*All the solutions to this I found online were either unnecessarily wordy or confusing due to formatting/syntactical issues. So, I'm adding mine which I will try to keep short and clear. *

**The Problem: You are given 12 balls. All of them are identical except one, which is either heavier or lighter than the rest. You are given a simple two-arm balance scale, which you can only use three times. How do you identify the odd ball, and whether it is lighter or heavier?**

**Solution:** Divide balls into three equal groups (A,B,C) and number them (1-4) in each group.

**Step 1. **Weigh Group A against Group B:

*They balance*=> Odd ball is in Group C**Step 2.**Weigh [C1, C2, C3] against any three normal balls (from groups A or B since they don't contain the odd ball).*They balance*=> C4 is odd**Step 3.**Weigh C4 against any other ball to see if it is lighter or heavier.*[C1, C2, C3] heavier*=> One of [C1, C2, C3] is heavier**Step 3.**Weigh C1 against C2. The heavier one is the odd ball, otherwise C3 is heavier.*Normal balls heavier*=> One of [C1, C2, C3] is lighter**Step 3.**Weigh C1 against C2. The lighter one is the odd ball, otherwise C3 is lighter.

*Group A heavier*=> Either Group A contains a heavier ball or Group B has a lighter one**Step 2.**Replace [B1, B2, B3] with three normal balls (let's say [C1,C2,C3] and swap A4 with B4. In other words, weigh [A1, A2, A3, B4] against [C1, C2, C3, A4].*They balance*=> One of the removed balls [B1, B2, B3] was lighter**Step 3.**Weigh B1 against B2. The lighter one is the odd ball, otherwise B3 is lighter.*[A1, A2, A3, B4] heavier*=> (Odd ball hasn't moved.) One of [A1, A2, A3] is heavier**Step 3.**Weigh A1 against A2. The heavier one is the odd ball, otherwise A3 is heavier.*[C1, C2, C3, A4] heavier*=> (The odd ball has switched sides.) Either A4 is heavier or B4 is lighter.**Step 3.**Weighing one of them against a normal ball should tell you which is the case.

*Group B heavier***=>**This case is the same as Group A being heavier, just swap the A/B labels.