Your solution that I quote below is not correct, since the fuses do not burn at consistent rates. So when you light from the middle and both ends, you may have one half that burns up in 2 minute, while the other half takes 58 minutes (or half of 58 minutes, since it lit from both ends).
You said:
Light one in the middle (or light both ends), it will take one half hour to burn.
When that is finished burning light each end and the middle of the next one and it will take 15 minutes to burn.
That should take 45 minutes.