CEMC Pascal, 2024

We begin by determining the number of inner toothpicks used to
make a $20$ by $24$ grid of squares.

A grid containing $20$ rows has $19$ horizontal lines of inner toothpicks,
each of which contains $24$
toothpicks (since there are $24$
columns).

Thus, the number of inner toothpicks positioned horizontally is $19\times24=456$.

A grid containing $24$ columns has
$23$ vertical lines of inner
toothpicks, each of which contains $20$ toothpicks (since there are $20$ rows).

Thus, the number of inner toothpicks positioned vertically is $23\times20=460$.

In total, there are $456+460=916$
inner toothpicks.

Next, we determine the total number of toothpicks used to make a
$20$ by $24$ grid of squares.

There are $21$ horizontal lines of
toothpicks, each of which contains $24$ toothpicks.

There are $25$ vertical lines of
toothpicks, each of which contains $20$ toothpicks.

Thus, there are a total of $21\times24+25\times20=1004$ toothpicks
used to make a $20$ by $24$ grid.

(Alternately, we could have determined that there are $88$ outer toothpicks, and so there are
$916+88=1004$ toothpicks in
total.)

The percentage of inner toothpicks used is $\frac{916}{1004}\times100\%$, which is
$91\%$ when rounded to the nearest
percent.