There are problems that are solved not with more effort, but with a single idea that inverts everything. This is one of them.
The impossible race
Riddle statement
A sultan promises his inheritance to one of his two sons, but imposes a strange condition:
it will be won by the owner of the horse that arrives the latest in the neighboring city.
As soon as they hear the rule, the two princes stop moving. Neither wants their horse to arrive before the other's.
Time passes and, seeing that the race does not start, they consult a wise woman from the desert. She tells them a single sentence.
Immediately, the two mount up and leave at full speed.
What did she say to them?
Show solution
Solution
If each prince rides his own horse, both want that horse to arrive later, so neither has an incentive to advance.
But if they swap horses, each starts riding the other's horse. To win the inheritance, what is best for each prince is for his own horse—the one his brother is now riding—to arrive later than the other's.
That means that each one has an incentive to make the horse he is riding arrive as soon as possible.
The wise woman told them to exchange horses.