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The Magic Cafe Forum Index » » Puzzle me this... » » An Everyday Puzzle of Dwarves and Dartboards (0 Likes) Printer Friendly Version

JParker
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An Everyday Puzzle of Dwarves and Dartboards
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by Jack Parker

So, the seven dwarves come round for tea. Snow White is on the booze again so she can’t make it. The little guys bring a big trunk with them containing their new “game”. You’re a little nervous after the chainsaw incident last time they came round, but you tentatively ask what it is. They show you that in the trunk are 6 motorcycle crash helmets and each one has had a dart board bolted to the top. This means that when your vertically challenged buddies pop the helmets on their heads the dartboards are horizontal, playing surface uppermost.

Six of the dwarves put on their helmets and the last one, good old Dopey, picks out a big golden dart from the bottom of the trunk. He’s laughing like a maniac and dribbling as usual as he proudly shows you the dart.

“Loooook meeester, iz gooold!”

The guys all line up in a straight column in front of you. Dopey is at the front facing you, behind him is Bashful who is facing away from you. Behind him is Sneezy who is facing you. Behind him is Happy who is facing away from you. You get the picture? They are alternating which way they are facing.

Dopey cackles like the Wicked Witch of the West and hurls the golden dart backwards over his head. It sails up into the air and lands on one of the dartboards.

The rules are:

Nobody can turn around or move.

The guys in the line can only answer “yes” or “no” to a question.

They can each only be asked one question, this question being….

“Do you know where the dart is?”

Dopey asks this question as many times as he needs.

If they can see the dart in front of them onn someones head or they felt it land on their own head they must answer “yes” to this question. They will definitely know if it landed on their own head – it’s a gold dart for Christ sake!

What’s the maximum possible number of questions it would take for Dopey to work out who has got the dart on their head?
Bennie90
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Is that this thing ( if I ask that other one what does he say bla bla )
*going crazy with Jeff Mcbrides Art Of Card Manipulation dvds*
Wolflock
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Only one.

“Do you know where the dart is?”
Wolflock
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Jonathan Townsend
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You just asked how dopey we are... cause Dopey don't need to figure out where the dart is... and probably couldn't cause he's DOPEY.

Doc, do you have the dart?
Doc, do you have the dart?
Doc, do you have the dart?

No wonder Snow hits the bottle.
...to all the coins I've dropped here
JParker
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Hi guys

Not sure I understand you're answers, but that may just be me. I had a heavy night out with Snow White last night

I guess the fact "you" were at the head of the line may be misleading. Dopey can't see anyone in the line. All he can do is ask this question "Can you see the dart?", and he can ask that to as many of the other guys as he likes. They can only answer "yes" or "no" to his question, nothing else. He has to work out who's head the dart is on.

He may be Dopey, but he 'aint stupid.

Cheers

Jack
Jonathan Townsend
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If none of them are dopey, three seems about right

It's a tree search.
Ie start by asking the guy in position 4, if yes, ask 2, if yes again, then it's 1, if not yes from 2, then it's 3

how's that work for you?
...to all the coins I've dropped here
JParker
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Hi Jonathan

Many thanks for your reply. You might want to have a rethink?
For instance, if it lands on no.3 and you asked questions of all the dwarves..what would their answers be and from them could you determine its location?
You would get a "Yes, I know where it is" from no.2, no.3, no.5 and no.7.

Cheers

Jack
dcobbs
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Five.
Magically yours,

Daniel Cobbs.
Jonathan Townsend
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Ah, okay, the "can you see the dart" means you have to check with the neighbor... right, takes one or two extra questions there.
...to all the coins I've dropped here
TomasB
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The question is "Do you know where the dart is?" not "Can you see the dart?" I believe.

/Tomas
Jonathan Townsend
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Thanks Tomas, that gives us ("do you see the dart?" OR "did the dart land on your board?"). Back to the truth tables. Smile
...to all the coins I've dropped here
norticat
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Jonathan is right the least amount needed is three.ask the middleish guy and depending on his answer you go back or forward, forward there is only two in front. so you ask those two=three questions. if ~3 says he cant see it then ask ~5 and you then ask ~4 or ~6 if they know depending on the answer. longer than jonathan but inspired by him.
TomasB
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Quote:
On 2005-02-25 16:03, Jonathan Townsend wrote:
Thanks Tomas, that gives us ("do you see the dart?" OR "did the dart land on your board?"). Back to the truth tables. Smile
Might be more complex than that since a dwarf asked could know where the dart is without seeing it and without feeling it land on his head.

/Tomas
idris
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Hello all,
I stumbled on this topic a few weeks ago, but wasn't a member of the Café at that time. This is my first chance since then to log in.

Given that only the dwarves who can see the dart and the dwarf the dart landed on are the ones who know where the dart is, this puzzle is not solvable if the question is "Do you know where the dart is?"

For simplicities sake, I refer to the dwarfs by their position number behind Dopey.
Dwarves 1 and 2 are facing each other, dwarves 3 and 4 are facing each other, and dwarves 5 and 6 are facing each other.

Let's consider possible answer sets if the dart lands on dwarf 1 or 2.
If it lands on dwarf 1, dwarves 2, 4 and 6 can see the dart and dwarf 1 felt the dart land on him so they would each answer the question yes, while dwarves 3 and 5 would answer no.
Landing on dwarf 2 would give dwarves 1, 4 and 6 seeing the dart and dwarf 2 feeling the dart on his head, so they would each answer yes, and dwarves 3 and 5 would answer no.

Therefore it is not possible to determine whether 1 or 2 got the dart, but we would know it was one of them.

Similar logic applies to dwarves 5 and 6, leaving dwarves 3 and 4.

If the dart lands on dwarf 3 or 4, then 1 and 6 will see the dart, either dwarf 3 or 4 will see it and the other will feel it, and dwarves 2 and 5 won't know where the dart is. So again we cannot diffrentiate between them.

Given the above, the best we can do with the give question is to localize the dart to two dwarves. The lowest maximum number of dwarves to ask to the question to do the localization is 2.

1)Start by asking dwarf 3 if he knows where the dart is.
If he says no, the we know the dart is on 1 or 2 and cannot proceed any further.

2) If he says yes, ask dwarf 4 if he knows where the dart is.
If he says no, then it is on 5 or 6 and we are done.
If he says yes, then it is on 3 or 4 and we are done.

Enjoy,
Jerry
Jerry
leonard
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Jerry,
First, welcome to the Café.
While I am not 100% certain, the way I read the initial setup, dwarves 1 and 2 are facing away from each other. This would solve the problem for 1 and 2, as well as 5 and 6. I think 3 and 4 could be worked out as well.
leonard
mike paris
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There is an easier way,dopey ,tell the rest to close their eyes, then turn round and have a look he he,mike
idris
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Thank you leonard.

The initial setup says that the dwarves line up behind Dopey, Bashful is facing away from Dopey (1) and next is Sneezy who is facing Dopey and you (2), followed by Happy who is facing away from Sneezy (3) etc. Therefore 1 and 2, 3 and 4, 5 and 6 are the pairs of Dwarves who are facing each other. <G>

If we reverse the facings for the dwarfs, 1 and 6 are facing nobody, 2 and 3 are facing each other and 4 and 5 are facing each other.

If the dart lands on 1 then 1, 3 and 5 know where the dart is.
If the dart lands on 2 then 2, 3 and 5 know.
If the dart lands on 3 then 2, 3 and 5 know.
If the dart lands on 4 then 2, 4 and 5 know.
If the dart lands on 5 then 2, 4 and 5 know.
If the dart lands on 6 then 2, 4 and 6 know.

Therefore we can get unique answers only if it is on 1 or 6. Otherwise we can only localize it to either 2 or 3, or 4 or 5.

The maximum number of questions needed is to get the best restriction is 2.
As dwarf 3 if he know where the dart is. If he says yes, as dwarf 1.
If dwarf 1 says yes then we know it is on dwarf 1. Otherwise it is on 2 or 3.
If dwarf 3 says no then ask dwarf 6. If he says yes then the dart is on him, otherwise it is on 4 or 5.

Enjoy,
Jerry
Jerry
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