I picked up a trek 820 to ride around and it's pretty great except for one small problem (get it? SMALL). The bike won't shift into the smallest back cog. In fact, the whole thing is one "gear" off. I suppose the bike could be all buggered up but I tend to think that some "minor adjustment" is more in order. The cable is plenty slack in the low and since thins bike was used daily before I got my hands on it, I tend to think that major screwing around is not necessary but I guess.

Yes, this in my one zillionth shifting question. I just can't get the hang of it.

1) Hanger is not bent?
2) Derailleur is not bent?
3) High limit screw is adjusted properly?

^ Disconnect the cable and https://www.parktool.com/repair/readhowto.asp?id=64 on setting the high limit - probably off.

If the cable's slack in the smallest cog it'll shift to, it's the high limit screw.

He's got the opposite problem with the cable being too tight and off by one click. The shifter-position for the smallest cog ends up dropping into the 2nd cog. He needs to unbolt the RD cable and loosen up about 3mm of cable. Depending upon the position of the barrel-adjuster, he might be able to spin it in enough to loosen up the cable enough to drop the RD into the 1st cog.

"The cable is plenty slack in the low"
I think he means the cable is slack in the high gear, which would indicate either the high limit screw needs to be adjusted or the derailer hanger is bent.

He DID mean "high" gear and sure enough adjusting the high limit screw did the trick.

Now, if any one is interested a little more reading.....

I re-set the high limit screw and the bike shifts fantastically right now. When I got it the bike it had a wheel with a 26" slick on it and another wheel with the stock tire on it. Both have 34t "mega-range" gears on them. After I adjusted the deraillure with the DIRT tire-wheel installed I put the slick tire-wheel back on and guess what. Yep, the shifting was off by one gear! I thought all the spacing on such things was standard. How can one wheel require a different setting/adjustment than another? Is it just that one is OEM and the other aftermarket?

Not unusual to find that two separate wheels require retrimming the settings.

Yup, just different geometry on different hubs and freewheels, sometimes its a big difference and sometimes everythings just a smidge off.

Measure the distance between the dropout and small cog for each wheel.

You can prolly make em the same if you can be stuffed arseing about (shuffling washers, locknuts, slight re-dish).