite like the whiteboard format too.

[@padraighill4558]

very nice! I didn't think of 'finite' or 'bounded' induction

[@OscgrMaths]

That's what I quite liked about it - it's a simple problem but not necessarily easy to see induction from the start. Glad you enjoyed!!

[@franolich3]

Cool explanation! You could also present the proof as follows
Let d[i] = a[i] - a[i-1]
Case 1:
a[i] = 2.a[i-1] - a[i-2]
=> d[i] = (2a[i-1] - a[i-2]) - a[i-1]
=> d[i] = a[i-1] - a[i-2]
=> d[i] = d[i-1]
Case 2:
a[i] = -a[i-1] + 2.a[i-2]
=> d[i] = (-a[i-1] + 2.a[i-2]) - a[i-1]
=> d[i] = -2(a[i-1] - a[i-2])
=> d[i] = -2.d[i-1]
Suppose d[i]=±1 for some i.
Case 2 implies d[i-1] = ±1/2. But d[i-1] must be an integer since by definition it is the difference of 2 integers. This contradiction means that case 2 cannot apply.
Instead d[i]=d[i-1]=±1. By induction, must also have d[1]=±1 ie a[1] and a[0] are consecutive integers.

[@OscgrMaths]

Absolutely, glad you enjoyed!

[@notmymain2256]

Isn't it enough to notice a_n-a_(n-1) is either a_(n-1)-a_(n-2) or -2*that, so if a_2024-a_2023 is odd, it must be for n

[@andrewbuchanan5342]

Hi this is a great channel keep it up.
Here's another approach that pretty much avoids induction:
Set b_i = a_i - a_(i-1) for i>0
b_i = b_(i-1) or - 2*b_(i-1)
So by induction so trivial, we don't need to write it down: b_i = (-2)^k(i) * b(1) for some k(i)>= 0.
If |b_2024| = 1, then k(2024) = 0
So b_2024 = b_1
I.e. if a_2024 & a_2023 are consecutive, so are a_1 & a_0
Thanks very much

[@OscgrMaths]

Yeah that's also a great approach! Thanks for the message.

[@Titanic_shirohige]

could you have done this using the theory of recurrence relations instead of using induction?
Using the characteristic equation of the relation and then showing that the difference is 1 or -1

[@OscgrMaths]

Yeah that was another acceptable solution. You can show that 1= |a2024-a2023| >= |a2023-a2022| >= ... >= |a1-a0|
Therefore |a1-a0|=1 and they must be consecutive.

[@debtanaysarkar9744]

@@OscgrMaths That's how I did, hehe😁

[@BreezeTalk]

🗣👌. 👍. Well spoken. I like.

[@BreezeTalk]

This is great coaching of advanced mathematics, I’m very grateful. 🙏

[@OscgrMaths]

@@BreezeTalk Thank you so much!!!

[@milos_radovanovic]

Also, a technical suggestion: The squicky whiteboard marker is a terrible experience for the audience, try using a Lavalier microphone, or something directional, or find a way to edit it out with software (RTX Voice comes to mind).

[@ryanng1905]

It's fine

[@Chris-5318]

@@ryanng1905 No it''s not. It's horrible.

[@waffles6132]

I recommend that you use manim math software
To help you explain solutions more clearly and better visually
Its the same software 3blue1brown uses in his videos, but overall love your videos, subbed.

[@OscgrMaths]

That's a nice recommendation, I'll check that one out thanks.