## Rome

A forum for good logic/math puzzles.

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Jayraj
Posts: 15
Joined: Sun Oct 21, 2018 11:06 am UTC

### Rome

In ancient Rome, there was a slave master who owned 25 slaves.

The slaves were kept isolated and had no form of contact with each other.

But the slave master was also a logician and valued intelligence and cooperation.

One day he gathered all 25 of his slaves and told them the following:

"There is a room in this house which contains many tools and supplies that would make your life much easier, but require a certain intelligence to wield."

"This room is only accessible through a master room, which also leads to a third room."

"The door to the supply room is securely locked, but the third room's door is closed but not locked."

"The master room and the third room are devoid of any furniture and the walls cannot be marked."

"At random times, I will bring one of you, chosen randomly, to the master room, but you cannot bring any tools with you."

"You will be allowed in for only a few seconds, and when you leave the room, I will make sure that you have left nothing inside."

"When one is brought to the master room, if he is certain that every other slave has been to that room, he may tell me so and this will end the game."

"However, failure to comply to any rules or ending the game before everyone has been to the master room will result in all of you being sold off."

"You have 10 minutes to discuss between yourselves before you are all back to being isolated again."

"If you successfully complete the game I will grant you access to the supply room."

How can they ensure that they win the game?

Yat
Posts: 131
Joined: Tue Feb 03, 2009 2:05 pm UTC

### Re: Rome

As I understand it, it is the classic switch puzzle, (viewtopic.php?f=3&t=73 viewtopic.php?f=3&t=13669 viewtopic.php?f=3&t=69986), the unlocked door playing the role of the switch.

If we can't count on the room to stay in the state it was left in, then I don't see how any information could circulate between the slaves. So the strategy used for the switch puzzle can be used here.

We could, however, speed up the process a bit by
Spoiler:
using the fact that the door position is not binary. As they have no tool to measure anything, I don't see a way to distinguish 26 separate states for the door and do it in one pass, but having a few different states can ensure the process goes much faster. My personal proposal would be to distinguish six easily distinguishable states:
0 closed
1 not closed, but only with a thin aperture
2 clearly open, but not close to 90° from the wall
3 at a roughly 90° angle from the wall
4 open less than 90° from the wall, but still far from against it
5 wide open against the wall
The process would then be, just as in the switch solution, to nominate a counter, and then every slave except the counter that goes in the room and sees the door not completely open against the door, progresses its state one step, but only the first time it is possible. The counter, when they enter the room, adds the current state of the door to their own mental counter, then closes the door again. When the mental counter reaches 24, they know the game is over.