The weights of four boxes are 70, 100, 20 and 40 kilograms. Which of the following cannot be the total weight, in kilograms, of any combination of these boxes if each box is used at most once?

Introduction / Context:
This is another subset sum problem involving four box weights. We are given several candidate totals and must determine which one cannot be obtained by summing the weights of some subset of the boxes, with the restriction that each box can be used at most once. Problems like this test systematic enumeration or logical reasoning with sums.

Given Data / Assumptions:

• The four box weights are 70 kilograms, 100 kilograms, 20 kilograms and 40 kilograms.
• We can choose any subset of these four boxes, possibly just one, several or all four.
• Each box can appear at most once in any chosen subset.
• The candidate totals are 230, 190, 160 and 200 kilograms.
• We need to find which total is impossible to obtain from any allowed combination.

Concept / Approach:
Because there are only four boxes, one direct approach is to list all possible subset sums. Alternatively, we can reason about each option by trying to express it as a sum of some combination of the four weights. If a total cannot be matched by any choice of 70, 100, 20 and 40 without repetition, then that total is the impossible one. We will use a mix of both methods for clarity.

Step-by-Step Solution:
Step 1: Start with simple combinations. Single box totals are 70, 100, 20 and 40 kilograms.Step 2: Consider combinations of two boxes: 70 + 100 = 170, 70 + 20 = 90, 70 + 40 = 110, 100 + 20 = 120, 100 + 40 = 140 and 20 + 40 = 60. The distinct two box totals are 60, 90, 110, 120, 140 and 170.Step 3: Consider combinations of three boxes: 70 + 100 + 20 = 190, 70 + 100 + 40 = 210, 70 + 20 + 40 = 130 and 100 + 20 + 40 = 160. The three box totals are 190, 210, 130 and 160.Step 4: Consider all four boxes together: 70 + 100 + 20 + 40 = 230.Step 5: Collect the distinct totals that are possible: 20, 40, 60, 70, 90, 100, 110, 120, 130, 140, 160, 170, 190, 210 and 230.Step 6: Compare these with the given options. We see that 230 is possible (all four boxes), 190 is possible (70 + 100 + 20), and 160 is possible (100 + 20 + 40).Step 7: The total 200 kilograms does not appear in the list of achievable sums, so it cannot be the sum of any subset of the given weights.

Verification / Alternative check:
We can also reason about 200 directly. The largest total less than or equal to 200 using the two largest weights is 100 + 70 + 40 = 210, which is already too large. So we cannot use all three of 70, 100 and 40 together. Trying 100 + 70 + 20 gives 190, which is close but not 200. Using 100 + 40 + 20 gives 160. Using 70 + 40 + 20 gives 130. No combination of three boxes gives 200, and clearly no combination of two or one box gives 200, since the possible one and two box sums are already listed. This confirms that 200 is impossible.

Why Other Options Are Wrong:
230 kilograms is the sum of all four boxes: 70 + 100 + 20 + 40 = 230. 190 kilograms is the sum of 70, 100 and 20. 160 kilograms is the sum of 100, 40 and 20. Because each of these totals can be formed by a valid combination of the given weights, they are not the required impossible total.

Common Pitfalls:
It is easy to overlook certain combinations or to assume that a total is impossible without a complete check. Some learners might try only one or two combinations for 200 and conclude too quickly. Systematically listing sums or using logical bounds, such as the maximum and near maximum totals, helps avoid such errors.

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