Re. the OP: I had viewed some "Scrabble" earlier this morning, and tried to see how the letters were displayed. To be honest, I'm stumped too. Rear projection would've been my guess, but such a system would probably have been too large and klunky to incorporate into a desk that size (especially with contestants sitting back there, and the various trays for the tiles being in the center). CRTs would be my best guess. Actually, CRTs would probably display scan lines at times (unless they were genlocked to the rest of the studio), and would probably not have looked as sharp as they had on the show. So I'm still stumped.
Your remark on rear projection got me thinking; you could probably rig up a quadruple mirror system and do it with RP. Projector behind the contestant backdrops, use four mirrors to reflect down under the rotating platform's floor, forward to under the desk, and up to a final mirror mounted behind the visible display. Hide it all inside black curtains to eliminate light penetration and it might work. That being said, I can't imagine any producer in his right mind (or even a batsh*t crazy one) saying "Yeah, that sort of complexity is really going to pay off..."
But there are definitely CRT's involved somehow; they'd lose sync every so often when switching the displays between contestants. See, for example, this clip; watch the right-side contestant's display after the stopper.
Also perhaps worth mentioning: they are, for all intents and purposes, monochrome displays. Yes, they're blue and white instead of black and white, but they're still just two-color displays; maybe sacrificing color gave them room to experiment with frame rates or resolution?