Thirty-five students are seated in five rows and seven columns. Is it possible for the students to change seats if every student must move exactly one seat to the left, right, front or back?
Source: NCTM Mathematics Teacher, January 2006
Suppose there are students named , and seated in rows and columns
One possible way for them to change seats
Suppose there are students seated in rows and columns
If of the students changed seats
then student is left alone unable to change seat with anyone.
This fact tells us that in order to change seats the number of student must be even. Since there are students, they cannot change seats if every student must move exactly one seat to the left, right, front or back.
Answer: No, it is not possible.
Imagine the seats are represented by a checkerboard made up of black squares and white squares arranged in rows and columns. Each of the students seated in a black square must move to a white square. However, only white squares are available.