To add to the sibling: if you have infallible advice that there's a solution, but you have ruled out all possible solutions, you know you've made a mistake, and

To add to the sibling: if you have infallible advice that there's a solution, but you have ruled out all possible solutions, you know you've made a mistake, and you can review your logic. It basically serves as a checksum on your thinking. If the advice isn't infallible (worst case of which is the stuck bit you describe), then you don't have that checksum.