Flop = '666'. What does it mean when someone bets?

It means he is the devil !!!

The most plausible explanation is that UTG had a low pocket pair (or, less probably, weak Ax) that he considered strong enough to overbet on the flop because it was improbable that anyone had a better hand. Given his preflop limp and the fact that he was playing NL2 without autorebuy to 100 bb, I doubt that he was aware of the concept of a value bet. He was likely betting according to the absolute strength of the hand - a full house or trips with the top kicker - or its strength relative to opponents' perceived overall ranges, as opposed to their perceived calling ranges.

Another possible explanation is that UTG likes to make overbet steals when he's missed the flop but everyone else has likely missed it too. Again, he might not think in ranges, so it wasn't necessarily a thought-out bluff, he just wanted to take the pot right away.

To tell which of the conjectures is true, we'd need to make more observations of that opponent, but his behavior in this hand is worth a text note (like 'limp UTG, bet 1.4x pot 3way on 666') whose implications can be figured out later.

Accidentally, an advanced player with a non-empty UTG limping range (there are very few of them, everyone else only has folding and openraising ranges UTG) facing other advanced players in the pot would likely overbet just like this opponent, but for a different reason - his range would be very polarised on this flop (consisting of few value hands - mostly full houses because the only 6x hands that aren't completely terrible candidates to be included into a preflop limping range are 65s, 76s and perhaps 86s - and some bluffs - mainly backdoor straight/flush draws whose high card is a ten or worse), and he'd want to give the opponents very bad pot odds in order to reduce their calling frequencies and be able to have a profitable bluffing range (which requires getting at least 1 fold most of the time, as 2 opponents are too likely to have at least 1 full house by the river, devaluing potential straights and flushes), as well as want to get maximum value from others' bluffcatchers at those times when he does have a strong hand.

A lower pocket pair scared of overcard coming on later streets makes sence to me. Or bluff with Ax, Kx. But he still can have JJ+ for example and hoping that the lesser pairs will call him. I think majority of people on boards like this even forgot that there is 4th six somewhere in game and focuses more on full houses than possible quads. If I have no reads on the aggressor I would continue maybe with 99+ to see the turn. On next bet I would consider fold depends on sizing. If he would slow down on turn (with no A or K), there can be some value.

Backdoor flushes I would exclude from reasons for beting on boards like this. That would be quite optimistic imho. I wonder if someone can say opinion how to play correctly e.g. middle pair in position here.

Originally posted by Ihuarraquax
Backdoor flushes I would exclude from reasons for beting on boards like this. That would be quite optimistic imho.

Yeah, that would be too optimistic at the micros, where having a bluffing range makes little sense because overcards call too often even with the large sizing I agree with your analysis. I was just making a bit of a joke about how UTG's sizing is unintentionally GTO-ish.

What is the correct possibility formula of calculating if some one will hit quads on flop? is this correct:

It's easy. There are 52 cards in the deck, we must substract 5 of them, the flop and your hole cards (unless the case 6 was in your hand but in that case the possibility is 0), so we have 47 unknown cards left and the players still in the hand hold 6 of them. So it is 6/47 or 12.8%. And this is if the players hold completly random hands, which is only true for the BB, for the rest we don't know, but this can change the calulations a lot.

About what hand UTG has, only God knows. He can have quads, full house, high cars, 32o etc. These guys are unpreditable, don't try to put him on a hand or range. Most of the time you'll be wrong.