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Four glasses are placed on the corners of a square table. Some of the
glasses are upright (up) and some upside-down (down). A blindfolded
person is seated next to the table and is required to re-arrange the
glasses so that they are all up or all down, either arrangement being
acceptable, which will be signaled by the ringing of a bell.
The
glasses may be re-arranged in turns subject to the following rules.
1.Any
two glasses may be inspected in one turn and after feeling their
orientation the person may reverse the orientation of either, neither or
both glasses.
2.After each turn the table is rotated through a random
angle.
3.The puzzle is to devise an algorithm which allows the blindfolded
person to ensure that all glasses have the same orientation (either up
or down) in a finite number of turns. The algorithm must be
non-stochastic i.e. it must not depend on luck.
Here is that algorithm!
What was the puzzle?
Below is the algorithm which makes sure the bell will ring in at most five turns.
1.On the first turn choose a diagonally opposite pair of glasses and turn both glasses up.
At this point, the position of other 2 glasses is not known.
2.On the second turn, choose 2 adjacent glasses. One of them was turned up in the previous step, so other may or may not in up position. If the other is down then turn it up and if remaining one X is also in up position then bell will be rung.
If the bell does not ring then there are now three glasses up and one down(3U and 1D).
3.On the third turn choose a diagonally opposite pair of glasses. If one is down, turn it up and the bell will ring.
And if you find both are up, then you must have chosen other diagonally opposite pair.
If so, then turn one down so that 2 glasses are up and other 2 are down.
4.On the fourth turn choose two adjacent glasses and reverse both. If both were in the same orientation then the bell will ring.
And in case, if you find one is up and other down like -
still reverse orientation of both as -
Now diagonally opposite pairs are either up or down.
5.On the fifth turn choose a diagonally opposite pair of glasses and reverse both.
The bell will ring for sure.
A solid, four-inch cube of wood is coated with blue paint on all six
sides.
Then the cube is cut into smaller one-inch cubes. These new
one-inch cubes will have either three blue sides, two blue sides, one
blue side, or no blue sides. How many of each will there be?
Here is solution of the puzzle!
What is the puzzle?
Apart from the 8 cubes at the center all 4 x 4 x 4 - 8 = 56 will have some paint on ones side at least. See below the 1/4 th cube is taken out.
The cubes at the 8 corners will have blue paint on three sides.
The cubes between corner cubes along 12 edges of big cube will have 2 sides painted. That is 12 x 2 = 24 cubes will painted with blue on 2 sides.
And 4 center cubes on each of 6 faces (left, right, top, bottom, front, back) will have only 1 side painted with blue. That is , there are 6 x 4 = 24 cubes having paint on one side only.
To conclude, out of 56 painted cubes,
24 cubes have paint on 1 side,
24 cubes painted with 2 sides,
8 are painted with three sides.