This was fun! Some great responses here although I thought more folks might have an opinion on the subject.

First, we had a couple of questions so I'll try to clear those up:

I feel like there should have been a stipulation added to the question such as "Assuming you are trying for max point pressing...", and then I also feel like there is maybe a clever answer to this question based on the premise of attaning maximum points from the scenario.

Yes, I was hoping the title of the thread would be enough to clarify that point. But, you're correct, the idea was to pick an option that should yield the most points on average -- with the assumption that both options carry the same level of risk. This is often a pretty big assumption and risk certainly should factor into our decisions while playing this game since dying unintentionally is almost always extremely costly. But given the assumption, we're trying to maximize our points here.

Question (also just wondering why no one does this in general): Can we forward jump the first barrel while grabbing the hammer and turn around and smash it (like most people do with the top hammer)?

Good question, and I had not thought about this option for this scenario, but I guess I could have added this as a choice. However, in my opinion, it turns out that this option would be significantly inferior to the other two options presented. First, we'd again have to assume that this option carries equal risk to the other two options -- this is a pretty big assumption in this case since this is a far more difficult move to pull off successfully (and without dying) for almost all players than the move required in the other two options. But even if we could imagine a player for whom this technique carries equal risk, it's still the worst option. The reason is because your horizontal position would end up much farther to the left on the screen after the grab and smash.

In general, with all other factors being equal (for example, you didn't feel a serious urge to chase down a fireball as quickly as possible), you want to begin your bottom hammer timeframe as far to the right of the screen as possible because steering barrels becomes significantly more efficient from that position. By starting so much farther to the left, you run the risk of missing one or more barrels with your bottom hammer much easier than you would if you began in the more ideal position toward the right side of the screen.

Ok, now, which is the better choice? Option A or Option B? In option A, we back jump the lead barrel and smash it, and then turn around just in time to smash the other two barrels which are stacked directly on top of each other. This maneuver alone yields 100 + 300 + 300 + 300 = 1000 points. In option B, we perform a standing jump over the lead barrel and then back jump the group of two barrels and smash them both. This maneuver alone yields 100 + 300 + 300 + 300 = 1000 points. We have scored 1000 points off of those three barrels in both situations. So, they are the same right?

Wrong.

With the second option, you have delayed your hammer grab by about a half of a "Kong cycle". This means that perhaps about half of the time you'll be able to incorporate one additional barrel smash into your bottom hammer. In addition, occasionally you may be able to smash one more fireball at the tail end of your hammer cycle that would have otherwise been just out of reach. Assuming you would have jumped over or leeched that last barrel if you were not able to smash it, you would still get 100 points if you come up short, but you'd get 300 points if you smash it. With Option A, you'll always get 100 points, but with Option B you'll get 100 points about half of the time and you'll get 300 points the other half of the time. So, on average, this barrel will yield about 200 points with Option B -- you've picked up 100 points (in terms of expectation) by making the decision to go with option B. This number could even be slightly higher if we realize that occasionally this last barrel could be a blue barrel...

On top of that, fireballs have an expected value of about 429 points (it's been determined recently that they yield about a 4:2:1 point distribution ratio, which may vary slightly based on their internal difficulty level). If even 10% of the time our delayed hammer grab allows us to smash a fireball that otherwise would have been just out of reach, we've gained about 43 more points from that AND we know that just barely missing a fireball as a hammer expires can cause problems that would often alter our most efficient line for playing out the board -- occasionally causing lost points (or death). At the very least let's estimate that we are "saving" an additional 57 points from avoiding these problems those times that we ARE just barely able to smash a fireball under Option B. So, that's perhaps another 100 points in expectation from the fireball (on top of the 100 points in expectation from additional barrel smashes).

Now, for those of you that are still following all of this, a sharp observer might point out "Yeah, but by delaying your hammer grab, you are allowing additional time to tick off of the Bonus Timer, so you might be losing just as much as you're gaining." ...

Ok, this concept is of critical importance to anyone who really wants to get good at the barrel screens. Conserving Bonus Time has virtually no importance. In fact, the phrase "Beating the Clock" is a pretty serious misnomer when it comes to barrel screens. This is because on barrel screens, the Timer is actually NOT a timer at all! It is a barrel counter. At the end of the screen, you get 100 Bonus points for every barrel that is still sitting in that pile to Kong's right, unused. When point pressing a barrel screen, your job is to make sure that you are scoring more than 100 points for every barrel that is released onto the screen, on average. The more efficiently you can score PER BARREL, the better you will score on the screen, regardless of how much time went by. In fact, you can finish a barrel screen with the same number of Bonus points remaining in the Timer despite a significantly different amount of real time elapsing!

Well then, how does that relate to our current situation in this quiz? Well, because of our position on the screen, we have NOT "lost" any barrels, so even if the Timer shows a lower number when grabbing the hammer, no points are actually lost. This is a change in thinking that is very important in order to really understand point pressing barrel screens.

So, if no points are lost due to what's going on with the Timer, and an equal number of points are scored off of those three initial barrels, but delaying the hammer grab can yield an additional 200 points from subsequent barrels -- then this really is a net gain of 200 points just from this one decision! That may not seem like much, but when you are going for really big scores, if you can "find" an additional 200 points on every screen throughout the entire game, over the course of 116 screens that can add up to a LOT of points.

BONUS -- So, what if one of our three initial barrels is a blue barrel? Well, if the blue barrel is part of the group of two, it makes no difference. The expected number of points scored from these three barrels would still be the same whether we used option A or option B, and option B would still be the best choice for the other reasons already explained. However, if the LEAD barrel is a blue barrel, the decision is now very close. A blue barrel has a point distribution ratio of 1:2:1 and will yield 525 points on average. With option A we will score 100 + 525 + 300 + 300 = 1225 points from the initial three barrels. With option B we will score 100 + 300 + 300 + 300 = 1000 points from the initial three barrels. This is a LOSS of 225 points up front. If we can still assume a net gain of 200 points by delaying our hammer grab in Option B, now there isn't much difference between the two options. Incidentally, in this case, because we know the location of a blue barrel (the lead barrel), it's now almost impossible for that last barrel on the tail end of our hammer cycle to be a blue barrel because of what we know about how the blue barrels are spaced out. On the other hand, missing the blue barrel now would result in a fireball spawning which could eventually be smashed some small percentage of the time (which would tend to favor option B, although this fireball might be just as likely to just cause other problems which might cancel this out). In the end, I think this scenario is probably very close and I would probably switch my answer in this case to Option A when the lead barrel is blue.

Hopefully some people enjoyed this quiz and my analysis of the situation. I'll try to come up with another one soon.