Every bulb can be switched by its previous bulb,by itself and by bulb following it .
So if we switch all bulbs sequentially,a bulb in between changes its state 3 times,which results in chng of its state from initial(3%2=1).
If all bulbs are switched from 1 to n sequentially all bulbs, including 1 and n
change there state .
Therefore to switch all bulbs on we only need to switch them on sequentially,
irrespective of their n value.

We want to use as little fuel as possible so we try minimize the number of trucks we use as we go along. Let’s say we start with all 50 trucks with full fuel (5000 miles range). For each mile, we lose 50 miles in range. After two miles, we lose 100 miles leaving us with 4900 miles. This can be supported by 49 trucks so we drop one truck.
As you can see for every 100 miles we lose in range, we drop a truck.

50 trucks: 100/50
49 trucks: 100/49
…

Total distance = 100/50 + 100/49 + 100/48 + … + 100/2 + 100/1 (harmonic series) = 449.920533833

The man in front knew he was wearing a black hat because he knew the first man did not see two white hats and he knew that the second man did not see one white hat because if he saw a white hat, the second man would have known that his hat was black from hearing the first man’s statement.