On a digital clock showing hours and minutes, how many different readings between noon and midnight contain at least one 3?
+2666 Lets count them case by case:
\(3: \text {__}\) - 60 cases, 1 for each minute
\(\text _ : 3 \text _ \) - 11 cases for the hour hand (we already counted for 3), 10 cases for the ones digit, so 110 cases
\(\text{_} : \text{_} 3\) - 11 cases for the hour hand (we already counted for 3), 6 cases for the tens digit( anything other 3), so 66 cases
In total, there are \(60 + 110 + 66 = \color{brown}\boxed{236}\)
