A grandfather clock strikes at regular intervalsIt strikes the number of hours on the hour, twice for every half hour (30 minutes past) and once for each of the remaining two quater hours (15 minutes and 45 minutes past).How many times does the clock strike in 24 hours?

Let's break this down:

1. The clock strikes the number of hours on the hour. This means it strikes 1 time at 1 o'clock, 2 times at 2 o'clock, and so on, up to 12 times at 12 o'clock. This cycle repeats twice in a 24-hour period (once for AM and once for PM). So, the total number of strikes for the hours in a day is 2*(1+2+3+...+12).

2. The clock strikes twice for every half hour. There are 24 half hours in a day (two in each hour), so the total number of strikes for the half hours in a day is 2*24.

3. The clock strikes once for each of the remaining two quarter hours. There are 48 quarter hours in a day (four in each hour), so the total number of strikes for the quarter hours in a day is 2*48.

Now, let's add these up:

1. The sum of the first 12 positive integers is given by the formula n*(n+1)/2, so the sum of the first 12 positive integers is 12*(12+1)/2 = 78. Therefore, the total number of strikes for the hours in a day is 2*78 = 156.

2. The total number of strikes for the half hours in a day is 2*24 = 48.

3. The total number of strikes for the quarter hours in a day is 2*48 = 96.

Adding these together, the clock strikes 156 + 48 + 96 = 300 times in 24 hours.

Let's break this down:

1. The clock strikes the number of hours on the hour. This means it strikes 1 time at 1 o'clock, 2 times at 2 o'clock, and so on, up to 12 times at 12 o'clock. This cycle repeats twice in a 24-hour period (once for AM and once for PM). So, the total number of strikes for the hours in a day is 2*(1+2+3+...+12).

2. The clock strikes twice for every half hour. There are 24 half hours in a day (two in each hour), so the total number of strikes for the half hours in a day is 2*24.

3. The clock strikes once for each of the remaining two quarter hours. There are 48 quarter hours in a day (four in each hour), but we've already counted the strikes for the half hours, so we only need to count the additional strikes for the 15-minute and 45-minute marks. This gives us 2*24 strikes.

Adding these all together, we get the total number of times the clock strikes in 24 hours.