There are 6400 seats in rows of 32. No row may contain more than four consecutive occupied seats. Find the maximum occupancy.

01 / Standard#Your turn

Official paper · jm01-2022 · I.11 · PDF 3

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Each block of five seats needs an empty seat.

Hint 2

Construct a pattern achieving the upper bound.

Worked solution

The first 30 seats contain six disjoint five-seat blocks, hence at least six vacancies per row.
$\text{occupied per row}\le32-6=26$
Repeat four occupied then one empty six times, then occupy the last two.
$\text{rows}=6400/32=200,\quad\max=26(200)=5200$

B: 5200.

Checks and common pitfalls: The final partial block need not contain another empty seat.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The first 30 seats contain six disjoint five-seat blocks, hence at least six vacancies per row.
$\text{occupied per row}\le32-6=26$
Repeat four occupied then one empty six times, then occupy the last two.
$\text{rows}=6400/32=200,\quad\max=26(200)=5200$

Think first. Reveal a hint when the class is ready.

2022 JM01