$\left.\quad \begin{array}{ll}{\left[\begin{array}{l}\text { Principles of divisibility by 3 and 9 } \\ \text { [Examples and counterexamples. Constructions] }\end{array}\right]}\end{array}\right]$

Write the number 2013 several times in a row so that the resulting number is divisible by 9.
The number 201320132013 is divisible by 9, since the sum of its digits is equal to $(2+0+1+3) \cdot 3=18$.

## Answer

Given a sheet of graph paper. Each node of the grid is marked with some letter. What is the smallest number of different letters needed to mark these nodes so that on any segment (going along the sides of the cells - ed.), connecting two nodes marked with the same letter, there is at least one node marked with one of the other letters?

## Solution

Answer: two letters. These letters should be arranged in a checkerboard pattern.