A puzzle mug is a novelty that is counter-intuitive. It will mainly have handful holes in the rim, authoritative it impossible to drink from in the normal way. Frequently the solution is to drop all the holes in the rim, and then sip via Photo Mugs another cistern in the striated handle.
A travel mug is a variation on the traditional mug that is more useful for transporting hot liquids. It may or may not be a vacuum flask, but is as usual well insulated and completely enclosed, with an easily closed opening on the top.
