This is a game I originally encountered in the form of its Game Boy port, which was on the 32-in-1 pirate cart I had as a kid, and have mentioned several times on this blog previously. Unfortunately, the Game Boy version is a pretty bad port, for reasons I'll get into later, so I'm reviewing the original arcade release instead.
Anyway, Palamedes is a "matching stuff as it descends from the 'bove"-type puzzle game. This time you're not matching colours, but sides of a die. In a similar manner to Magical Drop (though predating that series by a few years), you control a little character at the bottom of the screen, who holds a six-sided die above their head. Pressing one button cycles through the sides of the die, and the other throws it upwards, where rows of dice are steadily advancing downwards. You throw the die at other die showing the same face to make them disappear. When the advancing rows reach the bottom of the screen, it's game over. There's a solitary score attack mode, as well as versus modes where you can compete against another human player, or a series of AI opponents.
And, were I reviewing the Game Boy port, that's where the description would end, as that port omits the most interesting aspect of the game: the fact that by clearing dice in the right order, you create simple mahjong-esque "hands", that can be used to clear several lines of descending dice at once. There's a whole bunch of different hands to get, from simply getting the same number three or more times in a row, to getting all the numbers one to six in order, and a bunch of others in between.
Clearing lines in this way is by far the best way of scoring points, and in all modes, clearing a certain number of lines is the way to advance the level. In the competitive modes, clearing lines via your hands is also the way you attack your opponent: lines you clear are added to their field at the same time. Without this whole thing, the Game Boy port is not only a lot less interesting, it's also so difficult as to be almost unplayable.
This is the arcade version, though, and it's a game i definitely recommend giving a try. There's also a sequel on the Famicom that I haven't played, and there are a few ideas I feel would add to the game, so that's something I'll be looking into and probably covering here at some point in the future too.