Total buzzes made by a striking clock in 24 hours — A clock buzzes 1 time at 1 O’clock, 2 times at 2 O’clock, …, 12 times at 12 O’clock. How many total buzzes occur in a full day (24 hours)?

Introduction / Context:
A striking clock repeats a daily pattern: during each 12-hour cycle it chimes 1 + 2 + … + 12 times at the top of each hour. Over 24 hours, this exact 12-hour sequence happens twice, so the total in a day is just double the 12-hour sum.

Given Data / Assumptions:

• Within any 12-hour span, chimes follow the sequence 1, 2, 3, …, 12.
• The day contains two such spans: 0–12 and 12–24 (or a.m. and p.m.).
• We assume no extra chimes beyond the hour counts.

Concept / Approach:
Use the arithmetic series formula for 1 through 12, then double. Sum of first n integers is n(n + 1)/2. Here n = 12.

Step-by-Step Solution:
Sum(1..12) = 12 × 13 / 2 = 78.Daily total = 2 × 78 = 156.

Verification / Alternative check:
Spot-check: 1 + 2 + 3 + 4 = 10; 5 + 6 + 7 + 8 = 26; 9 + 10 + 11 + 12 = 42; total 10 + 26 + 42 = 78; doubling gives 156.

Why Other Options Are Wrong:
100 and 150 undercount; “None of these” is unnecessary because 156 precisely matches the calculation.

Common Pitfalls:
Accidentally adding 12 twice or forgetting to double for 24 hours. Another mistake is assuming 0 chimes at 12, but standard striking conventions use 12 chimes at 12 O’clock.

Final Answer:
156