Mastermind will long be a favorite game of mine. If you want a game to play with just one child at a time that develops their logical thinking, then Mastermind is the game you want!
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Not only do I have many fond memories of playing Mastermind with my father when I was a kid, but this game allowed me to watch my son’s logic develop! When he was about 5, I couldn’t find a used copy of Mastermind anywhere. We had been gameschooling for a few months – long enough for me to discover the beauty of thrift stores. So one evening, my husband went out and bought me a new copy of the game.
We brought it out and my son was eager to give it a try. At first, my husband set up a normal sort of code for him, which was too difficult for our son to guess. So I set up an easier code, with just two colors, and helped coach him through how to make guesses and interpret the feedback. What followed was maybe an hour of watching him develop his logical abilities.
How to play Mastermind

There are two players in Mastermind. A code maker, and a code breaker. The code maker has 4 spots to fill, and six colors to choose from (white, black, yellow, red, blue, and green). Colors may be used more than once. For a more advanced option, a blank space may be used, which is essentially another color.
The code breaker has 10 tries to guess the code. After each guess, the code maker provides feedback in the form of black or white pegs placed next to the code. A white peg indicates a correct color but in the wrong location, while a black peg indicates that both the color and location are correct.
Challenges
If you are playing with a child as a code maker, it is very helpful to have someone else double-check their scoring. My kids are getting better at correctly scoring a guess, but it can be very frustrating to follow the logic and not have any viable guesses, only to find out that a scoring mistake was made early on. That said, having your kids play the part of the code maker is also excellent for their development of logic.

We have run into difficulties over the number of scoring pegs. There simply are not enough. You are supposed to be able to keep score between rounds using these scoring pegs, in addition to them being used to score each guess. We have not done this, and we still sometimes run out of scoring pegs. Buying more scoring pegs on sites like ebay is usually very expensive. I think the best option would be to buy another copy of the game used, and just combine the scoring pegs.
How to teach your child to play Mastermind

Like with many other games, I teach my kids by doing a lot of thinking out loud. So when they’re scoring my guesses, for example, I’ll say, “Okay, for this guess I got one black peg and one white peg. This means I have two correct colors, and one is in the right spot.” (This also serves to help them double-check their own scoring.) “I’m going to try these other colors to see if any of them are used.” Or I might say, “I’m going to try all red, to see exactly how many are in the code.”
When my child is the code breaker, my goal isn’t to create a code they can’t guess. (At least, not yet – that will come!). Rather, I want to create a code in the sweet spot of difficulty. I don’t want it so easy that it’s boring, but I also don’t want it so difficult that it’s frustrating. I want them to play the game with me again.
With young children, this means I start with just one or two colors. If you’re playing for the first time against a teen, you can probably start with 3 or 4. If they guess it quickly (and it’s not that they just got lucky), up your difficulty level. If they’re not able to guess it at all, lower the difficulty level. As you play with your kids more, you’ll start to get a better feel for how to create a challenge in their sweet spot.

Variations
There are several different variations of Mastermind. I don’t remember the version I played as a kid having the ability to keep score between rounds. There are also Junior versions of the game, that have larger pegs. Personally, my kids have done great with the regular version. Often, their smaller fingers are better at handling the little pegs. If nothing else, it’s great for their fine motor development.