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Dragon Hunting (Posted on 2004-12-10)

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
answer	K Sengupta	2008-02-21 14:26:20
No Subject	Robyn	2005-04-05 21:38:53
re: Solution	Michael	2005-04-05 01:38:07
re: Solution-robyn	Michael	2005-04-05 01:36:56
Solution	artemis	2005-03-05 17:11:42
Solution	Robyn	2005-02-28 06:04:39
re: Solution	Hugo	2005-01-28 21:44:51
Solution	Stephen S.	2005-01-28 20:35:24
No Subject	adam	2005-01-13 22:28:54
re: Is it really this easy?	nikki	2005-01-10 19:52:55
re: Is it really this easy?	Canute	2005-01-10 15:27:20
Is it really this easy?	Canute	2005-01-10 15:24:23
i might have it	sarah	2004-12-22 05:24:28
re: A Solution	anton	2004-12-17 17:19:16
re: think i may have it	audrey	2004-12-17 09:42:34
think i may have it	audrey	2004-12-17 09:24:04
PETA	Penny	2004-12-13 09:04:38
Dear Nate: re(2): Biologic not math answer	Larry	2004-12-13 03:16:04
re(3): Biologic not math answer	Penny	2004-12-13 00:46:46
re(2): Biologic not math answer	Dustin	2004-12-12 22:31:59
re: Biologic not math answer	nate	2004-12-12 22:10:58
re(3): A stroke of genius	owl	2004-12-10 22:07:25
re(2): A stroke of genius	Larry	2004-12-10 22:06:24
re: A stroke of genius	owl	2004-12-10 21:44:50
re: A stroke of genius	Timothy Bard	2004-12-10 20:58:13
re: A stroke of genius	Michael Cottle	2004-12-10 16:57:57
If temporary incapacitation is not death, then...	Charlie	2004-12-10 16:49:54
A stroke of genius	Penny	2004-12-10 16:46:15
Biologic not math answer	Larry	2004-12-10 16:03:43
Proof of Minimum (I think)	nikki	2004-12-10 15:51:46
re(3): A Solution... I think	Timothy Bard	2004-12-10 15:40:50
re(3): A Solution... I think	Eric	2004-12-10 15:36:49
re(2): A Solution... I think	Erik O.	2004-12-10 15:28:26
re: A Solution... I think	Hugo	2004-12-10 15:15:10
A Solution... I think	Timothy Bard	2004-12-10 15:12:40
Solution	Eric	2004-12-10 15:01:10
A Solution	Dustin	2004-12-10 14:54:16

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