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One strategy I've heard of in clue is to name a card or two in your own hand when making a suggestion. I presume this is primarily so you can gain information about the other two factors that need to be determined (the three being murderer, location, and weapon), because if you name a place from your own hand, the player who disproves you must reveal either a weapon or a murderer.

When should I make a suggestion that includes a card from my hand? Would it potentially be advisable to make a suggestion that includes 2 or even 3 cards from my hand? Do I reveal extra undesired information to other players when I do so?

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You should almost always make suggestions that include one or two cards in your hand, especially after you know one or two of the real murder components. (Though this might change with the number of players. I'm used to playing with 3 or 4 people.)

The end goal isn't to know who holds which cards, it's to know which cards are not held. The most efficient way to gather information isn't when someone shows you a card, it's when they pass; especially when all other players pass. This does give information to other players, but more to you.

Illustrative Example

Let's say the real solution is Prof. Plum with the Rope in the Conservatory. It's Player A's turn and he's in the Conservatory. If A makes a random guess of cards he doesn't hold, maybe A gets "lucky" and accuses Prof. Plum (correct) with the Revolver, and of course the Conservatory. What A learns is who holds the Revolver. If he's lucky it won't be the person on his left, so maybe he'll see a pass or two, but it won't be everyone. However, if A suggests Ms. White (in hand) with the Knife (in hand) in the Conservatory, everyone will pass, and he knows that the Conservatory is the room! The other players will know that they desperately need to check out Ms. White and the Knife and the Conservatory.

Now, let's step into another player's shoes. Player B knows that Player A learned something big, so she runs to the nearby Billiard Room and suggests Ms. White with the Knife. If anyone shows her a card, Player A knows that it's the Billiard Room. Great! Player A learned as much as Player B did from her own suggestion. If people pass until Player A, he shows her, say, the Knife, and A knows that either B or someone sitting to his left before B holds the Billiard Room. Player A should now preferentially show the Knife rather than Ms. White to everyone. Meanwhile Player C can guess that Player A has one of the Knife or Ms. White, but Player C still has to make his own suggestion to find out which (and there's a possibility Player A has the Billiard Room).

Contrast this with Player B being a little more conniving: she decides to test Player A's knowledge more gradually. She accuses Col. Mustard (in her hand) with the Knife in the Billiard Room. Now if someone before Player A shows her a card, Player A can't be sure if it's the Knife or if it's Col. Mustard. If it gets around to Player A, he doesn't have a choice of what to show Player B, he must show the Knife, which is also better for B than if A gets to choose.

Other considerations

As for suggesting 3 cards from your hand, that can be a devious way to throw people off track, but you also essentially lose a turn of information gathering so I wouldn't recommend doing it often.

Your card distribution (especially what rooms you hold) can affect your ability to make suggestions involving cards from your hand. I try to make one of the first 3 rooms I visit a room I hold. It's especially nice if you hold two secret passage rooms to hop back and forth between the two (though pretty soon everyone will know what you're doing). It can be tough if you hold only one card (0 even worse!) of a category. I'll focus on that category first so that I can at least alternately suggest 2 items when I get to the other categories.

In the beginning of the game it's fine to make suggestions with only one card you hold, or the occasional 0-held-card suggestion. You'll learn something--maybe you'll get lucky and learn a lot. But as soon as you know one category there's no point in making a suggestion involving a card in that category that someone else holds. You already know somebody holds the card, but in most cases you shouldn't care who.

tl;dr You learn more from people passing, and people pass more often if you make suggestions that use some cards from your hand.

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In general, you should probably only do so before everyone else could deduce that you have it. For example, lets say that previously you disproved Col. Mustard, Candlestick, and Study. At some later point, you could not disprove Col. Mustard, Revolver, and Study. Your opponents should know at this point you have Candlestick in your hand.

If you choose to use Candlestick in your future suspicions, you are giving your opponents extra information. If no one disproves you, they will now know the correct Suspect and Room or you have either the Suspect and/or the Room (see all 3 cards below) card(s) from you suspicion. If someone disproves you they know what that person gave you the Suspect or the Room, since they know you have the Candlestick.

It is also sometimes useful to choose a Suspect that happens to be in your hand (or even in the envelope), even if other players know that you have it (or it is in the envelope), just to move that player pawn away from a room that you would prefer they didn't make a suspicion from. This is especially true near the end of the game, when usually it is only the Room that people don't know since there are 9 of those and only 6 of the other card types.

When should I make suspicions that include 1 card from my hand?

Usually, the reason for doing this is so that you can eliminate one or more possible cards that a player used to disprove another players suspicion. If at some point, a player 2-3 seats to the left disproved Col. Mustard, Candlestick, and Study. You have the Candlestick in hand, and you want to know exactly which card the Study/Col. Mustard they used to disprove the other players suspicion (perhaps, because this will let you know that persons final card. If you know their final card, you know that they don't have all other cards.) That will be useful later on when you want to make suspicions that get past that player.

When should I make suspicions that include 2 cards from my hand?

I usually do this less often (if ever). I could see using 2 cards from your hand if you needed to find out the final card that you where unsure of.

When should I make suspicions that include 3 cards from my hand?

Never. The point of a suspicion is to eliminate possible murderers, weapons, or rooms and deduce the correct cards within the envelope. If you make a suspicion that includes all the cards in your hand, you gain no new information (you know everyone else cannot disprove you). You have given everyone else extra information, because they all now know that no one else except for possibly you has those cards. The only possible time that you would do this, is if you already know who the correct answer, and in that case there would be no point since you can accuse immediately. If you are just messing around and have no intention of winning, it is easy enough for at least one opponents to verify if you have the Room and/or Weapon cards.

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I said all 3, and concealed my room card from everyone. Half the people accused using the room I hid, and that's because they believed it was in the clue folder, because no one could show me the person, weapon, and room card I asked for

They saw either my weapon, or person card, or both, but none saw my room card. I only did it once, nearing the end. They assumed I was using the 1-card or 2-card method, but no one conceived that I would ask for all 3 that I had!

The wrong accusations revealed to me who the person and weapon were. In my next two turns; one more to find the correct room, and one more to correctly accuse, I had won the game.

Lastly, when I said all 3 of my cards, I felt the others getting close. And I only narrowed it down to 3persons, 2weapons, and 3rooms. I was not too close.

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