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## Question

A very intelligent king protected his treasure by giving the keys to the locks of the treasure to his vizier and 4 slaves in such a way that vizier could open the treasure if he was with any of the slaves , and the slaves could open the treasure if 3 of them worked together.

How many locks do the treasure have?

## 40 answers to this question

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i myself have found the answer..... but i would like someone giving a general solution

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This is my solution...

1 lock. the key hole looks like a three prong hole. each slave has a third piece of a full key. the vizier has a two prong key with one side missing. so it require the use of one of the slave's keys with it to open the one lock.

Edited by Silas Nunnery

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5 locks and 5 types of key (A-E), distributed as such...

vizier ABCD

slave 1 ABE

slave 2 BCE

slave 3 CDE

slave 4 DAE

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i myself have found the answer..... but i would like someone giving a general solution

Hmm well I'm not sure how general this is, but this is how I found it.. (maybe the same way as you?)

For 3 of 4 slaves to be able to open it, you can distribute keys like

AB, BC, CD, DA

If it was any 2 slaves to open it you would do..

ABC, BCD, CDA

ect..

Then, to allow the vizier to be able to open it with any 1 of the slaves, you simply give him the full ABCD set, and then give each slave the key to an extra lock E.

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Hmm well I'm not sure how general this is, but this is how I found it.. (maybe the same way as you?)

For 3 of 4 slaves to be able to open it, you can distribute keys like

AB, BC, CD, DA

If it was any 2 slaves to open it you would do..

ABC, BCD, CDA

ect..

Then, to allow the vizier to be able to open it with any 1 of the slaves, you simply give him the full ABCD set, and then give each slave the key to an extra lock E.

"any 1 of the slaves" == "any 3 of the slaves"

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4 locks.. the vazir has 3 keys.. the forth key is common to all the slaves.. the 3 keys which are with vazir has its copy distributed among the 4 slaves.. so 3 slaves can open it and vazir can open it with any of the slave..

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5 locks and 5 types of key (A-E), distributed as such...

vizier ABCD

slave 1 ABE

slave 2 BCE

slave 3 CDE

slave 4 DAE

You have the right idea but one flaw.

With slaves having ABE, BCE CDE and DAE note that only two slaves are necessary to open the treasure as first and third slave have full key set and second and fourth slave have full set

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4 locks.. the vazir has 3 keys.. the forth key is common to all the slaves.. the 3 keys which are with vazir has its copy distributed among the 4 slaves.. so 3 slaves can open it and vazir can open it with any of the slave..

From my interpretation of your problem, at least three slaves should be required to open the treasure. How are you going to distribute 3 keys among four slaves so any three will have a full set but no two will have a full set? If your answer is to give slaves 1, 2 and 3 a key each and not give any keys to slave 4, then you have changed the problem since slaves 4 with any other 2 could not open the treasure.

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From my interpretation of your problem, at least three slaves should be required to open the treasure. How are you going to distribute 3 keys among four slaves so any three will have a full set but no two will have a full set? If your answer is to give slaves 1, 2 and 3 a key each and not give any keys to slave 4, then you have changed the problem since slaves 4 with any other 2 could not open the treasure.

you are completely right..... in fact you stole my words....

i'ld like to hear the answer from you......

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Hmm well I'm not sure how general this is, but this is how I found it.. (maybe the same way as you?)

For 3 of 4 slaves to be able to open it, you can distribute keys like

AB, BC, CD, DA

If it was any 2 slaves to open it you would do..

ABC, BCD, CDA

ect..

Then, to allow the vizier to be able to open it with any 1 of the slaves, you simply give him the full ABCD set, and then give each slave the key to an extra lock E.

the slaves with BC and DA can together open the door...

good effort though... think a bit more

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This is my solution...

1 lock. the key hole looks like a three prong hole. each slave has a third piece of a full key. the vizier has a two prong key with one side missing. so it require the use of one of the slave's keys with it to open the one lock.

the 3 slaves cant open the door this way.......

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HINT : the answer is between 5 nd 10

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7 locks.

Vizier has ABCDEF

Slave 1 has ACFG

Slave 2 has BDFG

Slave 3 has BCEG

No 2 slaves by themselves can open all the locks but any combination of 3 can.

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```V0 0 1 1 1 1 1 1

S1 1 1 1 1 0 0 0

S2 1 1 0 0 1 1 0

S3 1 0 1 0 1 0 1

S4 1 0 0 1 0 1 1```

Edited by curr3nt

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7 locks.

Vizier has ABCDEF

Slave 1 has ACFG

Slave 2 has BDFG

Slave 3 has BCEG

No 2 slaves by themselves can open all the locks but any combination of 3 can.

congo..... you got it....

but tell me honestly.... did u do that yourself?

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you have done this with hit and trial......

can sum1 prove it generally with help of maths...

i am trying to use sets for this puzzle....

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From my interpretation of your problem, at least three slaves should be required to open the treasure. How are you going to distribute 3 keys among four slaves so any three will have a full set but no two will have a full set? If your answer is to give slaves 1, 2 and 3 a key each and not give any keys to slave 4, then you have changed the problem since slaves 4 with any other 2 could not open the treasure.

the questn doesnt says any 3 slave, it just says '3 of them worked together' not 'any three of them' as it was explicitly mentiond in case of wazir that he can open with any slave. i think that answers your question.

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Mathematical explination

The mathematical key cam after you gave the vizier all the keys but one. With the remaining number of unknown locks you needed to have 2 keys for each lock among the remaining slaves. In order to not allow only 2 to have the right combination of keys, each lock had to be assigned to a differnt combination of slaves. (ie. Lock A to slaves 1 and 4, Lock B to slaves 2 and 3, etc) that would leave 3! extra locks in addition to the lock the vizier does not control. Hence 7. If there were 5 slaves there would have to be 4!+1 locks and so on.

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Perhaps the treasure has only one lock that must be opened by a sectioned, symmetrically notched cylindrical key. The key would be sectioned lengthwise in multiples of 120 degrees, so that the visier would have a two-thirds section and all four slaves would have identical one-third sections.

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the 3 slaves cant open the door this way.......

As far as mechanics go, they can. They all would have a key that was identical with a cylinder shaft, a flap on one side to twist and another flap that goes in the the lock to turn the mechanics. the lock would have to have an extra large cylinder hole to accommodate the three keys as they would have to be turn in unison. It is a completely viable answer. No one said what type of lock it was. So I went ahead and used a lock from the far past that only required something that fit in the whole and move the internal mechanics. I will from here on assume they are using modern locks.

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Mathematical explination

The mathematical key cam after you gave the vizier all the keys but one. With the remaining number of unknown locks you needed to have 2 keys for each lock among the remaining slaves. In order to not allow only 2 to have the right combination of keys, each lock had to be assigned to a differnt combination of slaves. (ie. Lock A to slaves 1 and 4, Lock B to slaves 2 and 3, etc) that would leave 3! extra locks in addition to the lock the vizier does not control. Hence 7. If there were 5 slaves there would have to be 4!+1 locks and so on.

thnx man....... that was the answer i was looking for

gr8 job....

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Perhaps the treasure has only one lock that must be opened by a sectioned, symmetrically notched cylindrical key. The key would be sectioned lengthwise in multiples of 120 degrees, so that the visier would have a two-thirds section and all four slaves would have identical one-third sections.

As far as mechanics go, they can. They all would have a key that was identical with a cylinder shaft, a flap on one side to twist and another flap that goes in the the lock to turn the mechanics. the lock would have to have an extra large cylinder hole to accommodate the three keys as they would have to be turn in unison. It is a completely viable answer. No one said what type of lock it was. So I went ahead and used a lock from the far past that only required something that fit in the whole and move the internal mechanics. I will from here on assume they are using modern locks.

In this way you are distributing the parts of key to the slaves while the question says that the king distributed the keys to the locks of treasure to the slaves

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From my interpretation of your problem, at least three slaves should be required to open the treasure. How are you going to distribute 3 keys among four slaves so any three will have a full set but no two will have a full set? If your answer is to give slaves 1, 2 and 3 a key each and not give any keys to slave 4, then you have changed the problem since slaves 4 with any other 2 could not open the treasure.

I interpreted the question the same way, that ANY combination of 3 slaves should be able to open the door.

If this were the problem, then I wonder what the answer would be?

Find a combination which allows any 3 of 6 slaves to unlock the chest, (if this is possible), then, stitch two slaves together (like a human centipede) and give them the job of vazir. Then you have a vazir who can open the chest with 1 more of 4 slaves.

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In this way you are distributing the parts of key to the slaves while the question says that the king distributed the keys to the locks of treasure to the slaves

My method has two keys (divided up: vizier+slave[key A], 3 slaves[key b]) and one lock. The riddle says the "king protected his treasure by giving the keys (I did figure in two keys) to the locks of the treasure." It never once states whether or not the keys where divided into pieces. If you want to fault me on semantics then simply say the riddle says locks (plural) so one should automatically be assumed not to be the answer. I admit my answer is wrong because of that fact. It is; however, the simplest answer as to how it would work with one lock. I was to eager to answer my first riddle on here. Next time I will pay closer attention to the riddle itself. Happy solving }: -)

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Instead of a series of hasps with a lock and key for each, what if the treasure chest is locked with a chain-like configuration? Within the links of the chain are nine larger rings (1-9). Each of the four slaves (A,B,C,D) and the vizier (V) have their own lock and key. Lock A goes through rings 1,2,3,4,7,8. Lock B goes through rings 1,2,3,5,6,8,9. Lock C goes through rings 1,2,4,5,7,8,9. Lock D goes through rings 2,3,6,7,8,9. Lock V goes through rings 3,4,5,6,7.

1-ABC-2-ABD-3-AV-4-CV-5-BV-6-DV-7-ACD-8-BCD-9

So if the vizier and any one slave or if any three slaves unlock their respective locks, the chain is broken.

Can't fully visualize this but think it would work

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