Best "fair play" mysteries?
The first story in Morrison's JLA is a fair play mystery that Batman solves. There are fair play mysteries in Morrison's Bat-run, too, but they're mostly misdirection (like the clues and mystery of "The Island of Doctor Mayhew"). Peter Milligan did some good fair play mysteries during his time with Batman, including "The Executioner Wore Stiletto Heels" which is just amazingly good. Kia Asamiya's Child of Dreams was a fair play mystery. "Run, Riddler, Run" and unless I'm confusing it with something else, "Room Full of Strangers".
Arkham Asylum: a Serious House on Serious Earth and Catwoman Defiant are both awkwardly fair play and not fair play mysteries, as we require information the detective on hand doesn't have to see it coming, same for "Last Rites".
Most Batman stories tend to be adventure stories before anything else, with mystery, horror, or soap opera relationship stuff taking a very secondary or assisting role. There are very few fair play whoddunits or howdunnits.