The Broken Packaging Test involves determining the minimum number of broken items that guarantees the test stops at the first box. The test proceeds by opening boxes one by one and sampling (k) items from each box. If all (k) items in a box are broken, the test stops immediately, concluding the entire batch is broken. Otherwise, the next box is opened. To ensure the test stops at the first box, the first box must contain at least (k) broken items. However, the distribution of broken items could be adversarial, meaning the worst-case scenario where broken items are concentrated in other boxes to avoid stopping early. Each of the other (m-1) boxes can hold up to (n) broken items. Thus, the maximum number of broken items that can be placed in the other boxes is ((m-1) \times n). The total number of broken items must be sufficient such that even after placing as many as possible in the other boxes, the first box still has at least (k) broken items. Therefore, the minimum total broken items required is: ]

This value guarantees that the first box contains at least (k) broken items, forcing the test to stop at the first box, regardless of how the broken items are distributed.

Example:

• (m = 2) boxes, (n = 10) items per box, (k = 5) sampled items.
• Minimum broken items to guarantee stopping at the first box: