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[1800GAMBLER]

There are 25 students in a tutorial. The classroom has 5 rows of desks, with 5 desks
in each row. The tutor asks each student to change his or her seat by going either to
seat in front, behind, to the left or to the right (of course, not all these possibilities
will be available to every student). Explain why this is not possible.

[ctide]

<font color="white"> The last student to move will never have a seat available. The easiest way to visualize this would be to form a concentric circle around the classroom, with each student shifting one seat. You will always have one student with nowhere to go. You're going to have this problem with any configuration of switches.</font>

[1800GAMBLER]

That's not really math proof.

[felson]

<font color="white"> Pretend that each student weighs the same, and consider the center of mass of the classroom. After everyone changes seats, the center of mass remains fixed.

The number of students moving right must equal the number of students moving left, or the classroom would tip over. (This would not be true if the students could move more than one seat, since one student moving 2 seats to the left could counteract two students each moving 1 seat to the right.)

The same is true for students moving forward or back.

So, the total number of students moving left/right is even. The same is true for the students moving forward or back. Everyone moves in one of the directions. Therefore, the total number of students must be even. But there are 25 students. Therefore, the rearrangement is impossible.
</font>