, 2023

Each group of four jumps takes the grasshopper 1 cm to the east
and 3 cm to the west, which is a net movement of 2 cm to the west, and 2
cm to the north and 4 cm to the south, which is a net movement of 2 cm
to the south.

In other words, we can consider each group of four jumps, starting with
the first, as resulting in a net movement of 2 cm to the west and 2 cm
to the south.

We note that $158 = 2 $\times$
79$.

Thus, after 79 groups of four jumps, the grasshopper is $79 \times 2 = 158$ cm to the west and
158 cm to the south of its original position. (We need at least 79
groups of these because the grasshopper cannot be 158 cm to the south of
its original position before the end of 79 such groups.)

The grasshopper has made $4 $\times$ 79 =
316$ jumps so far.

After the 317th jump (1 cm to the east), the grasshopper is 157 cm west
and 158 cm south of its original position.

After the 318th jump (2 cm to the north), the grasshopper is 157 cm west
and 156 cm south of its original position.

After the 319th jump (3 cm to the west), the grasshopper is 160 cm west
and 156 cm south of its original position.

After the 320th jump (4 cm to the south), the grasshopper is 160 cm west
and 160 cm south of its original position.

After the 321st jump (1 cm to the east), the grasshopper is 159 cm west
and 160 cm south of its original position.

After the 322nd jump (2 cm to the north), the grasshopper is 159 cm west
and 158 cm south of its original position.

After the 323rd jump (3 cm to the west), the grasshopper is 162 cm west
and 158 cm south of its original position, which is the desired
position.

As the grasshopper continues jumping, each of its positions will always
be at least 160 cm south of its original position, so this is the only
time that it is at this position.

Therefore, $n = 323$. The sum of the
squares of the digits of $n$ is $3^2 + 2^2 + 3^2 = 9 + 4 + 9 = 22$.

Each group of four jumps takes the grasshopper 1 cm to the east
and 3 cm to the west, which is a net movement of 2 cm to the west, and 2
cm to the north and 4 cm to the south, which is a net movement of 2 cm
to the south.

In other words, we can consider each group of four jumps, starting with
the first, as resulting in a net movement of 2 cm to the west and 2 cm
to the south.

We note that $158 = 2 $\times$
79$.

Thus, after 79 groups of four jumps, the grasshopper is $79 \times 2 = 158$ cm to the west and
158 cm to the south of its original position. (We need at least 79
groups of these because the grasshopper cannot be 158 cm to the south of
its original position before the end of 79 such groups.)

The grasshopper has made $4 $\times$ 79 =
316$ jumps so far.

After the 317th jump (1 cm to the east), the grasshopper is 157 cm west
and 158 cm south of its original position.

After the 318th jump (2 cm to the north), the grasshopper is 157 cm west
and 156 cm south of its original position.

After the 319th jump (3 cm to the west), the grasshopper is 160 cm west
and 156 cm south of its original position.

After the 320th jump (4 cm to the south), the grasshopper is 160 cm west
and 160 cm south of its original position.

After the 321st jump (1 cm to the east), the grasshopper is 159 cm west
and 160 cm south of its original position.

After the 322nd jump (2 cm to the north), the grasshopper is 159 cm west
and 158 cm south of its original position.

After the 323rd jump (3 cm to the west), the grasshopper is 162 cm west
and 158 cm south of its original position, which is the desired
position.

As the grasshopper continues jumping, each of its positions will always
be at least 160 cm south of its original position, so this is the only
time that it is at this position.

Therefore, $n = 323$. The sum of the
squares of the digits of $n$ is $3^2 + 2^2 + 3^2 = 9 + 4 + 9 = 22$.