An inspector of weights and measures uses a special set of weights to check the accuracy of scales. Various weights are placed on a scale to check the accuracy of any amount from 1 oz. to 15 oz. What is the fewest number of weights the inspector needs? What weights are needed to check accuracy of scales from 1 oz. through 15 oz.? From 1 oz. through 31 oz.?

To determine the fewest number of weights needed to check the accuracy of scales from 1 oz. to 15 oz., we'll first examine the pattern of weights required.

For each weight, we will denote it as its equivalent value in binary form. For example, the weight of 1 oz. can be represented as 1, 2 oz. as 10, 3 oz. as 11, and so on.

To cover the range from 1 oz. to 15 oz., we observe that the binary representation of these numbers covers the range from 1 to 1111 in binary.

To find the fewest number of weights, we need to determine the number of unique digits required to represent the range from 1 to 1111.

From 1 to 15 in binary: 1, 10, 11, 100, 101, 110, 111.
In this range, we require four unique digits: 0, 1, 10, 11.

Hence, the fewest number of weights the inspector needs is four.

The weights needed to check the accuracy of scales from 1 oz. through 15 oz. are: 1 oz., 2 oz., 4 oz., and 8 oz.

To extend to the range from 1 oz. through 31 oz., we consider the binary representation from 1 to 11111.

From 1 to 31 in binary: 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000, 10001, 10010, 10011, 10100, 10101, 10110, 10111, 11000, 11001, 11010, 11011, 11100, 11101, 11110, 11111.

In this range, we require five unique digits: 0, 1, 10, 11, 100.

Hence, the fewest number of weights needed to check the accuracy of scales from 1 oz. through 31 oz. is five.

The weights needed are: 1 oz., 2 oz., 4 oz., 8 oz., and 16 oz.

To find the fewest number of weights needed, we can start by observing the pattern of weights required to check the accuracy of scales from 1 oz. to 15 oz.

Let's list all the possible combinations of weights that can be used to measure every weight from 1 oz. to 15 oz:

1 oz: 1 oz
2 oz: 2 oz
3 oz: 1 oz + 2 oz
4 oz: 4 oz
5 oz: 1 oz + 4 oz
6 oz: 2 oz + 4 oz
7 oz: 1 oz + 2 oz + 4 oz
8 oz: 8 oz
9 oz: 1 oz + 8 oz
10 oz: 2 oz + 8 oz
11 oz: 1 oz + 2 oz + 8 oz
12 oz: 4 oz + 8 oz
13 oz: 1 oz + 4 oz + 8 oz
14 oz: 2 oz + 4 oz + 8 oz
15 oz: 1 oz + 2 oz + 4 oz + 8 oz

From the list above, we can see that the fewest number of weights the inspector needs to measure from 1 oz. to 15 oz. is 4. These weights are 1 oz, 2 oz, 4 oz, and 8 oz.

To check the accuracy of scales from 1 oz. through 15 oz., the inspector needs weights of 1 oz, 2 oz, 4 oz, and 8 oz.

To check the accuracy of scales from 1 oz. through 31 oz., we can extend the pattern above:

16 oz: 16 oz
17 oz: 1 oz + 16 oz
18 oz: 2 oz + 16 oz
...
30 oz: 2 oz + 4 oz + 8 oz + 16 oz
31 oz: 1 oz + 2 oz + 4 oz + 8 oz + 16 oz

So, to measure from 1 oz. through 31 oz., the inspector needs weights of 1 oz, 2 oz, 4 oz, 8 oz, and 16 oz.

In summary, the fewest number of weights the inspector needs to measure from 1 oz. through 31 oz. is still 4, but the weights required are 1 oz, 2 oz, 4 oz, 8 oz, and 16 oz.

a. 8 Weights.

b. 1, 3, 5, 7, 9, 11, 13, 15 oz.

c. Same as b.