Imprisoned Logicians

Two friends, logicians – Ein and Stein – get imprisoned in two distant cells in a castle. Both cells have just one door, and a window with 8 bars in the first cell, and 12 bars in the second cell. The first day both logicians get the same letter from the prison master:

“The total number of bars in the two prison cells in this castle is either 18 or 20. Starting tomorrow, every morning I will go first to Ein and then to Stein, and will ask how many bars the other logician has. If one of you answers correctly, I will immediately let both of you leave the castle. If one of you answers incorrectly, I will execute both of you. Of course, you can always decide not to answer and just stay imprisoned.
I have sent a copy of this letter to you and your friend. There is no point in trying to communicate with him – your cells are far away from each other, and he won’t hear you.”

Will the logicians manage to escape the castle eventually? When will they do it?

Solution coming soon.


Lost In the Forest

You are lost in the middle of a forest, and you know there is a straight road exactly 1 km away from you, but not in which direction. Can you find a path of distance less than 640 m which will guarantee you to find the road?

Imagine there is a circle with a radius of 100 m around you, and you are at its center O. Let the tangent to the circle directly ahead of you be t. Then, follow the path:

  1. Turn left 30 degrees and keep walking until you reach the tangent t at point A for a total of 100×2√3/3 meters, which is less than 115.5 meters.
  2. Turn left 120 degrees and keep walking along the tangent to the circle until you reach the circle at point B for a total of 100×√3/3 which is less than 58 meters.
  3. Keep walking around the circle along an arc of 210 degrees until you reach point C for a total of 100×7π/6 which is less than 366.5 meters.
  4. Keep walking straight for 100 meters until you reach point D on the tangent t.

An Ant’s Path

An ant is positioned at one of the vertices of a cube and wants to get to the opposite vertex. If the edges of the die have length 1, what is the shortest distance the ant needs to travel?

We unfold a cube to get a cross-shaped figure. Then, the problem is to find the shortest path between two points separated by a horizontal distance of 2 units and a vertical distance of 1 unit.

It is easy to see that the path in question is the one passing through the middle of the edge between the start and end points, and which has a distance of √2.