A wall clock gains **2 minutes** every **12 hours**, while a table clock loses **2 minutes** every **36 hours**. Both clocks are set to show the correct time at **12 noon on Tuesday**. When will both clocks show the same time again, and what will that time be?

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To solve this problem, we need to determine when the wall clock and table clock will show the same time again.

First, let's calculate how the **difference between the clocks** changes over time:

In **36 hours**:
- The wall clock gains: (36 ÷ 12) × 2 = **6 minutes**
- The table clock loses: **2 minutes**
- Total difference accumulated in 36 hours: 6 + 2 = **8 minutes**[2][5]

The clocks will show the same time when their difference is exactly **12 hours** (720 minutes). This happens because when one clock is exactly 12 hours ahead, both will display the same time (just with AM/PM difference).

To find when this occurs:
- Time needed = 720 minutes ÷ 8 minutes per 36 hours × 36 hours
- Time needed = 90 × 36 hours = **3240 hours**[2]

Converting to days:
- 3240 hours ÷ 24 hours/day = **135 days**[2][5]

Since both clocks were initially set at 12 noon on Tuesday, after 135 days, it will again be 12 noon (on a different day).

Therefore, both clocks will next show the same time at **12 noon, after 135 days**.

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