There's a fantastic children's game called Froggie Boogie. The game comes with a lily-pad track, a small frog for each player, eight larger frogs in two colors each, and two eyes for each frog. A fly is pictured on the bottom of one of the eyes on each frog; the other eye has a blank bottom face.
On your turn, you roll the dice. The two colors rolled on the dice indicate a specific large frog. You choose to look at the bottom of one of the frog's eyes. If it shows a fly, replace the eye in its place, move forward on the lily-pad track, and go again. If it shows a blank face, replace the eye in its place and pass the dice to the player on your left. The first player to reach the end of the track wins.
At the start of the game, picking the eye with the fly is a matter of luck. However, you should, in theory, always be able to pick the correct eye once you've "done" a particular frog. In practice, you tend to forget, again and again, which eye has the fly, much to the amusement of your children who win at least as often as the parents do. It's a fun game for kids, or for kids to play with adults.
Solving Froggie Boogie ...
About halfway through my first game of Froggie Boogie, I realized that the game is solvable. I could assign an order number to each frog, and a binary bit to each eye (for example, right eye is 0, left eye is 1), and thus, after every play, adjust the value of the bits in order, convert the binary number to a decimal number, and remember the decimal number. On my turn, reverse the process to determine the correct location of the fly.
... is Cheating
I could have ... but I didn't. Instead, I continued to play the game the way I believed it was "meant to be played": I tried to remember the locations visually. I got better as the game went on, but continued to make the occasional mistake. In turn, my memory got some exercise.
Why did I do it this way? Because using my solution would have been cheating. On the one hand, it wasn't against the rules. On the other hand, it was against the spirit of the game; solving the game wasn't what the game was "about" . It wouldn't have exercised my memory. And it would have made the game phenomenally boring for me, and for all of the other players. In other words, it would have taken a fun game and turned it into a waste of time. That's wrong.
Winning Should Be a Challenge
In a competition for $1,000 or a trophy, one is expected to try to win at all costs, so long as one plays by the rules and exhibits sportsmanship; in this case, my solution would not be cheating. In a family situation, there is a limit to how many personal resources you are expected to bring to bear in playing a game. You don't go all out against your four year old while playing tennis. You give a weaker player a handicap while playing Go or Chess. That makes it fun for all players, and that's the point; winning is besides the point.
In some games, information is hidden but trackable. For example, in Puerto Rico, you are aware of how many victory points a player gets when he or she gets them, but you are not "supposed to" keep track of exactly how many each player has . In It's Alive, you can track how much money and which tiles each player has, if you really want to. But you shouldn't. You should guess; it's part of the game. If I had wanted players to keep exact track exactly how many coins and which tiles each player had, I wouldn't have given each player a personal screen to hide them. Exactly keeping track of this information slows down the game, and also makes the game a little boring.
Over-analysis of a game situation may lead to better results, but it leeches away the fun and the game time from others. When you over-analyze, you get more play time than the other players do, and you may even prevent an additional game from being played later in the evening.
Over-analysis is spending 1/6 more time taking your turn than the average time spent by each other player taking his or her turn. If everyone else spends, on average, a minute to think through a move, you should spend no more than, on average, a minute and ten seconds. You may do better for thinking longer, but that's not the point of this particular game. You are expected, in this particular game, to think only so far, and estimate the rest by instinct.
Thinking for up to a minute or two should be able to get you to take the path with 33% chance of success over the path with 1% chance of success. If several more minutes of thinking gets you to a path with 55% chance of success, then you're over-analyzing, or you're specifically using the game to think through a particular type of strategy [1, 3], or you're playing Chess, where each player's turn is long and deep thinking is encouraged.
Froggie Boogie's game play is simple, light, and fairly quick. Its child-friendly theme acts to discourage playing the game in any way other than the canonical way.
Same games have open information; this makes tracking information simple, but additional concrete information can lead some players to additional over-analysis.
Some computer games and some board games prevent over-analysis by not providing you a lot of options, or by providing you many options, or by not providing complete information. Providing few options results in a narrower, but possible deeper, search tree. Providing many options results in a search tree that is so broad that anyone with sense should quickly realize that a complete analysis will take too long. Incomplete information is not much of a help on its own, if players can still work hard to analyze the odds.
Many computer game, and some board games, prevent over-analysis by requiring you to make your moves under a time constraint. You can institute your own time constraints (an hourglass or a kitchen clock works well) for any game, if you so desire.
 These sentiments do not apply for all occasions. For example, solving the game might serve as an instructive exercise for a class in math, logic, or algorithms.
 In a two-player Puerto Rico game, you can't help it, since you know how many you and the bank have, and you know how many there were to start with. To solve this, the bank's current total should be kept secret. Use twenty five 1 VP chips, and the remainder 5 VP chips. Place the 5 VP chips beside the board, and the 1 VP chips in an opaque bag. When you gain 1 VP chips, take them from the bag, tossing 5 of them back into the bag in exchange for a 5 VP chip when appropriate. Players may not count the chips in the bag.
 And for some people, that's the fun in a game, regardless of what the game play is "meant" to be.