Colored Hats, Round Train and Smashed Eggs

Let's have fun with hats of all colors, truth and lie tellers, round train, jailed prisoners and much more. They all are taking part in various seemingly irresolvable math riddles. Apparently these riddles have simple solutions. No math background is required, but an open-minded desire to think out of the box is most welcome.

(Recommended for 14 years old and up.)