On the planet Oz, there are 8 days in a week — Sunday to Saturday and another…

Concept

On any clock, the angle between the hour and minute hands at a given moment equals the absolute difference between each hand's own angular position. Each hand's position is found as (its own degrees-per-minute rate) × (elapsed time), where the rate comes from how long that hand takes to complete one full 360° revolution of the dial — this holds regardless of how many hours are in a day or minutes in an hour.

Application

1. Hour hand's rate: it completes the dial TWICE in Oz's 36-hour day, so one full revolution takes 36 ÷ 2 = 18 hours. Rate = 360° ÷ 18 = 20° per Oz-hour. Since 1 Oz-hour = 90 minutes, that is 20 ÷ 90 = 2/9° per minute.

2. Minute hand's rate: it completes one full revolution every 90-minute Oz-hour. Rate = 360° ÷ 90 = 4° per minute.

3. At 14:40 (14 hours 40 minutes), hour hand's position = 14 × 20° + 40 × (2/9)° = 280° + 8.89° ≈ 288.89°.

4. Minute hand's position = 40 × 4° = 160°.

5. Angle between the hands = |288.89° − 160°| = 128.89° ≈ 129° (already less than 180°, so this is the required angle).

Cross-check

Recompute from total elapsed minutes since 00:00: 14 × 90 + 40 = 1300 minutes. Hour hand position = 1300 × (2/9)° = 2600/9° ≈ 288.89° (matches). Minute hand position = (1300 mod 90) × 4° = 40 × 4° = 160° (matches) — confirming the angle is ≈129°.

Result

The angle between the hour and minute hands on Oz at 14:40 is approximately 129°.