If we combine the 51 and 49, we get two piles of 100 and 5 stones. Here, no matter what, we will have to end up with a number of stones divisible by 5 at a point (when you combine, that ensures 0 mod 5, and when you divide by 2, that doesn't change it mod 5) so we cannot end with 105 piles of one stone each. Similarly, if we combine the 51 and 5, we get two piles of 56 and 49, whose gcd is 7, and if we combine the 49 and 5, we get two piles of 51 and 54, whose gcd is 3. None of them have gcd of 1, so the result is impossible.

Notice that only when a number is a power of $2$, it can be divided in half repeatedly until we are left with $1$. So only piles with the number of stones being a power of $2$ can be split into some number of $1$-stone piles. It follows that it is impossible to get a $1$-stone pile from a pile that isn't a power of $2$. So since it is impossible to express $105$ as a sum of $2$'s raised to a positive power, it is also not possible to get $105$ piles with one stone in each.