You are in the desert and you have 3 buckets of water containing a,b,c liters respectively (a,b,c - positive integers).

You need an empty bucket for an unspecified purpose. Being in the desert you need the water and cannot just pour it away.

You have to pour the contents of one bucket into another one. But in any pouring, you must double the contents of the bucket which receives the water.

For example the sequence of bucket contents could be:

3 2 1

1 4 1

0 4 2

Now show that no matter what a,b,c are, you can always manage to empty a bucket under this constraint.

You may assume: ** **

a>b>c

&

**(capacity of each bucket)>(a+b)**

The famous

four-color-theorem states that every map can be colored with no more than four colors, such that no two adjacent regions receive the same color. Two regions are called adjacent if they share a border segment, not just a point.

There are certain types of maps however, where less than four colors are sufficient:

1. We partition the plane into regions by drawing a number of straight lines. In the resulting "map" there will be some infinitely large regions, but that shouldn't bother us. How many different colors do you need for such types of maps?

2. Instead of straight lines, partition the plane by drawing circles. How many colors are needed now?

3. What if you draw circles and chords?

4. Now consider 3-dimensional "maps", where you partition space into regions and two regions are called adjacent when they share a common (2-dimensional) face, not just an edge or a point. How many colors do we need if space is partitioned by planes?

5. As above, but with spheres instead of planes.

6. How many colors are required for general 3-dimensional maps?