Eight cards are numbered from $$1$$ to $$8$$. The cards are placed in two boxes $$P$$ and $$Q$$ so that the sum of the numbers on the three cards in box $$P$$ is equal to the sum of the numbers on the five cards in box $$Q$$. Which of the following statements must be true? 
Overseas Competition->Knowledge Point->Combinatorics->Logical Reasoning->Reasoning by Conditions
Note first that the sum of the numbers on the eight cards is $$36$$. Therefore the sum of the numbers on the cards in each of the boxes is $$18$$. There are only three cards in box $$P$$ and hence the possible combinations for the numbers on the cards in box $$P$$ are $$\left( 8,7,3 \right)$$, $$\left( 8,6,4 \right)$$ and $$\left( 7,6,5 \right)$$ with the corresponding combinations for box $$Q$$ being $$\left( 6,5,4,2,1 \right)$$, $$\left( 7,5,3,2,1 \right)$$ and $$\left( 8,4,3,2,1 \right)$$. The only statement which is true for all three possible combinations for box $$Q$$ is that the card numbered $$2$$ is in box $$Q$$. Hence the only statement which must be true is statement $$\rm D$$. 