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MathNet 2005 — Part 02 Question 3

Math Olympiad Vault · MathNet 2005 — Part 02

Question 3

If you want to color each positive entry with a color using the largest possible number of colors so that the following condition is met: If, in decimal notation, the number $B$ can be obtained from the number $A$, eliminating $A$ from the consecutive equal digits (aa) or By suppressing $A$ four consecutive digits that form two equal and consecutive pairs (abab), then $A$ and $B$ are of the same color. For example, 8, 833 and 22811 must be the same color and 72, 676772 and 173329898 must be the same color.
Determine the largest number of colors that can be used.