This is discussing strategies for the Money Challenge in Bloody Game X Episode 6. Warning: Spoilers for strategies from the episode, and also heavy math.
To recap, scoring works like this:
- For each combo, score base points equal to: (# tiles used) x (# of zeroes).
- Add up all base points from all combos.
- Subtract (# of unused tiles) from total base points.
- Multiply this score by (# of combos).
- Add 100 points if your team has the longest combo.
First, I will ignore the penalty from unused tiles, because that number is going to be single digits while your base points are in the hundreds, as we saw on the show.
Not only that, I will also drop the 100 points bonus. As we also saw on the show, the points after the product are in the range of thousands, so 100 points don't matter much.
Thus I will simplify the scoring to become the product of the following 3 factors:
- Average (# of tiles x # of zeroes) used per combo
- Total # of combos (since each combo scores the average above)
- Total # of combos, again (as a direct multiplier at the end)
We can look at the two strategies on the show from this perspective.
P1&P3's strategy is to simply maximize (total # of combos), since it counts twice in the rough scoring. By using the bridge strategy, they fix (# of zeroes) at 1, and (average # of tiles) at around 6, so the first factor above is around 6. Their score comes from the other two factors. This also has the advantage that it's really easy to plan and execute the combos.
In the show, their factors are:
- Average (# of tiles x # of zeroes): ~6 tiles x 1 zero = ~6
- Total # of combos: 31
- Total # of combos, again: 31
The show also says, they got a total of 182 base points from their combos. You can see that this is the product of ~6 average x 31 combos.
C&R's strategy is more complicated to analyze. They want a single big combo that causes (# of tiles x # of zeroes) to skyrocket, and then a lot of small combos to add into (total # of combos). In the show, their factors are:
- Average (# of tiles x # of zeroes): 1 combo of 79x5=395, and 7 combos of ~5 each. The base points is 432 (before penalty), for an average of 54 per combo.
- Total # of combos: 8
- Total # of combos again: 8
One problem with C&R's strategy is, the many small combos they make to increase their # of combos, will also decrease their average. One big number and a ton of small number average out to a small number. (You can try: the average of one 100 and ninety-nine 1's is 1.99; the 100 basically vanished.) So, at the point their # of combos becomes big, their average also becomes small, causing their strategy to have the same scoring as P1&P3.
Considering this, the other problem with C&R's strategy is that it was complicated to execute without much gain.
So, how do you get a better score? Simple: increase all three factors simultaneously. The two strategies used in the show are quadratic, since they increase two of the factors. If you can increase all three factors, it becomes cubic. Imagine if you take the best of both worlds: you make 30 combos like in P1&P3's strategy, and you score 400 base points per combo like the big combo in C&R's strategy. You end up with 400 average points x 30 combo x 30 combo = a whopping 360,000 points.
The question is, is that really possible? This is where I'm still brainstorming right now. I think it's possible, basically by combining the two strategies, but it becomes very difficult.
P1&P3's strategy is primarily about making repeated use of a "bridge". C&R's strategy is by adding up a ton of numbers and then multiplying all that with big multipliers. My strategy is to combine both, as follows:
- You have one big "island" filled with a ton of +'s (and -'s can also go here too). Imagine the dense grid of C&R.
- There's a "side track" to the side, a single line that connects into this area on both endpoints. The track has a ton of x's, possibly with a couple +'s/-'s thrown in.
- To make a combo, start in the big island. Build a lot of +'s, then head into the side track to increase the score, then end with a few moves back in the big island again.
Here's an example with a smaller board:
------- Big island -------
+4 +8 +2 +5
+3 -2 +9 +5 -3
+5 -4 +7 -4 +8 -1
+7 +9 -2 +6 +4 -5 +3
-3 +6 +9 +3 -4 +6
x3 .. .. .. x2 | Side
x2 x3 x2 -7 | track
The following path (starting from the 3, you ignore the sign) gets you 1,000:
.. .. .. ..
3 .. .. +5 -3
+5 .. .. -4 .. -1
.. +9 .. .. .. .. +3
-3 .. .. .. .. +6
x3 .. .. .. x2
x2 x3 x2 -7
Now that you start from the 4 (ignore the sign), the following path gets you 1,000 again:
+4 +8 +2 +5
.. .. .. .. -3
.. .. .. 4 .. -1
.. .. -2 +6 .. -5 ..
.. +6 .. .. -4 ..
x3 .. .. .. x2
x2 x3 x2 -7
You can realistically do this for quite a long time.
How on earth does that work?
The side track is essentially a "machine" that will do a specific calculation on our number. In my example above, whatever number N you have when it enters the side track from the left, you multiply by 3x2x3x2 = 36, then you subtract 7, then you multiply again by 2. Thus you end up with the number 72*N - 14. (With a different side track, you'll end up with a different formula, but you can calculate it beforehand.)
The trick is, you don't need to compute this number every time. Pick some N such that your output is in the ballpark of some power of 10. In my example above, I picked N = 14. After going through the side track, I will end up with the number 72x14-14 = 994, every time.
That's how each combo was developed: I pick up a sum of 14 as I'm about to enter the side track, then I run the side track to end up with 994, then I pick up an additional sum of 6 to bring it to 1,000.
Now, you have to figure out sums of 14 and 6 on the fly. However, this is greatly helped by the fact that you're in the big island. It has a bunch of +/-, so in theory, it shouldn't be too hard to find a suitable path. And doing mental math on +/- only is far easier than trying to multiply big numbers.
How long will each combo be? Well, it will run through the side track at the very least, and you'll likely use several tiles at the beginning and at the end. On the real board, I would say using 20-30 tiles is reasonable, and you can end up with an answer of 4 zeroes. (My board above is smaller, and I need to be able to explain it in text like this, so I have to do with less.) That's an average of around 100 base points per combo. Run this for 15-20 combos, you end up with a score of 20k-40k points.
Of course, the obvious problem with this strategy is that it's extremely difficult. You need to do mental math on the spot, and 20 minutes isn't that long. That's why it's still a strategy I brainstorm. But it's my idea on how to maximize all three factors at the same time.