Invariants and Monovariants

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Today's video is all about things that don't change, or that change in a predictable way. It's a nice introduction to the method of Invariants and Monovariants and how they can help in uncovering interesting aspects of problems.
#Invariants #Monovariants #ProblemSolving
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Опубликовано:

30 сен 2024

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Комментарии : 54

@leif_p 3 года назад

Very nice video showing a fundamental problem-solving technique using simple examples. Good job!

@ProfOmarMath 3 года назад

Thanks Leif

@andreben6224 3 года назад

This is such a nice problem solving technique. One of my favorites! For the last problem, we could have chosen any strictly convex function for the distribution of the weights, and it would have given the same conclusion, which is amazing to me. This distribution of weight always makes me think of entropy and how it can only increase. I wonder if it has some foundation and also how do people working in information theory think of entropy XP

@ProfOmarMath 3 года назад

Andre I think it actually is. Very cool introspection

@robertgerbicz 3 года назад

Another solution for the last problem: choose a minimal configuration where we can return to the original position, in this case every stone is moving at least once. Choose the leftmost stone, and mark it. Whenever we move it we can move it in the left direction(!), because it doesn't matter which stone (from the 2 stones) will move to left/right. Hence we won't return to the original configuration, since the marked stone's position is in the left side from the original configuration's leftmost stone.

@sumedh-girish 4 месяца назад

SO COOL!

@helo3827 3 года назад

I love your videos! Also, is it okay if you make a lecture on graph theory? Since I am pretty weak at it.

@ProfOmarMath 3 года назад

Actually …… I’m going to make a series …..

@helo3827 3 года назад

@@ProfOmarMath !! Thanks

@aziz0x00 3 года назад

very interesting video masha'Allah

@ShefsofProblemSolving 3 года назад

Nice video and a cool set of problems! Where did you find them?

@ProfOmarMath 3 года назад

I have so many problems in my head, I’m not sure of all the sources anymore!

@ShefsofProblemSolving 3 года назад

@@ProfOmarMath Thanks for the quick response. I usually take mine from a book, competition, or a handout. If you'd like to continue this series check out Engel Problem Solving Strategies chapter on invariance

@wesleydeng71 3 года назад

Interesting! For the last question, I used the sum of abs(n_i). Note that it does not decrease. So if I choose the origin as the square where the 1st step occurs, it increases by 2 at once and will never go back to the initial value.

@ProfOmarMath 3 года назад

I like this!

@elihowitt4107 3 года назад

You're answer is incorrect since it only shows that returning would be impossible if at any step you chose to split at 0, this however is not a requirement and the metric of absolute value otherwise fails to show how a series of moves that are not at zero would work.

@ProfOmarMath 3 года назад

@@elihowitt4107 Eli, do you suspect that the way you wrote that would want to make Wesley engage with you about this?

@elihowitt4107 3 года назад

@@ProfOmarMath I think you're right but I couldn't find a concise way to put it otherwise, how would you write it?

@ProfOmarMath 3 года назад

@@elihowitt4107 Give it a try 😀

@jyfortin 3 года назад

Good job as usual and interesting problems. I wonder for the last problem what would happen for a periodic chain or circle where the quantity X has to be modified at the periodic boundary.

@ProfOmarMath 3 года назад

Interesting modification…hmmmm

@BezNicka Год назад

Beautiful

@yoav613 3 года назад

Great and interesting video ,As always😃

@ProfOmarMath 3 года назад

Thanks yoav!

@aashsyed1277 3 года назад

wow

@ayoubabid8783 3 года назад

Easy, problem, very classic

@ProfOmarMath 3 года назад

There are three problems…

@tonyhaddad1394 3 года назад

Man wow 7:39 , it can be math magic trick 😍

@ProfOmarMath 3 года назад

Exciting!

@tonyhaddad1394 3 года назад

@@ProfOmarMath ❤❤❤

@rounaksinha5309 3 года назад

Thanks sir !

@ProfOmarMath 3 года назад

Thank you!

@aziz0x00 3 года назад

the solution of the first problem is so surprising!!

@ProfOmarMath 3 года назад

I love this topic!

@aziz0x00 3 года назад

@@ProfOmarMath thoese are so surprising solutions, i got other direct approches intuitively and expected them to be the ones u gonna provide as u were talking, but it surprised me so so so so much

@aziz0x00 3 года назад

@@ProfOmarMath it really opened new visions for me, made me notice that there are some other way way too much cool short solutions that are deep and no direct

@aziz0x00 3 года назад

@@ProfOmarMath awesome man ❤❤❤

@ProfOmarMath 3 года назад

@@aziz0x00 Variables are meant to be kept track of, but we never think to make them do so

@aziz0x00 3 года назад

this video is so good.

@ProfOmarMath 3 года назад

Thanks Aziz!

@aziz0x00 3 года назад

@@ProfOmarMath does the first problem have a name? im really wondering how someone could get that unexpected solution?

@ProfOmarMath 3 года назад

@@aziz0x00 Try and look up “checkerboard domino tiling problem”, and it might come up as that!

@aziz0x00 3 года назад

@@ProfOmarMath yeah i did it and i found it here en.m.wikipedia.org/wiki/Mutilated_chessboard_problem#:~:text=The%20puzzle%20is%20impossible%20to,square%20and%20one%20black%20square.&text=If%20the%20two%20white%20corners,dominoes%2C%20so%20this%20is%20impossible. thanks ❤

@ProfOmarMath 3 года назад

@@aziz0x00 Excellent!

@shohamsen8986 3 года назад

the video needs to be edited. The video and audio are out of sync.

@MrDowntownjbrown 3 года назад

I didn't notice any issues

@shohamsen8986 3 года назад

@@MrDowntownjbrown I think it has been re-edited and uploaded again.

@ProfOmarMath 3 года назад

@@shohamsen8986 Surprisingly I didn’t change anything. Maybe at a certain point it’s out of sync still?

@shohamsen8986 3 года назад

@@ProfOmarMath huh. Thats strange. I was getting the audio and video out of sync from somewhere between 8:55 to 9:15. The audio kept playing, but the video jumped to 10:32 ish, ie the next problem. Maybe it was just local to my computer? I wasn't noticing any issues with the other videos. Anyway, there is no problem anymore, so I guess its okay.

@ProfOmarMath 3 года назад

@@shohamsen8986 That’s wild!