I'll clear a couple of things up for you Cedric

I didn't play these hands they were being discussed in nitevenoms "hand of the day" thread, both hands are from community tournaments and during the discussions the check raise was suggested in both situations, I used those suggestions as the basis for the blog post to show how one situation is favourable and one is not

We are using the check raise to try and take this hand down now, if it's called then we will have to evaluate the turn, if our opponent shoves we have to evaluate our pot/hand odds and make the decision to fold or call, so for the purposes of the discussion we use a range of our opponents shoving hands

In the first hand we don't rule out AK because of the call to our raise, if he 4 bets pre flop we might as well fold now! what is our AQ beating that 4 bets? personally I think 4 betting pre flop with AK facing a raise from the blinds is over-playing AK

In the second hand I don't rule out AK for the same reasons, but if we do rule out AK then we now add 3K's to our outs! so 9 hearts, 3 tens and 3 kings gives us 15 outs and even better odds to call a shove

And this is where I think things are getting confused, we are discussing when to check raise and what to check raise with, we are looking at check raising to show strength with hands that give us calling potential if re raised or shoved to, but there is no guarantee that we can check raise here unless our opponent bets! if he checks behind we can't check raise, so we are giving our opponent the chance to bet out by checking from out of position so that we can come over the top of his bet in the hope that he folds, if he does fold that's perfect, if he calls we will have to re-evaluate on the turn and re calculate our odds of making our hand (outs times 2)

Whatever range we put our opponent on pre flop remains his range whatever happens with our check raise attempt, we are not attempting to narrow his range we are attempting to show strength to his bet, if he folds we don't need to worry about his range, if he calls we have to consider what he can pre flop raise with then call a re rasie, and then call a check raise with on the flop, but we have the turn card to re evaluate our options and should be ready to fold the turn to a bet (we certainly don't want to be leading out on the turn) if he shoves we have to put him on a made hand or a hand that is potentially winning now but fears the draw, and we need as many outs as possible if we're going to make the call

Removing AK from his range in either hand will improve out outs by 3, but we can't rule out all Ax hands, so in hand one we would improve to 6 outs 6x4=24% we can't add the 4 5's as that potentially splits the pot, in hand two we would now have 12-15 potential outs, 4x12=48% and 4x15=60%, so in hand two we have twice as much chance of improving to a winning hand if we are shoved on after aour check raise attempt

If we are going to attempt the check/raise we don't want to be check/raise/folding so flop texture and possible outs are where we base our decision, his action after our check raise will narrow his range