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Monkey Dance 1 (Posted on 2004-03-19) Difficulty: 3 of 5
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.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
answerK Sengupta2007-11-23 04:32:03
Solutionsolutiondaniel2004-12-18 00:15:09
3 groups of monkeysHugo2004-07-26 03:19:32
monkey dancechris2004-07-25 19:30:45
No SubjectHeather2004-06-03 17:30:43
answerkevin2004-05-17 00:39:23
re: solutionRobin2004-05-08 17:23:01
SolutionsolutionRobin2004-05-08 17:16:12
re(3): No SubjectAlec Lanter2004-04-28 12:22:34
re(2): No SubjectLisa2004-04-27 17:55:10
re: No SubjectAlec Lanter2004-04-27 16:52:23
No SubjectLisa2004-04-20 18:07:12
Solution?Mark Rees2004-03-29 10:13:11
re: Circus MonkeysSeven2004-03-28 19:20:31
Some ThoughtsCircus MonkeysSeven2004-03-28 19:06:11
Monkey Dance 1Philip2004-03-25 21:10:33
re(2): steve's solutionSteve Royer2004-03-25 00:30:01
re: steve's solutionDJ2004-03-24 11:51:08
Solutionsteve's solutionSteve Royer2004-03-24 00:50:19
SolutionNoviceJen2004-03-23 19:19:09
SolutionMy solutionpppy2004-03-22 23:14:55
SolutionMy SolutionGalendir2004-03-21 15:31:25
our solutionshea2004-03-20 15:02:27
re(5): My solutionCharlie2004-03-19 23:01:58
re(4): My solutionCharlie2004-03-19 22:57:41
re(3): My solutionGamer2004-03-19 16:57:13
re(2): My solutionCharlie2004-03-19 16:07:02
Solutionre: My solutionnikki2004-03-19 14:43:49
SolutionMy solutionIain2004-03-19 14:10:55
Some Thoughts10 seconds of thoughtLarry2004-03-19 13:51:42
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