Problem 1 · The class photo
Rows and columns: turning the two rearrangements into equations.
Integer answer, at most 4 digitsA school principal decided to take a photo of the class of 2026. He arranged the students in parallel rows, all with the same number of students, but the arrangement was too wide for his camera's field of view. To fix this, he realised he only had to remove one student from each row and place them in a new row. He didn't like this arrangement because the new row had 6 students fewer than the others. So he removed 1 more student from each row, placing them in the new row, and the students ended up arranged so that all rows had the same number of students; then he took the photo. How many students appeared in the photo?
Reasoned solution
Key idea: let there be $f$ rows of $n$ students. Follow the two operations.
First rearrangement: remove $1$ from each row → the new row has $f$ and the others $n-1$; the new one has $6$ fewer:
Second: remove $1$ more from each of the $f$ rows → the new row has $2f$ and the others $n-2$, and now everything matches:
Total students: $f \cdot n = 5 \cdot 12 = 60$. (Check: $6$ rows of $10$ ✓.)