So I love making little games within games, if you will.  I find that the players love it when they get to use their own thinking abilities on top of the skill sets of their characters.  Of course, if they want to make some kind of check using their character stats to either get the solution or hints, I of course allow them to do that, but it seems so much more fun to most of them to figure it out for themselves.

That brings me to my main question for this post.  What are your favorite ways of simulating these sort of things in game?  Do you have your players physically put together a puzzle or come up with an answer to the riddle?  When you have your players playing games like chess, marbles, or gambling games, how do you simulate them?  

Puzzles, riddles and such, doing the theater of the mind can be difficult yet awarding if you manage to explain them it succesfully. on the other hand if you don't they are left more confused. Having a physical puzzle as a prop for the table sounds perfect if you have one that works. And for the games like card games, there is a lot of different ways to manage. I will rather link the other discussion about that. 

http://www.dndbeyond.com/forums/dungeons-dragons-discussion/dungeon-masters-only/9274-making-ability-skill-checks-with-the-playing-cards#c8

you can use some of these general ideas for other games too. Hope its helpful.

I just did a big festival where my local tavern had 3 nights of games and gambling.

I did a mix. Real-world social games, straight dice-rolling skill games, luck games, and every mixture and variations I could come up with.

For instance, there was a "team riddles" which was straight, real-world ask a riddle, get an answer. I allowed the group to discuss, and i was flexible with my interpretations, but there was no dice rolling involved.

On the other end of the spectrum, there was an archery contest, roll 3 D% and add your dex stat... And a pie eating contest where you rolled a bunch of fate dice to see how many pies you could eat.

When I design games, my goals are

1. I want the game to be winnable by everyone. I don't want the archer to walk away with the archery trophy unless he earned it. I might slightly goose it in favor of character creation choices, but it's not gonna be a cake-walk
2. To keep it interesting, and loosely capture the idea that I'm going for. It's a vacation from DnD. There will be plenty of dex checks and wis saves in the future.

I tried this, and failed at it fairly badly.  One of the players was so against the idea he abstained from the game session.

I also enjoy adding riddles. Unfortunately, one of my players is dumb as a rock (I internally refer to him as a Chaotic Stupid. Nice guy, but a wizard trying to sneak up to a caravan of precious metals in transport, with armed guards, in the middle of the day with no cover with the intent to steal stuff is just... not smart). Not even role-playing, he's actually really not book smart what so ever. The one time I gave each member of the party a riddle to resolve, everyone other member of the six-member party figured it out in about ten seconds, and it took him more than half an hour, after hints and then eventually being guided there.

On one occasion I did actually create a physical game for them to work through, which resulted in one person sitting out and playing on his phone, the party leader working on it, and the dumb wizard enthusiastically trying to help while simultaneously doing the opposite. You really have to know your party before doing anything. 

So my advice is this: know your party, and either make the puzzles/riddles optional or group work. 

Some that I have used in the past. The beginning of a primal magic dungeon had written on the wall "Only when all the elements are brought together as one can balance be restored." In a later room were statues of elven women holding bowls of one of the elements at each of the corners, and the door was locked. There was also a fountain at the center of the room. The players had to take the bowl of dirt, light a candle and stick of incense in the bowl, then let the bowl float in the water to unlock the door. 

Another had an outdoor temple of the stars and moon, with a trap door in the middle, a floor made of mirror stone, and eight pillars. Each pillar had a gemstone set in it. The only way to tell what stone would unlock the trapdoor was by looking at the stones through the mirror. I am a fan of puzzle doors. 

As for card games, I have them make "luck" rolls with a plain d20. If they want to cheat or apply strategy, they run a sleight of hand or insight check, and if they pass they get advantage on the luck roll. Fail and they are either caught and kicked out of the game or get disadvantage on the luck roll (depending on if they tried to cheat or strategize). This has worked well so far. Archery tournaments they must compete with ranged attack rolls. 

For games of luck, I'll have my players roll a die, and if they are proficient in that game, then they may add their bonus to the d20.

For riddles, I generally follow a format. Two readings of the riddle, three guesses. You may spend a guess to have the riddle read again, or you can spend a guess to get the "secret line" (aka a hint). The party will then get all the time they need to try and answer the riddle. Should they fail, then something unexpected will happen. This may be a teleport spell triggering, a pit trap, teleporting in a monster or some other thing. When the party overcomes the challenge, if the dungeon leads them back to the riddle spot, then the riddle has changed at this point (hopefully to a riddle the party will find easier), or perhaps they now need to find a new way, as they are no longer on their path.

For a puzzle, I like to give an example of how it works, and then see if the party can get it. An example of this, there was a bowl with 5 dice in it. Thee dice were all dice that had blips in them rather than numbers. There was an engraving around the bowl that said something like "petals around the rose, it is even, or zero." How this worked is the on odd numbers on dice, there is a blip in the middle and blips surrounding it. The "rose" was the blip in the middle, and the "petals" were the blips surrounding the middle blip. The answer to any given roll, would be the number of petals. Even numbers rolled would not add to the total of "petals" because there was no "rose" in the center. Thus any answer had to be even, or zero (should there only be even numbers). To help the party, there was a person who answered the puzzle before the party tried it, and then that person entered the building.

Should the people in the party be desperate, then I will have some sort of "random" event that causes the party to be given a hint for the answer to the puzzle or riddle.

For riddles and puzzles, having a visual aid is pretty essential- either something on paper or something to put up on the screen. It's easier than re-reading a riddle or explaining a puzzle several times.

I've used many riddles and puzzles over the years (though sparingly over a single campaign). I generally expect players to solve them on their own, though I do prepare hints to give characters who might have specialized knowledge. I also generally encourage players to solve these problems as a group.

Thematic puzzles in dungeons can be entertaining. I once had an entire adventure themed around encounters with Fey that leaned heavily into puzzles and riddles. I think it's important that these encounters seem to make sense in the context they're encountered in in an adventure or campaign.

Games is something I've had less success with, as they can be pretty time-consuming to actually play out. I've sometimes used in-character card games or the like as social encounters or gambling, but I always model these more like skill challenges. I did recently include Liar's Dice in a campaign I'm running, though, and that went quite well, mostly because it's a game that's incredibly easy to learn, that plays quickly, and that doesn't seem too anachronistic in a Medieval Fantasy setting.

I've also run a couple adventures that were based on characters completing several trials in succession or in contests where victory was based on how many individual events they won. One example was a knight's tournament where there was a joust and a melee, as you'd expect, but also other events like riddles asked by the local noble's daughter and a more elaborate contest to capture special tokens hidden in the woods and hunt down a lucky white deer while avoiding obstacles and other competitors.