Lousy Management

Logic Level 3

You have just been hired for a new job. You are told that you will work with two other employees, Adam and Eve.

In your new office, there are three locked configuration boxes: one for electricity, one for plumbing, and one for gas.

Because your employer is cheap and unwilling to pay for repairs, you and your coworkers have to do routine maintenance on these boxes.

Originally, your employer bought three keys for each safe so each of you could access it. However, he has lost all of the keys and is only willing to pay for the absolute minimum number of keys to the boxes. Additionally, he doesn't trust you or your coworkers, so he will refuse to buy as many of any type of a key as there are of you.

What is the minimum number of keys that must be bought such that at any time, you or your coworkers are able to access all three configuration boxes, regardless of how many of you are at the office?

Note: Keys cannot be left out as your employer refuses to pay for minimal security and the janitors may throw them away by accident.

The answer is 6.

3 solutions

Haytham Connor
Oct 4, 2016

While the problem may appear difficult and confusing on the surface, it is actually rather quite simple. Stripping away all of the needless detail, we see that we have three containers, each of which can only be opened with its respective key.

The most efficient solution will be creating a complete cycle of A, B, and C, as this on its own would only require 4 keys:

one key to access the cycle,
one key that leads to the next box,
another key in that box that leads to the next box, and
finally the last key in the final box which leads back to the first box, creating a complete cycle.

Extending this to 3 people is essentially the same: 3 keys to access the infinite cycle and the 3 keys that create the cycle within the boxes.

So, therefore, the most efficient solution to this problem is a total of $\boxed6$ keys, where

3 of the keys are kept in the boxes at all times that establish a cycle and the remaining 3 keys, one to each employee, that allows each of them to access this cycle.

An interesting thing to notice that the above configuration poses a potential security risk, as it gives an intruder who somehow gets acces to one key acces to every box. If the manager is willing to accept such a risk then maybe the czeapest solution for him would be to get the same lock for all three boxes and just get three keys. I know this wasn't an option in the origal question, just additional thinking to come up with a potentially more secure solution. :-)

Or also they could just make an additional box, give each employee a key to acces it and put the three key for electricity, plumbing and gas in there.

Kai Ott - 4 years, 8 months ago

Thanks. I see that this problem has been edited. Those who previously answered 4 or 5 has been marked correct.

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Brilliant Mathematics Staff - 4 years, 8 months ago

5 Keys 3 Electricity And one Plumbing + one Gas in the local Electricity?

Philippe Pericard - 4 years, 8 months ago

After analyzing your guys solutions, it would appear 5 is in fact a valid answer and therefore a better solution than the original problem. Applying graph theory, your solution would be a "central hub" that can access all the other points while the original solution used an infinite loop. I adapted this problem from a logic riddle book originally, so apologies for overlooking the significant changes I did make. 5, is in fact, the correct answer given the way I stated it. If, however, I mentioned that you need an equal number of each type of key, then the infinite loop solution would be the most efficient. My apologies for my mistake.

Alek Nelson - 4 years, 8 months ago

Thanks. I see that this problem has been edited. Those who previously answered 4 or 5 has been marked correct.

Brilliant Mathematics Staff - 4 years, 8 months ago

I think 5 keys are exactly enough. The three workers will be given one key each, which is able to open A box. And besides that, the keys that are able to open B box and C box are stored in A box.

伟盛林 - 4 years, 8 months ago

Thanks. I see that this problem has been edited. Those who previously answered 4 or 5 has been marked correct.

Brilliant Mathematics Staff - 4 years, 8 months ago

I thought 5, one key each to access box a and the other two keys stored in box a or one in box a and the next in box b/c

Richard Mcdonagh - 4 years, 8 months ago

Haytham Connor - 4 years, 8 months ago

Thanks. I see that this problem has been edited. Those who previously answered 4 or 5 has been marked correct.

Brilliant Mathematics Staff - 4 years, 8 months ago

I did not realize the boxes were, or could be, connected. Can that be clarified?

Emily Namm - 4 years, 8 months ago

Y H
Oct 13, 2016

? Why not 5? Each employee gets a key to door #1, and behind that door are kept the two keys to the other two doors!

Thanks. I see that this problem has been edited. Those who previously answered 4 or 5 has been marked correct.

Brilliant Mathematics Staff - 4 years, 8 months ago

Vishal Mr
Oct 15, 2016

There are 3 boxes so box 1 is named electricity, box 2 is named plumbing box 3 is gas, Assume there are different keys inside each box ( ONE KEY ) Now 1st person has a key to open electricity box, now electricity box contains Plumbing key which he then opens the plumbing box and get the gas key,when he opens the gas key he gets the electricity box.

Now 2nd person default will have plumbing key therefore opens it and get gas key.......... now 3rd person the same 1st person - electricity key, 2nd person - plumbing key,3rd person - gas key so (3 keys) totally now 3 boxes contains inside (3 keys)so total of 3+3=6

Lousy Management

The answer is 6.

3 solutions

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