Re: what 3 calibers cover 0 to 1600 meters??
I think that it may pay to think in practical terms. I would weigh each selection based on the amount of shooting I can reasonably expect to be using it for, that it is best suited to do.
For me, ammo cost and availability are a key factor. In choosing shooting platforms, I would personally be front-weighting the 0-250yd end of the spectrum with special emphasis. That's where I'd be seeking to do the main bulk of my marksmanship maintenance/practice shooting, preferably with a .22LR. Can't beat the cost and the accuracy inherent in the basic cartridge; and there's no handloading time overhead, either.
For then next longer distance regime, I like the .223, but for versatility sake, I'd probably somewhat favor a .22-250 with a 1:8" twist, with the stipulation that this chambering can just as well be downloaded to .223 performance levels with no bore life or accuracy sacrifices, yet still pack a full .22-250 punch when called upon to do so.
For the longest stretch, I'd be asking myself just how much of the last 600yd of the 1600yd am I likely to be shooting on a regular basis. Personally, the next time I shoot past 1Km will be the first time, and there is no place I can drive to and do so without at least one overnight stay involved. That said, I think a .260 might be stoked to reach 1400yd, and anything between 1400yd and 1600yd will just have to take it's chances out there. Personally, I wouldn't be contemplating doing jumping jacks and thumbing my nose at the shooter out there just yet.
Bigger chamberings are just too costly to feed, IMHO; and the proper bits are getting just a little too difficult to find, as well.
Greg