There are 8 balls. 7 of them weigh the same. 1 of them has a different weight, (you don't know if it's heavier or lighter). How do you find the odd ball with 2 weighs?

To find the odd ball among 8 balls, where 7 of them weigh the same and one is either heavier or lighter, using only two weighings on a balance scale, you can follow these steps:

Step 1: First Weighing

Divide the 8 balls into three groups: C1 with 3 balls (e.g., balls 1, 2, 3), C2 with 3 balls (e.g., balls 4, 5, 6), and C3 with 2 balls (e.g., balls 7, 8).

• Weigh C1 against C2.

Possible Outcomes:

• C1 equals C2: This means the odd ball is in C3 (balls 7 and 8).
• C1 is lighter than C2: This means the odd ball is either lighter and in C1 or heavier and in C2.
• C1 is heavier than C2: This is similar to the previous case but with the roles of C1 and C2 reversed.

Step 2: Analyzing the Outcomes

If C1 equals C2:

• The odd ball is in C3. Weigh balls 7 against 8.
  • If one is lighter or heavier, you have found the odd ball and determined whether it is lighter or heavier35.

If C1 is lighter than C2 (or C1 is heavier than C2):

• Let's assume C1 is lighter than C2. This means the odd ball is either in C1 and lighter, or in C2 and heavier.
  • Take the heavier side (C2) and weigh two of its balls against each other (e.g., balls 4 and 5).
  • If they are equal, the remaining ball in C2 (ball 6) is the odd ball and it is heavier.
  • If they are not equal, the lighter one is the odd ball if it was in C1, or the heavier one is the odd ball if it was in C224.

Alternative Method for the Second Weighing

If you prefer a more streamlined approach after the first weighing:

• If C1 equals C2, weigh balls 7 and 8 as mentioned.
• If C1 is lighter or heavier than C2, take the three balls from the heavier side and one ball from the lighter side, and replace one ball from the heavier side with one from the lighter side.
  • For example, if C1 is lighter, take balls 4, 5, 6 from C2 and replace ball 4 with ball 1 from C1.
  • Weigh the modified group against the remaining two balls from the lighter side plus one of the balls from C3.
  • If the sides are equal, the ball that was replaced (ball 4 in this case) is the odd ball. If not, the heavier side will indicate which ball is the odd one14.

This method ensures you identify the odd ball and determine whether it is heavier or lighter in just two weighings.