(CAUTION- PLEASE DO NOT TAKE MY QUESTIONS VERY SERIOUSLY.
I received a ban from asking questions,I don't know what to say really,I am just a student trying to learn,not a professional mathematician,so of course the questions could have been,well weird or useless to the mathematical community...I really did not mean any harm or anything except just trying to understand something.I really wish a feature like,telling you exactly what to do with your questions so the ban would get lifted was there.But I guess this is just hard luck and me not taking this community very seriously.I do apologize to the community,and I do understand that asking a question is a privilege not a right here.because,like to stop wasting people's time and not spread wrong ideas?right? I REPEAT,I DID NOT MEAN ANY CONFUSION OR ANYTHING,I AM NO PROFESSIONAL,JUST A LEARNER.)
There are 25 people sitting around a table, and each person has two cards. One of the numbers 1 , 2, 3, . . . , 25 is written on each card, and each number occurs on exactly two cards. At a signal, each person passes one of her cards-the one with the smaller number-to her right-hand neighbor. Prove that, sooner or later, one of the players will have two cards with the same numbers.
I was reading this question from the book the art and craft of problem solving by paul zeitz.
now,it said in the book that 2 people can never have same cards for an even number.and went to show that with examples... what confused me was that why did the author say that each person has to be given a card from the set of even numbers from one half like {1,2} and the other card from {3,4}
this is an imitation of the book's solution since i don't know how to write it the way its written in book,but i hope you will get my point and confusion...
can't we have an arrangement of cards to 4 people like-
(1,2) (1,2) (3,4) (3,4)
which on first passing round would become
(3,2) (1,2) (1,4) (3,4)
now we sort the cards so top row number is less than bottom-
(2,3) (1,2) (1,4) (3,4)
which after another passing round becomes-
(3,3) (2,2) (1,4) (1,4)
now we have people having two same numbered card,but the books says that can't be true for even numbers,i can see so if you give each person the first card from say the first half set of even numbers and the second card from the second half set of even numbers that is {1,2} and {3,4} but the question doesn't really say it has to be that way only or have i missed or misinterpreted something?if you get the solution in book and my confusion,please do enlighten since i know there must be something wrong i am doing here but i don't know really how.
also it says that for odd numbers,the top and bottom row stop "mixing".what does that mean and how?
thank you for your time and answer.