Kreigaffe is correct, though few people will probably recognise that.
The reason for this, is the lack of recognition of the basic types/aspects of human behaviour words such as games, puzzles, competitions and even art represent.
Art and competitions are all about things done FOR, (and therefore TO), other people.
There are two types of puzzles - those we create, (which are things done FOR and TO other people aswell), and those we do not. An example of the latter would be figuring our how the universe operates, (covered by the sciences etc.). An example of the former would be a jigsaw, or crossword/sudoku etc.. All puzzles are about people interacting with things done TO them. The puzzles we create are therefore about creating things FOR other people to interact with.
Competition is about people TRYING to gain something, (either by something that happens TO them, or something they DO), at the expense of, or in spite of, the others. (The actual goal/reason for competing does not matter).
Competitions are about people competing (by whatever means) to have something done TO them. (To be told whether or not they have won).
Games, are about people competing by DOING something FOR themselves, which may involve something done TO other people or things. (Games are not, therefore about any goals that may be reached, just the process of trying to attain them). The most basic games of all, are a race, fighting, competitive movement and/or throwing - almost every game in history is either purely that, or an abstract derivation of such things, from snakes and ladders, (which if take away the race, just leaves a board/dice game and turn-based, which are types of games, not games in themselves), to football and World of Warcraft. If a product does not contain any of this, then the chances are extremely big - (if not certain) - that it is not actually a game to begin with - it's probably either a puzzle, competition or just a work of art instead.
Games are DEFINED by what the player DOES, and what they use to do it with!
Yes, the line between doing something FOR ourselves and interacting with something done TO us, is extremely fine, which is why people have problems... Though it's NOT helped by people taking things which are already considered to be puzzles OUTSIDE of a computer, and then calling them a GAME just BECAUSE it's ON a computer!
Note that because the behaviour the words puzzle and game represent are mutually exclusive - (either it's something you DO or it's something that happens TO you) - puzzles and games CANNOT co-exist!
(Note I've been working on another more fundamental reason for the problem here within the English Language, though I really need to re-write the paper about it - hint: it's because the way we learn and use the language to recognise human behaviour is generally subjective).