Problem Summary

We need to determine the smallest number of balls, denoted by $N$, that must be simultaneously picked from a box containing balls of three different colors (4 red, 5 green, 6 blue) to ensure that at least two balls among the picked ones share the same color.

Applying the Pigeonhole Principle

This problem is a classic application of the Pigeonhole Principle.

• The different colors (Red, Green, Blue) serve as the 'pigeonholes'. Thus, the number of pigeonholes is $k=3$.
• The balls being picked represent the 'pigeons'.
• The Pigeonhole Principle states that if you select more items (pigeons) than the number of categories (pigeonholes), at least one category must contain more than one item. Specifically, to guarantee at least two items in one category, you must select $k + 1$ items.

Worst-Case Scenario Analysis

To find the minimum value $N$ that guarantees a pair of the same color, we consider the least likely scenario (the worst case) where we pick balls of different colors for as long as possible:

• 1st Pick: Pick one ball (e.g., Red).
• 2nd Pick: Pick a ball of a different color (e.g., Green).
• 3rd Pick: Pick a ball of the third color (e.g., Blue).

After these 3 picks, we have successfully picked one ball of each distinct color. The next ball picked, which is the 4th ball ($N=4$), must inevitably match one of the colors already picked (Red, Green, or Blue).

Therefore, the minimum number of balls required to guarantee at least two are the same color is $k + 1 = 3 + 1 = 4$.

Conclusion

The smallest value of $N$ required to guarantee at least two balls of the same colour is 4.