Grandfather clock shines the number of times that correspond to that time. How many times does it chime in 24 hours?

Question

Each hour a grandfather clock chimes the number of times that correspond to that time of the day. For example at 3 pm it will chime 3 times. How many times does it chime in a 24 hour day?

Answer 1

I'm going to assume that it chimes once at 1am and twice at 2am and so on until it chimes 12 times at noon. I'm also assuming that it does the same thing when it starts over at 1pm - chiming once, 2pm - chiming twice, 3pm - chiming three times...

 

So we need to add the numbers 1 through 12 up twice, and I'll write it in a clever way:

 

1   +  2  +  3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12

12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 +    2 +   1

 

If we add vertically first, we'll get the same thing as we would if we went in order, because it doesn't matter what direction you add things. 

 

Notice, going from left to right, we get 1 and 12 make 13, 2 and 11 make 13, 3 and 10 make 13, all the way until we get to 11 and 2 make 13, and 12 and 1 make 13. 

 

We can rewrite this as 

 

13 + 13 + 13 + 13 + 13 + 13 + 13 + 13 + 13 + 13 + 13 + 13 

 

Well, that's the same thing as 13 twelve times or 13x12 = 156.

Answer 2

I assume that your grandfather clock is a 12-hour clock (not a 24 hour clock).

 

For the first hour, it chimes once, the second twice, etc.

 

This gives us the summation of:

 

1 + 2 + 3 + ... + 11 + 12

 

This happens to equal:

 

(12)(12+1)/2 (it's the summation for n items, but can also be derived by simply listing the terms and doing some counting)

 

6  * 13 = 78

 

Keep in mind, this cycle happens twice, as we have the AM and the PM cycles, so our overall number of chimes that the grandfather clock performs is twice as many as the summation from 1 to 12, giving us:

 

2 * 78 = 156 shines