Crossing the atlantic
Problem
It is believed that this problem was invented by Edouard Lucas and that he also invented the well known Frogs problem. He was a French mathematician who lived in the nineteenth century.
Every day at noon (Greenwich Mean Time) a boat leaves Le Havre
for New York while another boat leaves New York for Le Havre.
The ocean crossing takes seven days and seven nights.
How many boats on the New York to Le Havre route will the boat
leaving Le Havre today meet during its journey to New York?
Does it change your answer if the boats leave Le Havre and New York
at noon local time. How can you illustrate your answer?
Student Solutions
Was this one so difficult? No solutions came in for nearly a year then two arrived within a few days of each other from Gordon, Madras College, Scotland and Allan, Tao Nan School, Singapore, both well known to NRICH as ace problem solvers.
The boat leaving Le Havre will meet 13 boats from New York at sea, and one in Le Havre (arriving as it departs) and one in New York (departing as it arrives), that is 15 boats altogether.
One way to illustrate this is using a number line, the numbers from -7 to -1 corresponding to the boats that have left New York BEFORE our boat departs (actually giving the number of days earlier), the number 0 corresponding to the boat that leaves AT THE SAME TIME, and the numbers +1 to +7 corresponding to the boats that leave AFTER our boat departs from Le Havre.
Does it change the answer if the boats leave New York and Le Havre at noon local time?
Allan argues that the boat from Le Havre meets a boat coming in the opposite direction every half day, 14 boats, and in addition the one that is arriving in Le Havre when it starts its journey making 15 boats altogether.
Gordon argues as follows: Each boat travels at the same velocity, and starts its journey at a constant interval after the previous one. Consider two buoys in the Atlantic, one called New York and the other called Le Havre; consider the Atlantic to be infinite. When the ship from Le Havre sets off a satellite view might appear thus:
| NY | LH | |||||||||||||
| Boats leaving New York | ||||||||||||||
| 15 | 14 | 13 | 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |
After 7 times 24 hours the procession of boats from New York will have moved 7 to the right , the boat from Le Havre 7 to the left
| NY | LH | |||||||||||||
| Boats leaving New York | ||||||||||||||
| 15 | 14 | 13 | 12 | 11 | 10 | 9 | 8 | |||||||
If the boats leave the ports at the same instant (universal time) then the Le Havre boat meets 15 boats from New York because the boat from Le Havre meets boat number 1 in Le Havre, passes 13 boats and meets boat number 15 at New York. If the boats actually leave the ports at different times because of time zones it only meets 14 boats.
If the distance between Le Havre and New York is D nautical miles then the boats from New York are spaced out at distance D /7 apart travelling at D /(7x24) knots. The relative velocity is D /(7x12) knots and the boats pass every 12 hours. (Imagine the boats going from New York to Le Havre stopped and the boat from Le Havre going twice the velocity so covering a distance 2 D ).
This is like having a ruler of length 2 D with 15 marks for the procession of boats from New York to Le Havre at intervals of D /7. If you put one end on each port then 15 boats meet, otherwise only 14 (where there is any time difference between noon local time at the two ports).