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Say my players are in a dungeon of many keys and many doors, where if a wrong key is used in the wrong door, they will be electrocuted, ambushed, etc. This makes the players wary to just try every key in every door. The players have found a key that I describe as "a simple key, made of copper stained green with age." Our session ends, and next week the players find a locked door. A perception check has them see the lock as "relatively simple, made of aged copper, turning green." Due to my players lack of note taking and poor memory over a week, they do not associate the key with the lock. In this scenario the players may have forgotten, but the characters likely wouldn't have, but I do not want to just hand the players the answer.

After a few minutes of them unsuccessfully trying to get through the door that was never intended to be difficult puzzle, I want them to roll to recognize the similarities between the lock and the key they previously found.
An investigation check doesn't seem appropriate, as they are not investigating anything for this, just thinking about it. A perception check doesn't seem fitting either, as they have already seen the lock and key, they just need to recognize the similarities between them.

How should I have my players roll to deduce an answer to a simple question?

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    \$\begingroup\$ If it's important for the game that they figure it out (in other words, if you don't plan on having them fail in the end, or if failure carries no consequences), don't make them roll at all. \$\endgroup\$
    – user31662
    Aug 2 '20 at 22:57
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    \$\begingroup\$ "but I do not want to just hand the players the answer." Why not? Are you trying to punish the players, or put them through some bureaucracy, for forgetting about your game the other six days of the week? Or do you actually think the characters might not have cared about the key in-character? How did they describe handling the key? If it was never intended to be a difficult puzzle, then why are you adding difficulty now? Presumably the real reason your adventure involves a key at all is to require the players to obtain the key first - not to understand it.Keys are just keys, obviously. \$\endgroup\$ Aug 3 '20 at 6:22
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    \$\begingroup\$ If it's not meant to be a puzzle, why are you treating it like one? \$\endgroup\$ Aug 3 '20 at 17:40
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You should tell players what their characters would know.

For the players it has been a week between finding the key and finding the lock. Both key and lock are fictional elements of a game they play for fun. Their only experience of them is as verbal or textual descriptions you supplied to them. Why would they remember?

For the characters it has been a matter of minutes between finding the key and lock. Both key and lock are real physical objects involved in their real life-or-death struggle for treasure, glory, and perhaps the fate of everything they care about. They have held the key in their hands, felt its weight, observed the pattern of its teeth and wiped its tarnish off their fingers. How could they forget?

Part of the fun of roleplaying games is to associate with your character: to imagine what they see, live inside their mindset, and make their problems your problems. Finding the key, finding the lock, and figuring out the link between the two is the problem before the characters. Retaining information about the game world is a player-only problem. You, as the GM, are the sole bridge between the players and the world their characters inhabit. It's up to you whether you want to create additional problems for the players purely because of the distance between their own experience in real life and what the characters experience in the game fiction. Some do, and consider methodically retaining information on behalf of your character part of the skill of roleplaying. I wouldn't.

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    \$\begingroup\$ Also, keep in mind that, if the DC is low (which it probably should be), then a passive "take 10" intelligence check is perfectly reasonable here. So, unless your party consists entirely of orc barbarians then they should recognize this fact without needing a roll. \$\endgroup\$
    – Pace
    Aug 4 '20 at 21:02
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You shouldn't do anything, except say "check your inventory".

first it is not unreasonable to expect a player to keep notes. If they have a key in their inventory they should be checking it against every lock.

second if they are stuck they should be asking questions. Not sitting on their hands waiting for you to do something.

"What does the key look like?"

"Have I seen other locks that look similar?"

"Do all the locks look different?"

Encourage your players to be proactive. Try asking them if they have any questions to get them thinking. You can also ask them to describe what they think is the problem, or even just ask them to describe the scene, sometimes preconceived notions get in the way.

In this case you can just say "hint: check your inventory" because it sounds like your players forgot they had a key at all. And it is fine you remind them of things like that if it was several sessions ago.

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    \$\begingroup\$ "If they have a key in their inventory they should be checking it against every lock" - I think this is written for a very particular playstyle that not all players enjoy. The kind of brute-force mechanical checking of every lock they see that you're encouraging here gets old very fast in my experience, and it's generally much smoother to encourage taking general actions that take advantage of their character's senses and intelligence, like, "I keep an eye out for any lock that looks like it might fit my key," and the DM notifies the player if/when their character sees something like this. \$\endgroup\$ Aug 3 '20 at 17:25
  • \$\begingroup\$ @probably_someone you just described a different way of doing the exact same thing, that is just another way of checking every lock. If the players can't solve Key=>Lock either they are completely passive or forgot they have a key. \$\endgroup\$
    – John
    Aug 3 '20 at 18:21
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    \$\begingroup\$ @probably_someone It doesn't get old if you just say "I try all my keys on the lock", which after all is what the character is doing. If your group insists on you enumerating every key you intend to test, then yeah, that'll get tedious. \$\endgroup\$
    – Mark Wells
    Aug 4 '20 at 5:21
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Intelligence

Intelligence measures mental acuity, accuracy of recall, and the ability to reason.

Intelligence Checks

An Intelligence check comes into play when you need to draw on logic, education, memory, or deductive reasoning.

Straight from the Player’s Handbook, ask the player to make an intelligence check to see if they can recall or deduce the relationship between the appearance of the lock and the appearance of their key.

Personally, I would set the DC pretty low, probably 10, unless there was a particularly subtle clue relating the lock and key. But this is completely up to you how you set the difficulty class of the check. The same section of the basic rules includes a Typical Difficulty Classes table.

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How should I have my players roll to deduce an answer to a simple question?

You have two answers already that do a good job addressing your precise question head-on. I would argue the two answers that suggest an Int ability check are probably the best in terms of what you actually asked.

Two other answers suggest that you can simply lead or even directly reveal the connection. And frankly, those are appropriate options as well, if you as the DM and the players themselves enjoy the game that way. It's a game, after all. If those approaches fit into your play-style, by all means!

I will suggest a third option not mentioned so far:

Recapitulate before starting the play session

You're telling a story. Human memory is frail and recalling threads of narrative when there's a significant gap is difficult. It's a well-established practice when resuming story-telling after some interruption to summarize the story thus far.

Typically, this "recap" will focus primarily on the plot elements that occurred previously which have the most direct bearing on the soon-to-come parts of the plot in the current session. For example, your recap might include a retelling of the events that led to the discovery of the key and what the party members observed about it.

IMHO, this approach is especially useful if one wants to preserve a modicum of player-initiated problem-solving without being too leading. Rather than waiting until the moment when the previous information is pertinent, which can feel like too much of a direct hint if not outright reveal, the players start the session with a kind of "bag of relevant information" that they can keep in mind as they proceed.

Naturally, if they still fail to have that "a-ha!" moment you can still fall back on other techniques, such as an ability check or leading hints. But the recap approach is, I think, the most likely to preserve the original design of the story-telling as if the players had been playing continuously.

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    \$\begingroup\$ I DM for my wife and her friends, and there wouldn't be any continuity if I didn't recap at the beginning of every session. I find that it helps refresh everyone's memories, even myself. I go over everything relevant, not just last week's events. It's also a really nice way to transition from catching-up chit-chat into game mode. \$\endgroup\$ Aug 3 '20 at 17:39
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The answer is either Intelligence or Wisdom, but which will depend on what kind of process you are trying to represent. You aren't describing deduction, but rather induction, which is an important part of understanding which Ability to use.

Intelligence

The PHB describes intelligence (pg. 177) as:

... mental acuity, accuracy of recall, and the ability to reason.

If you are trying to represent the character's ability to recall what the lock or key looked like, or to logically determine a relationship between them, then Intelligence is the Ability to use.

Intelligence is almost the perfect answer for the question as you posed it. Deduction is almost definitely an Intelligence based activity without exception. Except that you aren't describing deduction, but rather induction.

Deduction may look something like, "All simple, aged copper locks can only be opened by simple, aged copper keys. I want to open a simple, aged copper lock, therefore I must require a simple, aged copper key." However, deduction requires assumptions that are absolutely true. Is it really true that all key-lock combinations share traits like this? Why can't an iron key open a copper lock? For example, suppose that a replacement key were created at some point. It may use different materials and styles.

So the kind of inference you are looking for is probably not deduction, and may not even be Intelligence based.

But Really Wisdom

The PHB describes Wisdom (pg. 178) as:

... how attuned you are to the world around you and represents perceptiveness and intuition.

If you want to represent whether a character has noticed something, or is able to reach a conclusion based on intuition rather than reason, then Wisdom is the way to go.

The logical process you are describing is inductive, and a much better match for Wisdom. An inductive process may look something like, "So far, all the locks I have seen are opened by keys made from the same material and in the same style as the lock they open. Therefore, I think that all locks are opened by keys in the same material and style as the lock."

To me, this sounds like what you are trying to describe. The character is trying to intuit a relationship between locks and keys based on their own experience and observation. Wisdom is a much better fit than intelligence.

Now, if the player still has a problem determining that they should use a simple, aged copper key for their simple, aged copper lock an Intelligence check would be appropriate, because this has become a deductive problem.

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I just want to bring attention to a part of your question that may be relevant:

Due to my players lack of note taking and poor memory over a week, they do not associate the key with the lock.

(Emphasis mine.) You need to be careful here, because you’re making a value judgement about your players’ play styles. There's nothing inherent in roleplaying games that says players have to take notes, and while many players prefer or expect to, many others do not. Some players want to show up to a game and enjoy that game in the moment, reacting as their character would to the world you portray. To those players, taking notes is a distraction that takes them out of the world they enjoy!

This may be an example of a difference of expectations — you expected that your game would involve a degree of bookkeeping and note-taking, and your players did not. Unless you set that expectation with your players in a “Session Zero” you shouldn't punish your players for not defaulting to your preferred play style.

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This is a matter of playstyle.

Are you playing a game where your players should take notes/be responsible for mapping/be responsible for trying to do well/expect that player exploratory skill matters?

If so, read this answer.

Are you playing a game where your players should make the right choices that best fit their character concepts/be interested in playacting, rather than creative problem solving/ expect their characters to always win in the end and see that not happening as prima facie a consequence of GM, rather than player, failure?

If so, read this answer, and if you insist on rolling for it, which is the wrong choice for this playstyle, yet common practice, read this answer but also this rant about why ability score determinations for mental tasks in D&D are fundamentally ambiguous and kinda dumb.

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Rather than rolling or just handing over the answer, you can use simple props to handle situations like this:

As the players gather significant items (where "significant" can mean plot-significant, or valuable, or encumbering (if you're using encumbrance rules), or interesting, or anything else that might make you want to remember that you have it) hand them index cards with "a simple key, made of copper stained green with age" written on it. Whichever player has the card, their character has the key, and they can look through their stack of "inventory" cards at any time to see what they have on them, including its description (and, if appropriate, encumbrance value, game stats, etc.). Once they have a couple "key" cards in their inventory, each with a different description, most players will realize that they should probably pay attention to them instead of assuming that one key is no different from another.

This is a common form of homebrew inventory tracking/management in RPGs, which also just happens to be very useful when implementing inventory-based puzzles, such as described in the question.

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