To determine which option is NOT a necessary condition for forming a triangle using the three pieces of the stick, we need to apply the triangle inequality theorem. For any three lengths to form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Given the stick is broken at points \(b_1\) and \(b_2\), we have three pieces: \(b_1\), \(b_2 - b_1\), and \(1 - b_2\).

First, let's apply the triangle inequality to these three lengths:

1. \(b_1 + (b_2 - b_1) > 1 - b_2 \Rightarrow b_2 > 1 - b_2 \Rightarrow b_2 > 0.5\)
2. \(b_1 + (1 - b_2) > b_2 - b_1 \Rightarrow b_1 + 1 - b_2 > b_2 - b_1 \Rightarrow b_1 + 1 > b_2 + b_2 - b_1 \Rightarrow b_1 + 1 > 2b_2 > b_1 \Rightarrow \frac{1 + b_1}{2} > b_2\)
3. \((b_2 - b_1) + (1 - b_2) > b_1 \Rightarrow 1 - b_1 > b_1 \Rightarrow b_1 < 0.5\)

Now let's evaluate the options:

• \(b_1 < 0.5\): Necessary as shown in step 3.
• \(b_2 < 0.5\): This is not possible as \(b_2\) must be greater than \(0.5\) to satisfy the inequality in step 1.
• \(b_2 < b_1 + 0.5\): A combination of inequalities implies this can be true.
• \(b_1 + b_2 < 1\): This doesn't affect the triangle formation based on given inequalities.

Concluding, the condition \(b_1 + b_2 < 1\) is NOT a necessary condition for forming a triangle using the given pieces.