Two trains set off at the same time from each end of a single
straight railway line. A very fast bee starts off in front of the
first train and flies continuously back and forth between the two
trains. How far does Sidney fly before he is squashed between the
If it takes four men one day to build a wall, how long does it take
60,000 men to build a similar wall?
Which segment on a digital clock is lit most each day? Which
segment is lit least? Does it make any difference if it is set to
12 hours or 24 hours?
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
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:
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
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).