The actual result is the same as the order you ask them. Since we know that two of three give one true (T) and one false (F) statement, and one of them gives two false statements, it can be shown as such:
First: "I am First (T) and Third is second (F)."
Second: "Third is third (T) and I am first (F)."
Third: "Second is third (F) and First is second (F)."
From First, we know that First is first/First (whether it's the position or the name, it works) and Third is not second. Since First is first and Third is not second, Third must be third. From Second, we know that Third is third (which confirms what we know from First) and that Second is not first, meaning that Second must be second. From Third, we know that Second isn't third (because they are second) and First is not second (because they are first). Therefore, First, Second, and Third finished first, second, and third, respectively.
