How many chimes in a day?

Question

A clock chimes once at one o'clock, twice at two o'clock, and so on up to twelve chimes at twelve o'clock. It also chimes once at every half hour. How many times does it chime in a full 24-hour day?

Accepted Answer

180 chimes. Hourly chimes. In one 12-hour cycle the clock chimes 1 + 2 + 3 + ... + 12. A quick way to add: pair 1 with 12, 2 with 11, and so on; six pairs of 13 make 78. Two cycles in a day: 78 × 2 = 156. Half-hour chimes. There are 24 half-hours in a day (one per hour), each with a single chime: 24. Total. 156 + 24 = 180. The pairing trick. Adding 1 to n by pairing the ends gives n(n + 1)/2, so 1 to 12 is 12 × 13 / 2 = 78. It is the same trick young Gauss reputedly used to add 1 to 100, and it turns up in almost every "how many handshakes" or "how many strikes" question. A clock that chimes …

How many chimes in a day?

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