This is one of my sillier endeavors but I think it will make sense to those other like-minded engineers/designers out there.
So, I have a pet peeve: wasting toothpaste at the end of the life of the tube. It bugs me when there's something leftover in any type of container for that matter. But I deal with toothpaste on a daily basis so it is a perpetual nuisance. The solid exoskeleton toothpaste containers seem like they're even worse than the soft tubes that you could squeeze or roll until you're blue in the face.
A couple of days ago I was getting to the end of another solid tube and the frustration got to the point that I had to take it apart just to see how much I was wasting. So I got my exacto and tore into it. To my surprise there was only a small glob of toothpaste at the very tip, only a single toothbrushing-worth.
It had felt like much more than that, not to mention I didn't know how it was possible to have that little in the rest of the foil-like sack. I was about to chock it up to "who knows" when I noticed something on the base.
That's right, whoever designed this achieved a one-way valve with a tiny piece of plastic and a heat-weld with a couple small interruptions for air to pass in only one direction. That allowed it to squeeze as much as possible out of the toothpaste sack. Something so cheap and simple yet innovative to solve man's toothpaste dispensing needs. It's the small things in life :).