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How Many Rectangles?

Age 11 to 14 Short Challenge Level:

Answer: 42


The sequence of patterns
 
Number of lines New rectangles
  1    0 
  2    0 
  3    0 
  4    1 
  5    1 
  6    2 
  7    2 
  8    3 
You add more rectangles horizontally, then vertically, so the pattern continues.
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.

 

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.