Answer :
Final answer:
The process involves adding up all the combined weights, finding the estimated weight of all the boxes and the lightest box, and then subtracting the lightest and the remaining boxes weights from the total to discover that the heaviest box weights 85 kg.
Explanation:
In order to find the weight of the heaviest box, we need to follow a couple of steps. First, remember that each box is weighed with every other box exactly once. Therefore, if we sum up all the weights obtained, we are in essence summing up twice the weight of all the boxes combined. The sum of the weights obtained when two boxes are weighed together is 1,056 kg.
Considering that each box is included in the total sum twice, we can divide the total by 2 to find the sum of weights of all boxes - 528 kg.
Now, let's find the weight of the lightest box. This box will produce the smallest measurement when combined with any other box. Hence, the weight of this box is less than the difference between the two smallest measurements. The two smallest measurements are 110 and 112 kg, so the lightest box weighs less than 2 kg.
From here, we calculate the weight of the rest of the boxes. Subtract the weight of the lightest box from the total weight of all boxes (528 kg - 2 kg), and we get 526 kg. Given that the weight of four boxes is smaller than the total weight (528 kg - heaviest box weight), we can calculate that the heaviest box must weigh more than 526 kg - (remaining four boxes combined). Using the five smallest weighing results (excluding the smallest we used before), we find that the remaining four boxes weigh 442 kg, leading us to conclude that the heaviest box must weigh more than 84 kg.
Further checking the weights will reveal that the heaviest box cannot weigh more than 86 kg, because then we would not be able to achieve the two highest weigh-in results. Hence, the heaviest box weighs 85 kg.
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