The weights of four boxes are 30, 70, 60 and 90 kilograms. Which of the following cannot be the total weight, in kilograms, of any combination of these boxes if each box can be used at most once in a combination?

Introduction / Context:
This question is a subset sum puzzle involving four box weights. You are given their individual weights and several candidate totals, and you must decide which total cannot be formed by adding the weights of some selection of these boxes. Each box can appear at most once in any combination. This type of problem checks your ability to reason systematically about all possible sums of a small set of numbers.

Given Data / Assumptions:

• The four box weights are 30 kilograms, 70 kilograms, 60 kilograms and 90 kilograms.
• Each box can be either included or not included in a combination, but cannot be used more than once.
• Candidate total weights are 250, 200, 190 and 220 kilograms.
• We need to determine which one of these totals cannot be obtained from any subset of the four weights.
• All additions are simple integer sums.

Concept / Approach:
Because there are only four boxes, we can list all possible subset sums. There are 2^4 = 16 subsets including the empty one, but we only care about positive sums. By calculating the sums of all one box, two box, three box and four box combinations, we obtain a complete list of achievable totals. Any candidate total not in this list is impossible. We can also reason quickly about the largest sum and values close to it.

Step-by-Step Solution:
Step 1: List single box sums: 30, 70, 60 and 90 kilograms.Step 2: List all two box sums: 30 + 70 = 100, 30 + 60 = 90, 30 + 90 = 120, 70 + 60 = 130, 70 + 90 = 160 and 60 + 90 = 150.Step 3: List all three box sums: 30 + 70 + 60 = 160, 30 + 70 + 90 = 190, 30 + 60 + 90 = 180 and 70 + 60 + 90 = 220.Step 4: List the four box sum: 30 + 70 + 60 + 90 = 250.Step 5: Collect distinct achievable totals: 30, 60, 70, 90, 100, 120, 130, 150, 160, 180, 190, 220 and 250.Step 6: Compare the candidate options. We see that 250 is achievable by taking all four boxes, 190 is achievable using 30 + 70 + 90 and 220 is achievable using 70 + 60 + 90.Step 7: The total 200 kilograms does not appear in the list of subset sums, so it cannot be formed by any allowed combination.

Verification / Alternative check:
As a faster check, note that the maximum possible total is 250 kilograms using all four boxes. To make 200, we would need to leave out exactly 50 kilograms worth of boxes in some combination. But none of the individual weights or pair sums equal 50 (the available individual weights are 30, 70, 60, 90 and their pair sums are 100, 90, 120, 130, 160 and 150). Since we cannot remove exactly 50 from 250, 200 cannot be reached. This reasoning agrees with the full subset enumeration.

Why Other Options Are Wrong:
250 kilograms is the sum of all four boxes: 30 + 70 + 60 + 90. 190 kilograms is the sum of 30, 70 and 90. 220 kilograms is the sum of 70, 60 and 90. Since each of these totals can be formed by valid combinations under the given rules, they are not correct answers to the question.

Common Pitfalls:
Learners sometimes forget certain two box or three box combinations and conclude too quickly that a total is impossible. Others may incorrectly think that because 200 is near 190 and 220, it must be reachable. Systematically listing all sums or using the idea of subtracting from the maximum total helps avoid these mistakes.

Final Answer:
The total weight that cannot be obtained from any combination of the four boxes is 200 kilograms.