I used to be able to access the mashes by clicking on "quiz" then scrolling down to index and then scrolling to "today's mashes". Ten were displayed and then the next hour 10 more were displayed.
Yesterday, I finally found a mash under each quiz type. I played one and was told no score would be recorded as I'd already played that day's mash!!
I played a different theme and was told the same thing.
Finally, I found a theme that actually scored for me.
How do I know whether I've played a mash? How can I ever hope to progress toward the "master masher" badge?

I don't think those are the ones being referenced. I believe they changed the name on you! What you need for this badge is to go to any category, sub-category, etc. and play the "mixed quiz game". Since the changes, there may no longer be a way to see if you've played it that day or not, wasn't one I could find since "recent scores" doesn't seem to show up until after you've played it. But there are literally hundreds from which to choose, so you'll probably be able to remember on a daily basis.