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 Monkey Dance 1 (Posted on 2004-03-19)
The director of a circus has decided to add a new performance, the monkey dance, to his show.

The monkey dance is danced simultaneously by 21 monkeys.
There are 21 circles drawn on the ground, and in the beginning, each monkey sits on a different circle.
There are 21 arrows drawn from circle to circle in such a way that exactly one arrow starts and exactly one arrow ends in each circle. No arrow can both begin and end at the same circle.

When the show begins, the monkeys dance in their circles until the ringmaster blows his whistle. At each whistle blow, the monkeys simultaneously jump from their circles to the next, following the arrows. The dance ends when all the monkeys have returned to the circles where they initially started.

The director wishes the dance to last as long as possible. What is the maximum number of whistle blows he can make before the dance ends?

 Submitted by Sandeep Rating: 4.1667 (6 votes) Solution: (Hide) 420 If all of the monkeys' circles were in just one large group with 21 circles all connected to each other, then the dance would of course last only for 21 steps. Suppose, however, that there are 10 circles in one group and 11 in another group, with the two groups of circles independent of each other. Then, one group of monkeys would be in their original circles after 10 steps, and on the next whistle blow, they would move on to the next circle while the second group of monkeys returned to their original positions. Thus, it would take 110 steps for all 21 monkeys to be in their original spots at the same time. In general, the total number of steps for a given arrangement is the LCM of the numbers of circles in each group (the LCM of 10 and 11 is indeed 110). For the case of 21, the LCM is maximized with groups of 2, 3, 4, 5, and 7, for a total of 420 steps in the dance.

 Subject Author Date answer K Sengupta 2007-11-23 04:32:03 solution daniel 2004-12-18 00:15:09 3 groups of monkeys Hugo 2004-07-26 03:19:32 monkey dance chris 2004-07-25 19:30:45 No Subject Heather 2004-06-03 17:30:43 answer kevin 2004-05-17 00:39:23 re: solution Robin 2004-05-08 17:23:01 solution Robin 2004-05-08 17:16:12 re(3): No Subject Alec Lanter 2004-04-28 12:22:34 re(2): No Subject Lisa 2004-04-27 17:55:10 re: No Subject Alec Lanter 2004-04-27 16:52:23 No Subject Lisa 2004-04-20 18:07:12 Solution? Mark Rees 2004-03-29 10:13:11 re: Circus Monkeys Seven 2004-03-28 19:20:31 Circus Monkeys Seven 2004-03-28 19:06:11 Monkey Dance 1 Philip 2004-03-25 21:10:33 re(2): steve's solution Steve Royer 2004-03-25 00:30:01 re: steve's solution DJ 2004-03-24 11:51:08 steve's solution Steve Royer 2004-03-24 00:50:19 Novice Jen 2004-03-23 19:19:09 My solution pppy 2004-03-22 23:14:55 My Solution Galendir 2004-03-21 15:31:25 our solution shea 2004-03-20 15:02:27 re(5): My solution Charlie 2004-03-19 23:01:58 re(4): My solution Charlie 2004-03-19 22:57:41 re(3): My solution Gamer 2004-03-19 16:57:13 re(2): My solution Charlie 2004-03-19 16:07:02 re: My solution nikki 2004-03-19 14:43:49 My solution Iain 2004-03-19 14:10:55 10 seconds of thought Larry 2004-03-19 13:51:42

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