Let's assume that we have a cryptocoin with a 3-block retarget period and 10 minutes between blocks.

A first blockchain looks like:0th block: timestamp 00:00, difficulty 1 <- first block after retarget

1st block: timestamp 00:05, difficulty 1

2nd block: timestamp 00:10, difficulty 1

3rd block: timestamp 00:20, difficulty 2 <- first block after retarget

4th block: timestamp 00:30, difficulty 2

5th block: timestamp 00:40, difficulty 2

Cumulative difficulty = 1 + 1 + 1 + 2 + 2 + 2 = 9

Hashes calculated (excluding 0th block) = N + N + 2N + 2N + 2N = 8N

Time spent = 00:40 - 00:00 = 40 minutes

Hashpower wasted = 8N / 40 = 0.2N hash/min

A second one looks like:0th block: timestamp 00:00, difficulty 1 <- first block after retarget

1st block: timestamp 00:10, difficulty 1

2nd block: timestamp 00:20, difficulty 1

3rd block: timestamp 00:30, difficulty 1 <- first block after retarget

4th block: timestamp 00:40, difficulty 1

5th block: timestamp 00:50, difficulty 1

6th block: timestamp 01:00, difficulty 1 <- first block after retarget

7th block: timestamp 01:10, difficulty 1

8th block: timestamp 01:20, difficulty 1

Cumulative difficulty = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 9

Hashes calculated (excluding 0th block) = N + N + N + N + N + N + N + N = 8N

Time spent = 01:20 - 00:00 = 80 minutes

Hashpower wasted = 8N / 80 = 0.1N hash/min

Which blockchain is better? If you take into account the cumulative difficulty you can choose the second one, which looks a bad choice coz it requires hardware with lower "intensity" of hashing. But if we took

**squares** of the difficulty of each block we would definitely choose the first blockchain (15 vs 9).

You may ask why squares? Well... I think there is a mystical connection to

http://en.wikipedia.org/wiki/Least_squares.