### Brain Teasers

# Three Statements X

Which of these three statements are true?

1. Using exactly three colors it is possible to color the sides of a cube in such a way that two sides having the same color never touch.

2. A regular octagon can be divided into 8 identical triangles by drawing 4 straight lines.

3. Three equally sized circles can be drawn in such a way that they overlap and create 9 separate compartments.

1. Using exactly three colors it is possible to color the sides of a cube in such a way that two sides having the same color never touch.

2. A regular octagon can be divided into 8 identical triangles by drawing 4 straight lines.

3. Three equally sized circles can be drawn in such a way that they overlap and create 9 separate compartments.

### Answer

Statements 1 and 2 are true, statement 3 is false.In statement 1, use each color on each pair of opposite surfaces, and they do not touch.

In statement 2, connect each vertex with its opposite one, this will be 4 straight lines, and the octagon is divided into 8 identical triangles.

In statement 3, the first circle divides the plane into 2 parts, the second circle at most divides each of the 2 parts into two, making 4 parts, and the third circle at most divides each of the 4 parts into two, making 8 parts. It's impossible to get more than 8 parts.

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