March 2000 - Puzzle of the Month!
| March 23, 2000 | A mutiny takes place on a
pirate ship. Wanting to punish the mutineers and regain control of the ship, the captain
puts the disloyal crew members off on different small islands. The mutineers soon discover
that communication between themselves is possible by sending messages attached to carrier
pigeons. However, each mutineer can only communicate with one other on any given day. Suppose each of them has some unique piece of information he would like all the others to know and that on any given day he can pass along all the information he has to his chosen recipient. There are eleven mutineers. How many days will it take before each of them has gathered all the information? |
| Answer: Karen Kolecke, Dusty Grundmeier |
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 their 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. |
| March 16, 2000 | A two digit number is divided by the sum of its digits. The answer is 6. What is the number? |
| Answer: NOBODY :( |
Let the digits be a, b, so the 2 digit
number is 10 x a + b, and the sum of the digits is a + b thus 10a + b = 6(a + b) � 4a = 5b The only integer solutions with 0 � a, b � 9 are a = 0, b = 0 (which is not acceptable) and a = 5, b = 4. Thus the original number is 54. |
| March 9, 2000 | (30 + 25)2 = 3025. What other 4-digit occurrences of this are there? |
| Answer: Dusty Grundmeier |
(20 + 25)2 = 2025 (98 + 01)2 = 9801 |