Your logic is missing a temporal element check. Before any runners have completed the race, there are 0 runners in the set of runners that have completed the r

Your logic is missing a temporal element check. Before any runners have completed the race, there are 0 runners in the set of runners that have completed the race (empty set/null set). As soon as the first runner crosses the finish line 1 runner has finished the race, increasing the counter by one increment and it is customary to assign them the designation of the upper end of the unit range rather than the lower number. Thus, they are the First (1st) runner to have completed the race. We count rooms the same way probably out of a similar convention, the first room on a floor is numbered 1, rather than 0, because it is more useful to annotate the end of the range than the start. It makes a lot of math easier too. If there are 10 rooms on a floor and 10 guests, there are 0 rooms free.