Maths olympiad question · P5–P6★★★★ Expert
Clocks and time: olympiad question 10
An expert question on clocks and time for P5–P6, with a step-by-step solution, diagrams and the answer. Every year of the Hua Xia Cup has clock questions, from time passed in P1 to the angle between the hands in P6.
The question
A digital clock shows 24-hour time:
7:05 am appears as 07:05, and 9:30 pm as 21:30.
From 03:00 to 08:59 it shows a new time every minute.
How many of those times have four different digits adding up to 10?
Show the hints (3)
- Hint 1: Fix the hour first: with its two digits known, what must the minute's two digits add up to?
- Hint 2: The minutes' tens digit can only be 0 to 5 and the units 0 to 9; and all four digits must differ — from the hour's digits too.
- Hint 3: For example at 03, the minutes that work are 16, 25, 52. List every hour like this.
Worked solution
Have a go first, then read the solution step by step.
- 1.Hour by hour: once the hour's digits are known, the minute's digits must add to 10 − the hour's digit sum. The minutes' tens digit is 0 to 5, and all four digits must differ.
- 2.03: minute digits must add to 7 — 16, 25, 52: 3; 04: minute digits must add to 6 — 15, 51: 2; 05: minute digits must add to 5 — 14, 23, 32, 41: 4; 06: minute digits must add to 4 — 13, 31: 2; 07: minute digits must add to 3 — 12, 21: 2; 08: minute digits must add to 2 — none.
- 3.In all 3 + 2 + 4 + 2 + 2 = 13.
- ⚠️Common mistake: checking the sum only and forgetting "all different" (at 03, for example, drop minutes repeating an hour digit); or letting the minutes' tens digit be 6 to 9.
Answer: 13 times
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