Question: You are provided with $8$ identical balls and a measuring instrument. $7$ of the eight balls are equal in weight, and one of the eight given balls is defective and weighs less. The task is to find the defective ball in exactly two measurements.

1. **Problem Statement:** You have $8$ balls, $7$ of which have the same weight, and $1$ is defective and lighter. You need to find the defective ball using exactly $2$ weighings. 2. **Key Idea:** Use a balance scale to compare groups of balls. Each weighing can have three outcomes: left side heavier, right side heavier, or balanced. 3. **Step 1: Divide the balls into three groups:** - Group A: $3$ balls - Group B: $3$ balls - Group C: $2$ balls 4. **Step 2: First weighing:** Weigh Group A against Group B. - If they balance, the defective ball is in Group C. - If they do not balance, the lighter side contains the defective ball. 5. **Step 3: Second weighing:** - If Group C contains the defective ball (from step 4), weigh the two balls in Group C against each other. - The lighter one is defective. - If Group A or B contains the defective ball, take any two balls from the lighter group and weigh them against each other. - If one is lighter, that is the defective ball. - If they balance, the ball not weighed is defective. **Final answer:** You can find the defective ball in exactly two weighings by dividing the balls into groups of $3$, $3$, and $2$, and following the weighing strategy above.