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Dragon Hunting (Posted on 2004-12-10) Difficulty: 3 of 5
Prince Valiant went to fight a 3-headed, 3-tailed dragon.

He has a magic sword that can, in one stroke, chop off either one head, two heads, one tail, or two tails.

This dragon is of a type related to the hydra; if one head is chopped off, a new head grows. In place of one tail, two new tails grow; in place of two tails, one new head grows; if two heads are chopped off, nothing grows.

What is the smallest number of strokes required to chop off all the dragon's heads and tails, thus killing it?

  Submitted by SilverKnight    
Rating: 3.7778 (9 votes)
Solution: (Hide)
All of the tails must be converted to heads in such a way that the dragon is left with an even number of heads. The way to do this in the smallest number of strokes would be to first convert the 3 tails to 6 tails by using 3 1-tail strokes, then converting the 6 tails to 3 heads via 3 2-tail strokes. Finally, the now 6-headed dragon can be killed with 3 2-head strokes, for a total of 9 strokes.

Another solution is to let the number of 1-head strokes be a, 1-tail strokes be b, 2-head strokes be c, and 2-tail strokes be d. Then the number of heads h after these strokes is h = 3-2c+d; tails t is t = 3+b-2d. If both are 0 we have that 2c = d+3 and 2d = b+3. It's clear that d must be odd, so let d = 2m+1, then c = m+2 and b = 4m-1. The least number of strokes occurs when a=0 and m is the smallest integer that makes b, c, d non-negative, and so m=1 implies b=3, c=3, d=3, giving us the same answer as before.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
answerK Sengupta2008-02-21 14:26:20
SolutionNo SubjectRobyn2005-04-05 21:38:53
re: SolutionMichael2005-04-05 01:38:07
re: Solution-robynMichael2005-04-05 01:36:56
SolutionSolutionartemis2005-03-05 17:11:42
SolutionSolutionRobyn2005-02-28 06:04:39
re: SolutionHugo2005-01-28 21:44:51
SolutionSolutionStephen S.2005-01-28 20:35:24
QuestionNo Subjectadam2005-01-13 22:28:54
re: Is it really this easy?nikki2005-01-10 19:52:55
re: Is it really this easy?Canute2005-01-10 15:27:20
SolutionIs it really this easy?Canute2005-01-10 15:24:23
i might have itsarah2004-12-22 05:24:28
Hints/Tipsre: A Solutionanton2004-12-17 17:19:16
re: think i may have itaudrey2004-12-17 09:42:34
think i may have itaudrey2004-12-17 09:24:04
PETAPenny2004-12-13 09:04:38
Dear Nate: re(2): Biologic not math answerLarry2004-12-13 03:16:04
re(3): Biologic not math answerPenny2004-12-13 00:46:46
re(2): Biologic not math answerDustin2004-12-12 22:31:59
re: Biologic not math answernate2004-12-12 22:10:58
re(3): A stroke of geniusowl2004-12-10 22:07:25
re(2): A stroke of geniusLarry2004-12-10 22:06:24
re: A stroke of geniusowl2004-12-10 21:44:50
re: A stroke of geniusTimothy Bard2004-12-10 20:58:13
re: A stroke of geniusMichael Cottle2004-12-10 16:57:57
SolutionIf temporary incapacitation is not death, then...Charlie2004-12-10 16:49:54
SolutionA stroke of geniusPenny2004-12-10 16:46:15
Biologic not math answerLarry2004-12-10 16:03:43
SolutionProof of Minimum (I think)nikki2004-12-10 15:51:46
re(3): A Solution... I thinkTimothy Bard2004-12-10 15:40:50
re(3): A Solution... I thinkEric2004-12-10 15:36:49
re(2): A Solution... I thinkErik O.2004-12-10 15:28:26
re: A Solution... I thinkHugo2004-12-10 15:15:10
SolutionA Solution... I thinkTimothy Bard2004-12-10 15:12:40
SolutionEric2004-12-10 15:01:10
SolutionA SolutionDustin2004-12-10 14:54:16
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