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.
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