1.

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?

Answer»

Solution :Four sittings are needed to fulfill the requirements. A sitting must include all possibleordered 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, thenumber 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 werequire 30 ordered pairs. To show that four concerts are sufficient, number the people 1 to 6and 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


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