×

# Perimeter-like Grids

Page 9

## Perimeter and Semiperimeter

In general, the perimeter of a $$a\times b$$ grid is $$2a+2b$$, the semiperimeter of a $$a\times b$$ grid is $$a+b$$.

For example, the perimeter of a $$3\times 5$$ grid is 16, its semiperimeter is 8.

## Perimeter-like Grids

If two grids have the same perimeter, then these 2 grids are "perimeter-like grids".

For example, $$5\times 2$$ grid and $$3\times 4$$ grid are perimeter-like grids.

## Maximum number of unit squares

Suppose there's a grid, $$a+b=10$$, if the number of unit squares in the grid reaches its maximum, then what is the value of $$a$$ and $$b$$?

Let's write out the possibilities: $1\times9,~~~~2\times8,~~~~3\times7,~~~~4\times6,~~~~5\times5,~~~~6\times4,~~~~7\times3,~~~~8\times2,~~~~9\times1$ The corresponding numbers of unit squares are: $9,~~~~~~16,~~~~~~21,~~~~~~24,~~~~~~25,~~~~~~24,~~~~~~21,~~~~~~16,~~~~~~9$ We see that when $$a=b=5$$, then the number of unit squares reaches its maximum.

We say that the maximum number of unit squares it can form when $$a+b=10$$ is 25.

What if $$a+b$$ is an odd number? Let's try this out...

Suppose $$a+b=9$$, the possibilities are: $1\times8,~~~~2\times7,~~~~3\times6,~~~~4\times5,~~~~5\times4,~~~~6\times3,~~~~7\times2,~~~~8\times1$ The corresponding numbers of unit squares are: $8,~~~~~~14,~~~~~~18,~~~~~~20,~~~~~~20,~~~~~~18,~~~~~~14,~~~~~~8$ We see that when $$a=b\pm1$$, then the number of unit squares reaches its maximum.

We say that the maximum number of unit squares it can form when $$a+b=9$$ is 20.

## The Greatest Rule

If $$a+b$$ is a fixed even number, then when $$a=b$$, the number of unit squares it can form in the grid will reach its maximum.

Another way of thinking about this is among all the perimeter-like grids which $$a+b$$ is even, then the grids which has the property of $$a=b$$ will have the most number of unit squares.

If $$a+b$$ is a fixed odd number, then when $$a=b\pm1$$, the number of unit squares it can form in the grid will reach its maximum.

Another way of thinking about this is among all the perimeter-like grids which $$a+b$$ is odd, then the grids which has the property of $$a=b\pm1$$ will have the most number of unit squares.

Proof:
Consider $$a+b$$ is even. Suppose $$a+b=2k$$, $$a=k+x$$, $$b=k-x$$ ($$k$$ is a positive integer while $$x$$ is an integer). The number of unit squares in the grid is \begin{align} a\times b&=(k+x)(k-x) \\&=k^2-x^2 \end{align} Since $$x^2\geqslant 0$$, to maximize the value of $$k^2-x^2$$, $$x$$ must be equal to 0, thus $$a=b=k$$.
Consider $$a+b$$ is odd. Suppose $$a+b=2k+1$$, $$a=k+x$$, $$b=k+1-x$$. The number of unit squares in the grid is \begin{align} a\times b&=(k+x+1)(k-x) \\&=k^2+k-x^2-x \\&=k^2+k-x(x-1) \end{align} Since $$x(x-1)\geqslant 0$$, to maximize the value of $$k^2+k-x(x-1)$$, $$x$$ must be equal to 0 or 1, thus $$a=k$$, $$b=k+1$$ or $$a=k+1$$, $$b=k$$, $$a=b\pm1$$.

Generalization:

If $$a+b$$ is a fixed even number, then when $$a=b=k$$, the number of unit squares in the grid will reach its maximum, which is $$ab=k^2=\left (\frac{a+b}{2} \right )^2$$.

$$\bullet$$ Hence if $$a+b$$ is even, the maximum number of unit squares it can form is $$\left (\frac{a+b}{2} \right )^2$$.

If $$a+b$$ is a fixed odd number, then when $$a=k+1$$, $$b=k$$ or when $$a=k$$, $$b=k+1$$, the number of unit squares in the grid will reach its maximum, which is $$ab=k(k+1)=\left (\frac{2k}{2} \right )\left (\frac{2k+2}{2} \right )=\frac{(a+b-1)(a+b+1)}{4}$$.

$$\bullet$$ Hence if $$a+b$$ is odd, the maximum number of unit squares it can form is $$\frac{(a+b-1)(a+b+1)}{4}$$.

## Worked examples

If $$a+b=14$$, what is the maximum number of unit squares it can form?

Solution:

Since 14 is even, according to the greatest rule, the maximum number of unit squares it can form is $$\left( \frac{a+b}{2} \right)^2=7^2=49$$.

If $$a+b=19$$, what is the maximum number of unit squares it can form?

Solution:

Since 19 is odd, according to the greatest rule, the maximum number of unit squares it can form is $$\frac{(a+b-1)(a+b+1)}{4}=\frac{18\times 20}{4}=90$$.

This is one part of Grids and Quadrilaterals.

Note by Tan Kenneth
2 years, 6 months ago