Which Hands Keep Bluffing the River? Solver+ Rules for Give-Ups and Triple Barrels

You bet the flop, you bet the turn, and the big blind called twice. Now they check the river to you with a pot that is bigger than the stack you started the hand hoping to win. Every player knows this moment. Most decide it the same way: "I have nothing, I have to bet." The problem is that "nothing" is not one category. Some nothing wins at showdown. Some nothing blocks the hands that call you. Some nothing blocks the hands that fold. Solver+ treats those as three different hands, and on the four rivers below it bets one of them 100% of the time and checks another 100% of the time while they look identical from your seat. This is the bettor's side of the river, and it is the side almost nobody studies. If you want the caller's side, the math is in bluff-catching is math, not instinct.

The spot for the whole article. 100bb cash, single raised pot. You bet a quarter pot on the flop, overbet the turn, and the big blind called both. Now they check the river with 30.6bb in the middle. Holding Q♠6♠, Solver+ bets 100% of the time. Holding A♠4♠ it checks 100% of the time. Both are queen-high or ace-high with nothing. The rest of this post is about why.

1The line, and why the river is where bluffs get sorted

The spot is the most common two-player pot in poker: the button opens to 2.5bb at 100bb effective, the big blind calls, and the flop comes with 5.5bb in the middle. The big blind checks, the button bets small, the big blind calls. The big blind checks the turn, the button bets big, the big blind calls again. River, big blind checks. What the button does now, with a 30.6bb pot and 85bb behind, is the whole question.

Before the river, notice what Solver+ actually does with the turn size, because it explains everything that follows. After a quarter-pot flop bet gets called, the turn barrel is not a "medium-big" bet. It is an overbet, 11.2bb into 8.3bb, and it is by far the most common barrel size on all four boards. The 80% pot option exists in the tree and is barely used. That is the definition of a polarized line: you are telling the big blind that you hold either a hand that beats top pair or a hand with no pair at all, and you are charging the maximum for the information. The full logic of choosing sizes by range rather than by hand is in range-based bet sizing.

Two things in that table should change how you play the river before you look at a single hand. First, the river bluff size is 90% pot or all in. The 35%-pot bet is used with 1 to 6% of the range, almost always with a specific value hand that wants a call from a worse pair. There is no such thing as a small river bluff in this line. Second, the big blind folds about half the time to the 90% bet on every board, which is almost exactly the minimum defense frequency for that size (52.6%). They are not over-folding. Your bluff does not print because they fold too much. It prints because the hands you bluff with were chosen to make their indifferent bluff-catchers a fraction worse than indifferent.

2Four rivers, hand by hand

The same line was run on four flops chosen for what the river does: a brick, a straight-completing card, an overcard ace, and a card that pairs the board. In each tab the grid shows every hand the button can arrive with, coloured by how often it bets. Orange fires, blue checks back. The ledger underneath sorts the no-pair hands into the ones that keep bluffing and the ones that give up, with the value line alongside so you can see where the bluffs are being counted against.

3The heuristics, in the order you should apply them

Across four boards with different river dynamics, the same sorting rules produced every decision above. Here they are as questions you can ask in the ten seconds you get at the table.

4Where the value line moves, and why that matters for your bluffs

Bluff selection is downstream of value selection, so it helps to know where the thin end of value sits on each texture. On the brick, KQ bets 97% and KJ checks 100%: top pair with a queen kicker is the last value bet. On the straight-completing river, every top pair below AJ and KJ becomes a check-back, and top set (JJ) bets the small size 62% of the time because it blocks the Jx that would call a bigger one. On the ace river, A2 and A4 that were pure bluffs on the turn bet 27.5bb for value. On the paired river, quads bets 90% pot instead of jamming so trips can pay it off, and AA checks. Each time that threshold moves up, count your bluffs down. Each time it moves down (the ace hits your range), you get to add them back. The same interaction on paired textures, from the flop onwards, is worked through in attacking paired boards, and the two-player flop dynamics that set up this line are in the K-Q-5 deep dive.

5The river checklist

Run this in order when the big blind checks the river to you after calling two barrels. Stop at the first answer that decides it.

1. Do I have a pair, or ace-high on a board where ace-high beats their busted draws? Yes: check. You are a bluff-catcher now, not a bluffer.
2. What is the hand BB calls with and cannot fold? Name it: the straight, trips, top pair plus. If it does not exist on this board, thin value is your bet and bluffs are few.
3. Do I hold a card from that hand? Yes: bet 90% pot, and jam if I block the very top of their range. No: go to the next question.
4. Do my cards sit in their folding range instead? Broadway air on a low board, low cards when BB's folds are low pairs and busted low draws. Yes: check. You are bluffing hands that were already folding.
5. Am I over budget? Roughly one bluff for every two value combos at 90% pot. If the board gave you many missed draws and few value hands, the weakest-blocker bluffs check.
6. Never bet small with air. If the hand cannot justify 90% pot, it cannot justify a bet.

6The mistakes this fixes

Frequently asked questions

Should I always bluff the river after betting flop and turn?

No. In this exact line, Solver+ checks back 43% to 62% of its range on the river depending on the board. Bluffs are a minority of the betting range (24 to 33%), and they are chosen by blockers, not by whether you "have to follow through". Following through with a hand that has showdown value or the wrong blockers is a leak, not a story.

Which hands make the best river bluffs?

Hands with zero showdown value whose cards remove combos from the big blind's calling range: a ten or queen when the straight is QT, an eight or nine when trips are T8 or T9, a six when the only straight is 86. The hand does not need to have been a draw earlier. Q3s on J95-2-8 bluffs 100% having never had a draw, because the queen blocks the nuts.

Why does ace-high check on the river after barrelling the turn?

Because the big blind called the turn with busted draws that ace-high beats and folds all of them to a river bet. On K72-5-9, BB check-called the overbet with A3, A4, 64 and 43, and folds every one of them to 27.5bb. Betting ace-high folds out the hands it beats and gets called only by pairs. On an ace river the answer flips: A2 and A4 bet for value.

What size should a river bluff be?

In this line, 90% of the pot or all in. The all-in is reserved for bluffs holding a blocker to the very top of the big blind's range, such as KT on a river that completes QT and T7. The small river bet (35% pot) is a value size used with a few specific hands, and never a bluff.

The takeaway

"I have nothing, I have to bet" is two decisions pretending to be one. The first is whether your nothing can win a showdown; if it can, it is a bluff-catcher and it checks. The second is whether your cards live in the big blind's calling range or their folding range; only the first kind bluffs. On four different rivers Solver+ used those two tests and nothing else to decide between hands that look identical from your seat, and the result was a betting range of two parts value to one part bluff, sized at 90% pot or a jam, with no small bluffs at all. Run the six-question checklist above for a week and you will find that the bluffs you stop making were never getting through, and the ones you add were folding out hands you used to check behind. Practise the river decision until the blocker read is automatic in Postflop+, and check the exact frequencies for your own runouts in Solver+, which holds every flop, turn and river in this line.