## Changing Seats

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

SOLUTION
Suppose there are $6$ students named $A,B,C,D,E$, and $F$ seated in $2$ rows and $3$ columns
$A\:B\:C$
$D\:E\:F$
One possible way for them to change seats
$A\leftrightarrow B$ (left, right)
$D\leftrightarrow E$ (left, right)
$C\updownarrow F$ (front, back)

Suppose there are $9$ students seated in $3$ rows and $3$ columns
$A\:B\:C$
$D\:E\:F$
$G\:H\:I$
If $8$ of the students changed seats
$A\leftrightarrow B$ (left, right)
$D\leftrightarrow E$ (left, right)
$C\updownarrow F$ (front, back)
$G\leftrightarrow H$ (left, right)
then student $I$ 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 $35$ 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.

Alternative solution
Imagine the seats are represented by a checkerboard made up of $18$ black squares and $17$ white squares arranged in $5$ rows and $7$ columns. Each of the $18$ students seated in a black square must move to a white square. However, only $17$ white squares are available.

Advertisements

## About mvtrinh

Retired high school math teacher.
This entry was posted in Problem solving and tagged , , , , , . Bookmark the permalink.