There is a king and there are his ten prisoners. The king has a dungeon in his castle that is shaped like a circle, and has ten cell doors around the perimeter, each leading to a separate, utterly sound proof room. When within the cells, the prisoners have absolutely no means of communicating with each other. The king sits in his central room and the ten prisoners are all locked in their sound proof cells. In the king's central chamber is a table with a single chalice sitting atop it. Now, the king opens up a door to one of the prisoners’ rooms and lets him into the room ... always only one prisoner at a time! So he lets in just one of the prisoners, any one he chooses, and then asks him a question, "Since I first locked you and the other nine prisoners into your rooms, have all of you been in this room yet?" The prisoner only has two possible answers. "Yes," or, "I'm not sure." If any prisoner answers “yes” but is wrong, they all will be beheaded. If a prisoner answers “yes,” however, and is correct, all ten prisoners are granted full pardons and freed. After being asked that question and answering, the prisoner is then given an opportunity to turn the chalice upside down or right side up. If when he enters the room it is right side up, he can choose to leave it right side up or to turn it upside down, it's his choice. The same thing goes for if it is upside down when he enters the room. He can either choose to turn it upright or to leave it upside down. After the prisoner manipulates the chalice (or not, by his choice), he is sent back to his own cell and securely locked in. The king will call the prisoners in any order he pleases, and he can call each prisoner as many times as he wants. The only rule the king has to obey is that eventually he has to call every prisoner in an arbitrary number of times. So maybe he will call the first prisoner in a million times before ever calling in the second prisoner twice, we just don't know. But eventually we may be certain that each prisoner will be called in ten times, or twenty times, or any number you choose. Here’s one last monkey wrench to toss in the gears, though. The king is allowed to manipulate the cup himself, ten times, out of the view of any of the prisoners. That means the king may turn an upright cup upside down or vice versa up to ten times, as he chooses, without the prisoners knowing about it. Assume that both the king and the ten prisoners have a complete understanding of the game as I have just explained it to you, and that the prisoners were given time beforehand to come up with a strategy they can all use. Now you must figure out not only how to keep the prisoners alive, but how to also ensure their eventual freedom. When can any one of them be certain they've all been in the central chamber of the dungeon at least once? And how? Don't try to imagine any trickery like scratching messages in the soft gold of the chalice. The problem is as simple as it sounds. The prisoners have absolutely no way of communicating with each other except through the two orientations of the chalice. If any of them attempts any trickery at all they will all be beheaded. All the prisoners can do is turn the chalice upside down or right side up, as they choose, whenever they are called into the chamber. That is all.
The king will call the prisoners in any order he pleases, and he can call each prisoner as many times as he wants. The only rule the king has to obey is that eventually he has to call every prisoner in an arbitrary number of times. So maybe he will call the first prisoner in a million times before ever calling in the second prisoner twice, we just don't know. But eventually we may be certain that each prisoner will be called in ten times, or twenty times, or any number you choose.
There's a dry well in a garden in which 2 planks are standing crossed from wall to wall.
One of the planks is 2 meters long, the other is 3 meters long.
The crossing point is 1 meter above ground of the well.
(cheesy ascii art:
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Question: How broad is the well?Ummm, your w is greater than 4. It's very unlinkely that a 2 meter plank is standing diagonally in such a well.
In this you should see the following equations:
w = w2+w3
2² = w²+a2²
3² = w²+a3²
And then:
w2/1 = w/a2
w3/1 = w/a3
Put them together and you should get:
1 = 1/√(9-w²) + 1/√(4-w²)
which is too complicated for me to believe there is a simpler way of arriving at this value.