Flush on a Double Paired Board, what do you do here?

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DECISION POINT:
In a Tournament, a Middle Position player calls, and the Cutoff and Button call behind. You call from the Small Blind with 2♦8♦, and the Big Blind checks. The Flop comes 6♦9♦J♦. You check, and it gets checked around. The Turn is the 6♣. You bet, and it folds to the Button who calls. The River is the 9♠. You check, and the Button bets. Action is on you, what do you do here?

PRO ANSWER: As played, we should fold to the river overbet.

However, we could have bet 400 or 500 on this river instead of checking. This is a great spot for a block bet/value bet of around 35-40% pot. Most opponents bluff-raise rivers infrequently, so we can safely fold if we get raised. We also can get value from some Jx hands or weaker flushes at that price.

Moreover, betting denies our opponent the ability to make a big overbet as they did here. Once we check the river, we allow them to bluff somewhat freely since we are not very likely to check a big hand on the river. If they hold something like AdTs, they have the green light to make an overbet and take down the pot very frequently against our now weaker range.

Continued below...

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We should still simply fold to this overbet, but we can avoid this spot by betting the river instead of checking.

Betting out on the river is the best play. Now that you have checked the river, you should fold to this overbet.

At this point folding is the best play.

How would you play it?
Share your answer in the comments below!