Question

8.

(a) Your local Be-Computer store has received 10 containers with 100 bePads in each container. Each bePad is marked to identify the container it is from. Other than that, all bePads are identical and weigh exactly one pound, except for those in one container that have a manufacturing defect. Those defective bePads weigh exactly 17 ounces each. You are asked to identify the bad container. Of course, you could weigh one bePad from each container until your scale measures 17 ounces. But this process may take 10 weighings and you are asked to instead use only one single weighing. The good news is that your scale is an industrial grade (single platform) scale and you can place any number of bePads on its platform. How then do you identify the problematic container by finding the weight of just one collection of bePads selected from various containers?

(b) Having heard of your success with single weighings, the next town’s Be-Computer store calls you for help. They have received 7 containers (with 100 bePads in each) and know that some of these containers may be coming from the lot of defective 17-ounce bePads. However, they don’t know how many containers are bad. It could be any number, including all or none. How can you in a single weighing identify all the bad containers?

Answer 1

(a)

Select one bePad from container 1, two bePads from container 2, . . . , ten bePads from container 10 and weigh this collection. Had no bePads been defective, the total weight would have been 1 + 2 + 3 + . . . + 10 = 55 pounds. But some container, say container n, has defective bePads. This container contributes n pounds + n ounces, instead of just n pounds. That is, the container number n is given by n = measured weight − 55 expressed in ounces.

(b) Select powers of two, that is, from container 1: 1 bePad, from container 2: 2  bePads, from container 3: 4 bePads, . . . , from container 7: 64 bePads. If no container is bad, the weight of this collection is 1 + 2 + 4 + 8 + 16 + 32 + 64 = 127 pounds. If container n is bad, it contributes 1 × 2 n−1 additional ounces. That is, to identify the bad containers, you find the binary representation of the number measured weight − 127 expressed in ounces. The 1’s in this representation indicate the bad containers. For example, a binary 0010101 would indicate that containers 1, 3, and 5 are bad.

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