I've played a flop game at a home game a few times recently that involves playing 3 hole cards against a typical flop/turn/river board. You can play 0, 1, 2, or 3 hole cards to make your 5-card hand, allowing for some interesting possibilities.

I don't know of any simulator for this type of game, nor do I wish to do this by manual calculations, but I'd like to know more about equities for preflop match-ups. For example, just how much of a favorite is rolled-up trips over trips? How strong is 678 against AKK?

(Yes, I know these premium match-ups are unlikely with only 3 hole cards, but we play this variant as part of a roll-your-own-ish split-pot game, so powerful 3-cards hands are pretty common.)

Lazy Pineapple? (That's what we call it). Pineapple but no discard.

Interesting. In either case, I'd generally want to see the flop before making much of a move. I think with three cards you want to see a flop with at least one card you can use. Otherwise you missed a fit with the flop. That is, after the flop in this game, I think you either want to be looking at a pat five card hand or have a four card hand with a multi out draw for a pat five card hand.

With 8 cards possible, I think you need a straight or better to win. What I'm saying is trips or two pairs is probably not good enough for high. With trips after the flop, you'd be drawing for quads or a full house.

What wins in this game is probably closest to 5-card Omaha.

We have played this and call it "Lazy Pineapple-8." Do you require low have an 8 or better, do you play the game as a declare game with no requirement for low, is it no requirement and cards speak, or something else? (Just curious).

Buzz

I'll go for the low-hanging fruit: For a close answer we can ignore straights and flushes.

41/46 + C(5,2)/C(46,2) =~ 90%

This is very different from Lazy Pineapple, at least the way I've played that game. In LP, you get 3 hole cards with no discard, but you ultimately can only use 0, 1, or 2 of your hole cards against the board at showdown. ("Discard" effectively happens when all cards are out.) In this game, you can also use all 3 hole cards at showdown, which changes things pretty dramatically from LP.

Seeing flops before committing much is a solid strategy, and yes, in a multi-way pot, anything less than a pat hand is unlikely to be the winner. In fact, flushes are always possible and so are quads and boats, regardless of board texture. All of this factors into hand-building.

Eh, not so much. The game where we've played this also plays 5-Card PLO and 5-Card PLO8 very often, and it's not really comparable. In Omaha with any number of cards, players will frequently hit the nuts and/or draws to the nuts. Catching the nuts in this game rarely happens. It's probably more comparable to 8-Card Stud (but with stronger hole cards).

Referring back to hand-building, this isn't a high-low game. It's a high-only split-hand game, played pot limit, cards speak. You get 5 hole cards, split them permanently into a 2- and 3-card hand preflop, and then play it out with those cards. Best 2-card and 3-card hand split the pot. I've thought about proposing a declare, but I think the game is complex enough as-is, for now anyway.

Between the roll-your-own aspect and the pot-limit betting, I feel like this game has the potential to create some insane all-in pots preflop. We play mega-deep too, so huge hand vs. huge hand is likely if players are all-in. My main interest here, really, is in figuring out which spots are most ripe for all-in confrontations. I know my all-in equities for the 2-card match-ups, but the 3-card match-ups obviously have much more potential.

Close answers are all I really need anyway.

What I'd really like to figure out are some benchmark equity comparisons like you frequently see for hold'em (pair over pair is about 4:1, dominating no-pair hand is about 3:1, two overcards to a pair are about 1:1, etc.). I'll add trips over trips as 9:1 for a start.

Obviously there are more intricacies to the 3-card hands, but only a handful of them really matter for this comparison: 3-card flushes, straight flushes, (to a lesser degree) straights, pairs with danglers, and rolled-up trips. Other match-ups are possible but far less likely if we're talking about all-in equity.

A simulator would probably be ideal for this, but I have no idea how to create such a thing.

I already had a simulation written (in ugly but functional code) for hold'em. To add a 3rd hole card I barely had to tweak the code. Here are the equities I found so far, with 1 million trials each:

KKK vs QQQ (all same suits) = 0.8936495
KKK vs 666 (different suit) = 0.872274
KKK vs 876s = 0.7695495

222 vs AKQs (one flush blocker) = 0.7880975
222 vs AKQo = 0.864189
222 vs AAK(s) = 0.721042
222 vs 987o = 0.777397
222 vs 987s (one flush blocker) = 0.716077
222 vs 987s (no blocker) = 0.718729 (obv should be <above, but small enough factor that 1M trials is not enough)

765s vs AKQo (villain one flush blocker) = 0.596256
765s vs AKQs (diff suit) = 0.468741
765s vs AAKs (villain no blockers) = 0.484154
765s vs AAKo (villain one blocker) = 0.492343

Unless I am mistaken, excluding two three-card hands leaves 46 cards from which the board comes. And C(46,5) is 1,370,754 which is only slightly larger than one million.

So if it is fairly easy to tweak your code to do an exhaustive enumeration over all possible boards, you can become the "Father of Modified Lazy Pineapple Equities". Or some other clever title that has a suitably regal acronym.

I dubbed the split-pot game Tomato.

The 3-card game on its own is sometimes called "Super Hold'em" and some other names (we originally called it "Suicide Pineapple"). Honestly, though, it doesn't play very well. You very rarely get 3-card openers that are even playable, never mind strong hands. Even when you do get good openers, the fact that such big hands can be hit on any board tends to stifle action in a big-bet game.

Awesome! Thank you. This is exactly what I was hoping for.

Some of these are pretty interesting. Suitedness and connectedness add a significant amount of equity against pair-type hands, and rolled-up trips versus an overpair plus overcard kicker is not as much of a favorite as I'd expect. KKK versus 876s is only slightly more of a favorite than KK versus 87s (about 0.78).

One of the things I'd worry about in this game would be facing a ton of preflop action with AA with a half-decent speculative hand on the 3-card side. If you get all-in preflop with deep stacks, you're likely to be looking at rolled-up trips, which I figured would make you a huge dog even if Villain has rags in hold'em. But if you have AA / 78T or something comparable, getting it all-in against rolled-up trips with a meh 2-card hand wouldn't be so bad. You might be pretty close to even money, from the look of it.