What are Invariants? Smashing Competition Math Problems with Invariants

An introduction of what invariants are and how invariants are used in math problems, with an example problem from USAMTS and extra problems at the end that can serve as practice.

More information on invariants and similar topics:

[oshinsamuel]

very appropriate video. most math videos talk abt monovarients. glad you’re taking about INvarients, the TRUE stars of the show. Michelangelo could truly never create such a masterpiece.

[ll-svuo]

this is so educational my brain doubled in size!

[avyakthaachar.]

Thank you so much for this amazing video ❤🙏

[DrickRT]

thanks for this man. i'll source this in my homework

[oshinsamuel]

omg this makes so much sense😫i love mangoes!!!!

[tcolbert]

(writing out my solution for myself or anybody else that wants to see it)
for the arranging numbers in swaps
let a useful swap be a swap in which two numbers end up in the right spot after the swap
let a useless swap be a swap in which the swap doesnt put any numbers in the right spot
let a normal swap be a swap where after the swap one number was put in the right spot

Arrangement is 43152
At the start no useful swaps can be done since no consecutive numbers are adjacent in their corresponding adjacent spots.

The last swap will always be a useful swap since its the last swap and both numbers have to be in the right space at the end.

There are an odd number amounts of normal swaps to get 3 numbers in the right spot then 1 useful swap to finish it off. So an even number of swaps. Even if a useless swap is made then it will either provide an opportunity for a useful swap or have no effect but cause another useful swap in the future. If the useless swap is made then a useful swap is made two useful swaps will be made so you have odd number of normal swaps plus 1 useless swap makes an even number plus 2 useful swaps. So it will always be an even number of swaps.