On a digital clock showing hours and minutes, how many different readings between noon and midnight contain at least one 3?
In each hour between 12:00 pm and 12:00 am [excluding 3:00 pm to 3:59 pm], the minute hand shows: 16 "threes" x 11 =176 times - repeats counted.
Between 3:00 pm and 3:59 pm, the hour hand + the minute hand show: 76 "threes" - repeats counted.
Total number of "threes" that appear between 12 pm and 12 am =176 + 76 =252 "threes" - with all repeats such as 1:33, 3:03, 3:33....etc. counted.
