Cross the pond, but there's a catch!

Question

There is a square pond, conveniently divided into segments, with coordinate $(0,0)$ in the bottom left and $(10,10)$ is the top right.

You have planks length $2$ and $3$. You start at $(0,0)$ and must get to $(10,10)$. Each plank is to be laid either horizontally or vertically, and bottom-left to top-right (no going backwards).

A similar question, with an answer as to how to approach this problem, is here.

The catch is that this pond has a very dangerous whirlpool at coordinates $(5,5)$, and this square cannot be landed on, but it can be passed over.

How many ways can we build a bridge across this pond?

Answer 1

So, there is actually an approach by hand (without calculating via a computer program or creating a $10 \times 10$ table of a recurrence relation).

Observe that to get a distance of $10$ by using planks of length $2$ and $3$:

We are using exactly $4$ planks, except a case of using $5$ $2$'s. If we are using $4$ planks, we will use exactly $2$ $2$'s and $2$ $3$'s, making $\binom{2+2}{2}=6$ combination cases.

Thus, from $(0,0)$ to $(10,10)$, the number of ways — including going into the whirlpool — is:

- Using $5$ planks on horizontal and $5$ planks on vertical: $\binom{5+5}{5} = 252$.
- Using $5$ planks on horizontal and $4$ planks on vertical: $\binom{5+4}{5} \times 6 = 756$.
- Using $4$ planks on horizontal and $5$ planks on vertical: $\binom{5+4}{4} \times 6 = 756$.
- Using $4$ planks on horizontal and $4$ planks on vertical: $\binom{4+4}{4} \times 6 \times 6 = 2520$.

Overall, there will be $252+756+756+2520=4284$ ways.

Now, we may then count how many ways of going from $(0,0)$ to $(10,10)$ and landing on $(5,5)$. It's simply:

Count the number of ways going from $(0,0)$ to $(5,5)$, multiplying the number of ways going from $(5,5)$ to $(10,10)$.

Again, using a similar observation to get a distance of $5$ and going from $(0,0)$ to $(5,5)$ (and also $(5,5)$ to $(10,10)$):

We are using exactly $2$ planks: $1$ $2$'s and $1$ $3$'s, making $2$ possible cases.
Using $2$ planks on horizontal and $2$ planks on vertical: $\binom{2+2}{2} \times 2 \times 2 = 24$ ways.

So, the final answer will be:

$4284 - (24 \times 24) = 3708$ ways.

Answer 2

Using the following program I get

3708

let result = 0;
go(0, 0);
function go(x, y) {
    if(x === 10 && y === 10) {
        result++;
        return;
    }
    if(x > 10 || y > 10 || (x === 5 && y === 5)) {
        return;
    }
    go(x + 2, y);
    go(x + 3, y);
    go(x, y + 2);
    go(x, y + 3);
}
console.log(result);