The biggest digit sum a 24-hour clock ever shows

Question

A digital clock displays times from 00:00 to 23:59 using four digits. What is the largest possible sum of the four digits ever shown, and at what time does it occur? Explain why nothing larger is possible.

Accepted Answer

24, at 19:59. Minutes. The minutes run from 00 to 59. The tens digit is at most 5 and the units digit at most 9, so the minutes contribute at most 5 + 9 = 14, achieved at :59. Hours. The hours run from 00 to 23. If the tens digit is 2, the units digit can only be 0 to 3, so the best is 2 + 3 = 5. If the tens digit is 1, the units digit can be 9, giving 1 + 9 = 10. So the biggest hour digit sum is 10, at 19, not 23. Total. 10 + 14 = 24, at 19:59. Since the two parts were maximised independently and both maxima can occur at the same moment, 24 is the true maximum. In particular, a digit sum of …

The biggest digit sum a 24-hour clock ever shows

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