The weights of four boxes are 80 kg, 60 kg, 90 kg and 70 kg. Which of the following values cannot be the total weight, in kilograms, of any combination of these boxes if in a combination a box can be used only once?

Introduction / Context:
This problem is another example of a box weight combination question from arithmetic reasoning. You are given four box weights and several possible total weights. The task is to determine which total cannot be achieved by adding the weights of some or all of the boxes, with the condition that each box can be used at most once. Such questions assess systematic thinking and arithmetic accuracy.

Given Data / Assumptions:
- Box weights: 80 kg, 60 kg, 90 kg and 70 kg. - Each box may either be included or excluded from a combination, but cannot be repeated. - We can form totals using one, two, three or all four boxes. - Options: 300 kg, 230 kg, 220 kg, 290 kg and 210 kg. - We must identify the total weight that is impossible to obtain.

Concept / Approach:
The straightforward method is to list all distinct sums obtainable from the given weights. With only four boxes, the number of subsets is small, and full enumeration is practical. After generating all possible totals, we compare them with the answer options. Any option absent from the list of achievable totals is the correct answer. This method prevents oversight and ensures accuracy.

Step-by-Step Solution:
Step 1: Single box totals are simply: 60, 70, 80 and 90 kilograms. Step 2: Two box combinations and their sums: - 80 + 60 = 140 - 80 + 90 = 170 - 80 + 70 = 150 - 60 + 90 = 150 - 60 + 70 = 130 - 90 + 70 = 160 Step 3: Three box sums: - 80 + 60 + 90 = 230 - 80 + 60 + 70 = 210 - 80 + 90 + 70 = 240 - 60 + 90 + 70 = 220 Step 4: Four box sum: 80 + 60 + 90 + 70 = 300. Step 5: Collect achievable totals: 60, 70, 80, 90, 130, 140, 150, 160, 170, 210, 220, 230, 240 and 300. Step 6: Compare with options: 300, 230, 220, 290 and 210. We see that 300, 230, 220 and 210 appear in the achievable list, but 290 does not.

Verification / Alternative check:
Try to construct 290 kg explicitly. The total weight of all boxes is only 300 kg, so 290 kg would require leaving out 10 kg. However, none of the box weights equals 10 kg, so you cannot obtain 290 kg by excluding a single box. Check other possibilities: any sum of three boxes is at most 240 kg and any sum of two boxes is at most 170 kg. Therefore, no subset of the given weights sums to 290 kg. This confirms that 290 kg is impossible to achieve.

Why Other Options Are Wrong:
- 300 kilograms is achievable with all four boxes together: 80 + 60 + 90 + 70. - 230 kilograms is achievable with 80 + 60 + 90. - 220 kilograms is achievable with 60 + 90 + 70. - 210 kilograms is achievable with 80 + 60 + 70. Since these totals appear among the computed sums, they do not satisfy the requirement of being impossible.

Common Pitfalls:
A common error is to attempt mental calculation without writing down all combinations, leading to omitted sums. Some candidates also incorrectly assume that the largest or smallest option must be the impossible one without checking. Others mistakenly reuse a box weight, violating the rule that each box can be used only once. Listing combinations systematically is the safest method for these problems.

Final Answer:
The total weight that cannot be formed by any combination of the four boxes, using each box at most once, is 290 kilograms.