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Number of lines | New rectangles |
1 | 0 |
2 | 0 |
3 | 0 |
4 | 1 |
5 | 1 |
6 | 2 |
7 | 2 |
8 | 3 |
$a$ | $b$ | $(a-1)(b-1)$ |
---|---|---|
$1$ | $14$ | $0 \times 13 = 0$ |
$2$ | $13$ | $1 \times 12 = 12$ |
$3$ | $12$ | $2 \times 11 = 22$ |
$4$ | $11$ | $3 \times 10 = 30$ |
$5$ | $10$ | $4 \times 9 = 36$ |
$6$ | $9$ | $5 \times 8 = 40$ |
$7$ | $8$ | $6 \times 7 = 42$ |
Therefore, the largest number is $42$ rectangles, formed by having seven lines in one direction and eight in the other.
Alternatively, you can use completing the square to maximise the quantity:
Since $a+b=15$, $(a-1)(b-1) = (a-1)(14-a) = -a^2+15a-14$. Then, by completing the square, this is $-\left(a-\frac{15}{2}\right)^2 + \frac{169}{4}$.
This is maximised when the square is minimised, which occurs when $a=7$ or $a=8$ (since $a$ must be an integer). This gives $6 \times 7 = 42$ rectangles.