MATH 64091
Jenya Soprunova, KSU
Colorings
Example 1. The surface of a wooden cube is painted white. Each of the cube’s edges
is divided into 5 equal segments and then the cube is cut into cubicles whose edge is
one fifth of the edge of the initial cube. How many cubicles are there with at least
one face painted white?
We have a 3 by 3 by 3 cube inside that consists of 27 cubicles none of whose faces
are painted. All the remaining cubicles have at least one painted face. The overall
number of cubicles is 125, so the number of small cubicles with at least one white
face is 125-27=98.
Example 2. The plane is colored in two colors, black and white. Show that one can
always find two points in the plane of the same color that are one foot apart.
Construct a triangle of side 1 foot. At least two of its vertices are of the same color,
which gives a segment of length one foot with endpoints of the same color.
Example 3. In a hexagon, each side and diagonal is colored either red or blue. Show
that there is always either a red or a blue triangle formed by the sides and diagonals.
Explain how this implies that in a group of six people there are either three who are
friends with each other or three who are not friends with each other.
Let’s consider all the segments that start at one of the vertices (say, vertex A).
Since there are five such segments, at least three of them are of the same color (say,
blue). Let these segments be AB, AC, AD . If one of BC, CD, DB is also blue we
have found a blue triangle. If all BC, CD, DB are red, we have found a red triangle.
Draw a hexagon whose vertices represent the six people. Connect two vertices with
a red segment if the corresponding people are friends, and with a blue segment if they
are not. We have shown that there is always either a red or a blue triangle formed
by the sides and diagonals, so there are either three people in group of six who are
friends with each other or three who are not friends with each other.
Example 4. Is it possible to color four vertices of the cube red and four vertices blue
so that every plane that passes through any three vertices of the same color contains
a vertex of the opposite color?
Yes, this is possible. Here is the coloring.
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Homework Problems
Problem 1. Color the map in four colors so that neighboring countries are of different
colors. Would three colors suffice? If not, explain why not.
Problem 2. Color the squares of the 5 by 5 grid in five colors, so that in each row,
column, and block (see the picture below) there is a square of each of the colors.
Problem 3. You want to color each of the 1 × 1 × 1 cubicles of 3 × 3 × 3 cube so
that cubicles that have at least one common vertex are of different color. What is the
smallest number of colors that you would need to use? (As usual, explain why that
many colors would work and why a smaller number would not.)
Problem 4. A 4 × 4 × 4 cube is constructed out of 64 1 × 1 × 1 cubicles. After that,
3 faces of the cube are painted red and three faces are painted blue so that there are
no cubicles with three red faces. How many cubicles have both red and blue faces?
Problem 5.
(1) Color the
colored in
(2) Color the
colored in
plane in three colors using all three colors so that every line is
at most two colors.
plane in three colors using all three colors so that every line is
exactly two colors.
Problem 6. A line is colored in two colors. Show that you can always find a segment
on this line such that its endpoints and its midpoint are all of the same color.
Problem 7. The plane is colored in two colors and both colors are used. Show that
one can find two points of opposite colors distance 1 apart.
Problem 8. Each square of an infinite sheet of graph paper is colored in one of eight
colors. Prove that one can place a pentomino on that sheet of paper of the shape
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shown below so that it contains at least two squares of the same color. When you
place the pentomino, you can rotate it around or flip it, if needed.
Problem 9. Each square of an infinite sheet of graph paper is colored either black
or white. Show that there are four squares of the same color that “form a rectangle”,
that is, these four squares are at the intersection of two rows and two columns.