Correct solutions were also received from Jan Siwanowicz and Thomas Teo. Two solutions were received which distributed the information correctly, but required more than the minimal number of steps.
First note that the number of people who know any one piece of information can at most double on any given day. This means that the information cannot be distributed in fewer than 4 days, since 4 is the smallest positive integer, k, satisfying 11 < 2k.
There are various schemes for exchanging the information. Here is one possible one.
Think of the islands as being numbered from 1 to 11, as on a clock, so that 1 follows 11. On the first day, everyone sends his (I think all mutinous pirates are men, aren't they?) information to the next island, that is, numbered one greater; on the second day, everyone sends his two pieces of information two islands further on; on the third day, four islands further on; and on the fourth day, eight islands further on. For example, the pirate on island 8 would send his various pieces of information to islands 9, then 10, then 1, and finally 5.
Why does this work? Since the scheme is the same for all the participants, we only need to see that the first piece of information is universally distributed. Look at island i, and write i -1 as a sum of powers of 2. The powers appearing will tell you how the message got from island 1. For example, if i = 6, write 5 = 4 + 1; the message from 1 arrived at island 6 by going first to island 2 via a single island flight, followed by a four island flight.
Exactly the same procedure solves the problem for 12 mutineeers.
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