Clock Problem, Number of Chimes

Question

An old fashioned clock chimes as many times as the number of hours it is when it hits a new hour. For example, the clock ticks two times when the clock reads two or the clock ticks 12 times when the clock reads 12. Additionally, it ticks exactly one time in the middle of each hour (i.e. exactly half past twelve, half past one, etc.). What is the largest number of chimes the clock can make in 2015 minutes straight?

Answer 1

English clock idioms: old clocks strike or chime the hour (sometimes also half-hour and quarter hour); a tick is the sound of the escapement mechanism, typically once per second.

Let's abstract away the clock: Chiming events occur every 30 minutes. $2015 = 67(30) + 5$. That $+5$ minutes means it doesn't matter whether the 2015 minutes is an open or closed interval (if it were $2010=67(30)$ minutes, we'd have to decide whether chimes at the start and end of the interval were in the interval).

How many chiming events can we fit in $2015$ minutes? In a $5$ minute interval, we can fit $0$ events or $1$ event, depending on when the interval starts. Each additional $30$ minutes is another event, so $2015$ minutes can give us $67$ or $68$ events. Since we're trying to maximise the total number of chimes, choose $68$ events. That gives $34$ on-the-hour events and $34$ on-the-half-hour events. The number of half-hour events is fixed, allowing us to put aside the half-hour events (subtotal 34 chimes) and focus on the hourly chimes.

What's the largest total of 34 consecutive terms from the repeating sequence $1,2,\cdots,10,11,12,1,2,3,\cdots$? Every 12 consecutive terms sums to $\dfrac{12(12+1)}{2} = 78$ regardless of where we start. $34 = 2(12) + 10$, so the problem reduces to finding the largest total of 10 consecutive terms in $1,2,\cdots,11,12,1,2,\cdots$. This will be $3+4+\cdots+11+12=75$.

So the maximum number of chimes is $34 \text{ (half-hour events) } + 78 + 78 + 73 \text{ (hour events) } = 265$ chimes. It could occur in, say, the interval starting 2:55 ending 12:40, or the interval starting 2:25 ending 12:10.