Briefly

The seven messengers

The seven messengers

In prince of an extensive kingdom, he left the capital of his kingdom on the day of his 30th birthday to know his domains. He carries with him seven faithful servants, which he will use as messengers to communicate with his father, the king, during the trip. The seven messengers are Alessandro, Bartolomeo, Caio, Domenico, Ettore, Federico and Gregorio.

At 24 hours after the game, he sends Alessandro with a message for his father. The messenger arrives in the capital, delivers the message, picks up the one they had prepared there for the prince and without major breaks part to reach him, who has continued his journey.

As soon as he reaches it, the prince reads the message and immediately sends his response through Bartolomeo, who follows the same process as Alessandro.

Each messenger marches at a speed that is one and a half times that of the prince's entourage, so he receives increasingly spaced news from the capital of the kingdom.

Time passes, and the prince is wearing away his life without ever reaching the end of his kingdom. At the moment of sending Gregorio, the last messenger, it is certain that both are seen for the last time in his life.

How long would it take from the start of the trip until Gregorio's return?

Solution

The time that the persecutor (the messenger) takes to reach the persecuted (the prince and his entourage) is the distance that initially separates them divided by the difference in speeds (since each unit of time brings them an amount equal to that difference).

As the first messenger must go and return to the capital of the kingdom, it is as if the initial distance between the persecutor (Alessandro) and the persecuted (the prince) was twice that existing at the time of the separation of both. Therefore, the time it will take to produce the match will be 4 times that distance. In other words, the scope is produced in 5 times the time of departure.

It is immediate that this result is generalizable. Therefore, Bartolomeo reaches the prince at 5 × 5 = 52 = 25 days since he leaves at 24 hours of travel.

Caio, at 53 = 125 days

Domenico, at 54 = 625 days. Twenty months!

Ettore at 55 = 3,125 days. Eight and a half years!

Federico at 56 = 15,625 days, that is almost forty-three years! The prince is already an old man, and he certainly has no hope of seeing Gregorio's return, which would occur, if he came to live so long, at 57 = 78,125 days, that is 214 years!