I own a digital 24-hour clock (i.e., a military time clock) that displays in HH:MM format. Each digit uses the standard seven-segment display in which seven lights toggle on or off to form numbers, as shown below. This involves 28 total light segments: four digits on the display times seven segments per digit. On my clock, some of these light segments are broken and never turn on. Despite this, with some practice, I can always tell what time it is—the displayed pattern of lights is never ambiguous. What is the maximum number of segments that can be broken?
The optional hint tells you how many segments can be broken. Your task is to then find which ones.
Up to 13 segments can be broken. A few patterns work; try to find one of them.
Up to 13 segments can be broken. One possible allocation is depicted below, with yellow segments representing broken lights.
The leftmost digit on a 24-hour clock only cycles from 0 to 2. When the digit is a 0, both lights shown are illuminated. When the digit is a 1, neither light is illuminated. And when it’s a 2, only the top light is lit up. So we can distinguish between the three possibilities. It doesn’t have to be these two segments specifically; several other pairs work as well.
The second and fourth digits on the clock have cycles of 0 to 9. Breaking any one of the seven segments creates an ambiguity except for the two broken ones shown above. For example, if the top segment breaks, then 1 and 7 will look the same. And if the middle segment breaks, then 0 and 8 will look the same.
The third digit has cycles of 0 to 5. In principle, you need a minimum of three functioning lights to distinguish the six possible digits. And in practice, this is achievable with the pattern shown. It would also work to break the top light instead of the bottom one.
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