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    • Nosferatu

      Kill the vampire

      18 June, 2012
      Nosferatu

      There is a vampire who lives in a castle with four underground vaults arranged in a line. A vampire hunter is on the prowl and to avoid her, the vampire sleeps in a different vault every day. But he is bound by a magic spell: he can only choose a vault that is directly adjacent to the vault he slept in the previous day. The vampire hunter can go to one vault every day. If she finds the vampire, she'll kill him. If not, she'll have to wait another day.

      Is there a sequence of vaults the hunter can choose to guarantee she'll eventually find the vampire? Can you find a strategy for the general problem with n rooms?



      This puzzle was suggested to us by Christian Perfect. He was told about it by David Cushing who traced it back to Mathoverflow.

      Solution link
      Kill the vampire: Solution
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      Anonymous

      18 June 2012

      Permalink
      Comment

      Let's note the vaults 1, 2, 3, 4.

      Day 1: Visit vault 2
      Day 2: Visit vault 2
      Day 3: Visit vault 3
      Day 4: Visit vault 3
      Day 5: Visit vault 2

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      Anonymous

      19 June 2012

      In reply to Maximum 5 days? by Anonymous

      Permalink
      Comment

      Having applied the abovementioned strategy, you'll get the vampire on day 3 (at the worst case).

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      Anonymous

      17 August 2012

      In reply to Having applied the by Anonymous

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      Comment

      not in three days; the worst case is in 5 days
      mina 2-2-3-3-2
      vampire 4-3-2-1-2

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      Anonymous

      23 November 2013

      In reply to of course, not in three days by Anonymous

      Permalink
      Comment

      The worst case is in 4 days

      mina 2-3-3-2
      vampire 3-2-1-2 or 1-2-1-2

      is it right? Or did I forget a case?

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      Anonymous

      19 June 2012

      Permalink
      Comment

      Let's call the vaults as A,B,C,D
      assuming you cannot return back to A from D

      Day 1, check vault B, the vampire must be in A,C,D
      Day 2, check B again, the vampire must be in D, since if it was in A it must go to B, if it was in C it can only go to B or D.
      Day 3, check C, and KILL.

      IF you can return from D-A, but not A-D

      Day 1, check B, the vampire must be in A,C,D
      Day 2, check B again, the vampire must be in D.
      Day 3, check C, the vampire must be in A
      Day 4, check B, and KILL. :P

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      Anonymous

      21 June 2012

      In reply to 3 Days Or 4 days by Anonymous

      Permalink
      Comment

      Day 2, if it was in D previous day, then now it is in C ;)

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      Anonymous

      25 June 2012

      Permalink
      Comment

      If there are n holes,then

      sequence of inspection should be,
      Part 1: 2,3,4...,(n-1);
      followed by
      Part 2: (n-1),(n-2),...2.

      if vampire started in an even numbered hole, then it will be found in part 1.
      otherwise
      if vampire started in odd numbered
      hole , then it will be found in part 2.

      Proof: The end holes 1 and n are not checked because, before being caught in these end holes, vampire will be found in hole 2 or hole n-1. in both parts we are traversing from one end to other and parity of our movement is different in part 1 and part 2. Hence hermit will be caught in whichever part our parity matches with that of vampire.

      Thanks

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      Anonymous

      5 September 2012

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      Comment

      couldn't the hunter just do 1313 or 2424

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      Anonymous

      25 September 2013

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      Comment

      no, you couldn't just check 1313 as if the vampire was in 2 when you checked 1, he could then move to 1 when you checked 3 and back to 2 again when you checked 1.

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      Anonymous

      29 August 2014

      Permalink
      Comment

      2 days tops. 50% chance 1st day 100% 2nd

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      Anonymous

      17 November 2014

      Permalink
      Comment

      put a trap in each vault each day then when the vampire goes to that one KABOOM he's dead

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      Ayaan Dutt

      21 April 2015

      Permalink
      Comment

      Why doesn't the just hunter wait in any one of the rooms till the vampier eventually shows up??

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      Silver_tulip111

      28 December 2022

      In reply to WHY NOT WAIT ?? by Ayaan Dutt

      Permalink
      Comment

      This wouldn't work when, for example the sequence:
      vampire: 4-3-4-3, assuming that the hunter only is in room 1

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      Anonymous

      8 June 2015

      Permalink
      Comment

      Hunter can do 2233, eventually she will get the vampire.

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