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The strategy guide is correct. When combining the prime boxes, we are allowed to take only the MINIMUM information that MUST be true. The two prime boxes have information that potentially overlaps, so we have to strip out that overlap before we combine them.

To see the overlap, imagine this scenario. You, Meryl Streep, and I are sitting in a room. I go into the next room (you can't see into this room) and I come back and tell you there's an orange sitting on the table in the next room. Then, Meryl Streep goes into the next room, comes back, and tells you there are two oranges sitting on the table in the next room.

What's the minimum number of oranges that MUST be on that table? Three? There might be three, but there don't have to be three. There might only be two - because the orange I told you about might be the same as one of the oranges that Meryl told you about. You know there are at least two, because Meryl reported two. But you don't know whether there's a third.

Same thing with the prime boxes. You've got 2,2,3 in one and 2,5 in the other. What if that 2 in the second box is the SAME 2 as one of the 2's in the first box? It's possible. So we don't know that we have three 2's. We only know for sure that we have at least two 2's. We might have three 2's, but we don't know.

Make sense?