Situation 1: You have eight billiard balls. One of them is "defective," meaning that it weighs more than the others. How do you tell, using a balance, which ball is defective in two weighings?

Situation 2: You have five jars of pills. All the pills in one jar only are "contaminated." The only way to tell which pills are contaminated is by weight. A regular pill weighs 10 grams; a contaminated pill is 9 grams. You are given a scale and allowed to make just one measurement with it. How do you tell which jar is contaminated?

Situation 1: To determine which ball is defective using a balance in just two weighings, you can follow the steps below:

1. Divide the eight balls into three groups: three balls in one group, three balls in another group, and two balls in the third group.
2. Weigh the first group of three balls against the second group of three balls on the balance.

- If both sides balance, it means the defective ball is in the group of two remaining balls.
- If one side is heavier, it means the defective ball is in the heavier group of three balls.

3. Take the group of two balls identified in step 2 and weigh them on the balance.

- If both sides balance, the remaining ball is the defective one.
- If one side is heavier, that ball is the defective one.

By following these steps, you can identify the defective ball by using the balance in just two weighings.

Situation 2: To determine which jar of pills is contaminated using a scale in just one measurement, you can do the following:

1. Assign each jar a number from 1 to 5 for easier reference.

2. Take a different number of pills from each jar, matching the jar number with the number of pills taken.

- For example, take 1 pill from the first jar, 2 pills from the second jar, 3 pills from the third jar, 4 pills from the fourth jar, and 5 pills from the fifth jar.

3. Place all the pills you have taken on the scale.

- If all the jars were clean, the total weight would be equal to (1 * 10) + (2 * 10) + (3 * 10) + (4 * 10) + (5 * 10) = 150 grams.

4. Check the weight measurement on the scale against the expected weight of 150 grams.

- If the total weight is less than 150 grams, it means the jar corresponding to the number of pills taken is contaminated, as the contaminated pills weigh 9 grams each.
- If the total weight is equal to 150 grams, it means all the jars are clean, as none of the pills in any jar is contaminated.

By following these steps, you can determine which jar of pills is contaminated using the scale in just one measurement.

Situation 1: To find the defective ball among the eight billiard balls using a balance in two weighings, you can follow these steps:

1. Divide the eight balls into three groups: Group A with three balls, Group B with three balls, and Group C with two balls.
2. Weigh Group A against Group B using the balance.

- If Group A and Group B balance each other, the defective ball must be in Group C.

3. Take one ball from Group C and weigh it against another ball from Group C.

- If the balance is even, the ball left off the balance is the defective one.
- If the balance is uneven, the heavier ball on the scale is the defective one.

Situation 2: To identify the contaminated jar among five jars of pills using just one measurement on a scale, you can follow these steps:

1. Take one pill from the first jar, two pills from the second jar, three pills from the third jar, four pills from the fourth jar, and five pills from the fifth jar.

2. Weigh all the collected pills on the scale.

- If the total weight of the pills is a multiple of 9 (e.g., 9 grams, 18 grams, 27 grams, etc.), the first jar is contaminated since it would have contributed 9 grams less if all the pills were regular weight.

- If the total weight is not a multiple of 9, one of the other jars is contaminated. To find the specific jar, subtract the total weight by the nearest multiple of 9, and the result will indicate the contaminated jar number.

For example, if the total weight is 35 grams, the nearest multiple of 9 is 36 grams. Subtracting 35 from 36, we get 1, indicating that the first jar is contaminated.