How many Rectangles?
By drawing 5 horizontal and four vertical lines, one can form 12 rectangles. What is the greatest number of rectangles that can be formed by drawing 15 lines?
Age
11 to 14
Challenge level
Problem
Image
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Answer: 42
The sequence of patterns
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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 |
9th line adds 3 rectangles (already seen that 9 lines make12 rectangles)
10th, 11th lines +4 each total 12 + 8 = 20
12th, 13th lines +5 each total 20 + 10 = 30
14th, 15th lines +6 each total 30 + 12 = 42
An expression for the number of rectangles
Suppose there are $a$ horizontal and $b$ vertical lines. The grid of rectangles formed is then $a-1$ rectangles high, and $b-1$ rectangles wide. This means there are $(a-1)(b-1)$ rectangles.
If there are a total of $15$ lines, the aim is to make $(a-1)(b-1)$ as large as possible with $a+b=15$.
This can be done by considering the different combinations that add to make $15$:
| $a$ | $b$ | $(a-1)(b-1)$ |
|---|---|---|
| $1$ | $14$ | $0 \times 13 = 0$ |
| $2$ | $13$ | $1 \times 12 = 12$ |
| $3$ | $12$ | $2 \times 11 = 22$ |
| $4$ | $11$ | $3 \times 10 = 30$ |
| $5$ | $10$ | $4 \times 9 = 36$ |
| $6$ | $9$ | $5 \times 8 = 40$ |
| $7$ | $8$ | $6 \times 7 = 42$ |
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.