Few distance into the rough sea, they decide to call it a day. Tintin and five of his comrades decide to take turns in controlling their ship. In each
Few distance into the rough sea, they decide to call it a day. Tintin and five of his comrades decide to take turns in controlling their ship. In each ‘sitting’, some of them sleep while the others control the ship. How many such ‘sittings’ are needed so that every person has a chance to control the ship to every other person sleeping?
1 Answers
Four sittings are needed to fulfill the requirements. A sitting must include all possible ordered pairs (a, p), where a are the ones sleeping while p are the ones controlling the ship. There are 6×5 = 30 such ordered pairs. In any sitting, if exactly m people to control the ship, the number of ordered pairs covered is m(6 − m). This is maximised if m = 3 and m(6 − m) = 9. Hence three sittings can cover at most 3 × 9 = 27 ordered pairs. This is insufficient since we require 30 ordered pairs. To show that four concerts are sufficient, number the people 1 to 6 and use the following construction. Controlling the Ship:
456
235
136
124
Sleeping:
123
146
245
356
It is easy to check that every ordered pair is covered by this construction.
Hence, the Answer is 4