Spire Changes

[Pauly7]

Interesting @Sir Derf, but do you factor in the fact that there are three possible outcomes with any one guess, ie correct, not needed and wrong place? If it comes up as not needed that's almost as good as correct and much more beneficial than wrong place. However if it comes down to just two left to find then wrong place can be better.

So many different variables and of course any random chance is greatly reduced in the second round of guesses.

 

[Pauly7]

OK as I was typing you addressed some more complications haha.

 

[Sir Derf]

Actually, I think red is more useful than yellow. Red eliminates an option from all other active spirits, yellow eliminates from the current spirit, and only eliminates other options from other spirits when combined with other information.

 

[Pauly7]

Really, though, the only percentages you can easily calculate are on the first round of guesses. It's much much harder to give a figure on the chance of failing altogether. There's just too many factors.

 

[Deleted User - 1759805]

Actually, I think red is more useful than yellow. Red eliminates an option from all other active spirits, yellow eliminates from the current spirit, and only eliminates other options from other spirits when combined with other information.

Usefulness of yellow vs red goes up with the number of choices. Think about more extreme examples of, say, 100 choices. Red variant only drops one from each 5 choices, leaving you with 99 that you don't know anything about. Yellow, on the other hand, tells you that at least 1 in 5 definitely has this particular choice (out of 100). Which one do you think gives you more information?

 

[CrazyWizard]

Usefulness of yellow vs red goes up with the number of choices. Think about more extreme examples of, say, 100 choices. Red variant only drops one from each 5 choices, leaving you with 99 that you don't know anything about. Yellow, on the other hand, tells you that at least 1 in 5 definitely has this particular choice (out of 100). Which one do you think gives you more information?

One of the worst outcomes I hate is if on the first try you have 4 green and 1 red, the it becomes a true gambling house instead of any sort of strategy.
It doesnt matter if I right or wrong I just hate that outcome.

 

[Sir Derf]

5 Spirits, 100 options each. 100^5 arrangements, 10,000,000,000.


You guess A on Spirit 1, and are told Green. 4 Spirits, 100 options each, 100^4 arrangements left. 100,000,000 reduction by 9,900,000,000, or 99.0%.

You guess A on Spirit 1, and are told Yellow. A=99 options, one of the others 1 option, the other 3 100 options. 99 * 4(100^3) = 398,000,000 reduction by 9,602,000,000, or 96.0%.

You guess A on Spirit 1, and are told Red. 5 Spirits, 99 options each. 99^5 arrangements left. 9,509,900,499 reduction by 491,000,000 or 4.9%.



Yes, usefulness of yellow vs. red goes up with more options. But, at what point did they cross over. We're not dealing with 100 options, we're dealing with 7.


5 Spirits, 7 options each. 7^5 arrangements, 16,807.

You guess A on Spirit 1, are told Green. 4 Spirits, 7 options each. 7^4 arrangements left. 2,401 reduction by 14406 or 85.7%.
You guess A on Spirit 1, are told Yellow. A=6 options, one of the others 1 option, the other 3 7 options. 6 * 4(7^3) = 8,232 reduction by 8,575 or 51.0%.
You guess A on Spirit 1, are told Red. 5 Spirits, 6 options each. 6^5 arrangements left. 7,776 reduction by 9,031 or 53.7%

With 7 options, taken on it's own, a Red is slightly more helpful than a Yellow.

 

[Sir Derf]

One of the worst outcomes I hate is if on the first try you have 4 green and 1 red, the it becomes a true gambling house instead of any sort of strategy.
It doesnt matter if I right or wrong I just hate that outcome.

Strategy is simply optimizing the gambling. It's always still gambling, just hopefullly gambling with improved odds.


However, your scenario does describe a point, which I have realized that my above analyses, interesting as they are, is actually deceptively inappropriate.

Assuming a 7-option round, being told 4 Green and one Red on the first go has drawn down the total possibilities from 16,807 to 6, a reduction of 16,801 or 99.96%. However, those 6 options have no choice but to be tested individually in up to 6 additional rounds; there's no way to eliminate multiple possibilities at once. As I posited in an earlier post, we're not really looking a reduction of options, or on early or total individual wins; we want to operate with a strategy that minimizes the rounds, which is not necessarily the same as minimizing the number of possibilities or the number of Spirits.

 

[Sir Derf]

Fooling around with a spreadsheet looking at this. A few interesting early observations.

As pointed out earlier, regardless of what your first guess, whether 11111 or 12345 or anything in between, you have a 46.3% chance that you will get none right. Equally, regardless of your first guess, you have the same odds to get 1 right (38.6%), 2 right (12.9%), 3 right (2.1%), 4 right (0.2%) and a single possible of getting all 5 right (0.01%).

On the other hand, the number of Red and Yellow responses are different.

	11111	12345
0 Red	00001 (000.01%)	00120 (000.71%)
1 Red	00030 (000.18%)	02400 (014.28%)
2 Red	00360 (002.14%)	07500 (044.62%)
3 Red	02160 (012.85%)	05700 (033.91%)
4 Red	06480 (038.56%)	01055 (006.28%)
5 Red	07776 (046.27%)	00032 (000.19%)
Total	16807 (100.00%)	16807 (100.00%)

You start out with each Spirit having 7 options, and independently requiring up to 7 rounds to guess all successfully.

With the 11111 guess, you get an average 0.71 Greens, so eliminate an average of 0.71 Spirits, leaving an average of 4.29 Spirits. But, every remaining Spirit has only eliminated 1 of the 7 options, leaving 6 options for each Spirit, requiring up to another 6 rounds to guess all successfully.

Also with the 12345 guess, you get an average 0.71 Greens, so eliminate an average of 0.71 Spirits, leaving an average of 4.29 Spirits. But, every Red removed a possible options from all remaining Spirits. The average number of the Reds received above is 2.31, meaning that, on average, you eliminated 2.31 of the 7 options, leaving an average of up to 4.69 rounds to guess all successfully, an improvement on the 11111 guess by 1.31 rounds.

 

[Sir Derf]

Guessing 12345

Pattern	Count	Percent	Depth
GGGGG	1	00.01%	0
GGGGR	30	00.18%	2
GGGGY	0	00.00%	0
GGGRR	250	01.49%	2
GGGRY	100	00.59%	2
GGGYY	10	00.06%	1
GGRRR	640	03.18%	2
GGRRY	1080	06.43%	3
GGRYY	420	02.50%	2
GGYYY	20	00.12%	2
GRRRR	405	02.41%	2
GRRRY	2220	13.21%	3
GRRYY	2910	17.31%	4
GRYYY	900	05.35%	3
GYYYY	45	00.27%	3
RRRRR	32	00.19%	2
RRRRY	650	03.87%	3
RRRYY	2840	16.90%	4
RRYYY	3260	19.40%	5
RYYYY	950	05.65%	6
YYYYY	44	00.26%	4

Depth	Count	Percent
0	1	00.01%
1	10	00.06%
2	1897	11.29%
3	4895	29.12%
4	5794	34.47%
5	3260	19.40%
6	950	05.65%

Depth 3 (4895) - GGRRY (1080) GRRRY (2220) GRYYY (900) GYYYY (45) RRRRY (650)
Depth 4 (5794) - GRRYY (2910) RRRYY (2840)
Depth 5 (3260) - RRYYY (3260)
Depth 6 (950) - RYYYY (950)

 

[CrazyWizard]

I know someone is trying to build a calculator.
If you look up mastermind there is some math on the wiki page, i think, it shows how some mathmetition proved you could solve it in x rounds always or so.

so technically it is possible to build a spire calculator that gives the most optimal spire options based on.
I would say an awesome AI based project for a college graduate to build a system that can play the spire in the most optimal fashion.

 

[Sir Derf]

The Spire (and Mastermind) comes nowhere close to being a complicated game. There are a fixed number of arrangements, a fixed number of guesses, and while these numbers are larger than can be encompassed at once, they come nowhere near overwhelming. This is not Go, or Chess, or Poker, or Blackjack. When you eliminate the stupidity of random guessing that allows repeated guesses of the same option for the same Spirit, a 7 option game resolves in 7 rounds. While you could, it's probably overkill to set an AI to optimizing the strategy.

 

[Sir Derf]

Thinking about our Depth 6 scenario. 950 arrangements.

The R has eliminated one option. 6 remain. The Y's have each eliminated 2 options, The R and its own Y, so each have 5 remaining options.

But, you also know that across all 5 Spirits, if there are any of the 2 unguessed options, there can only be a singular occurrence of 1 of them. And that 3 of the 4 Y's have to be 3 of the 4 Y's.

I think you have to guess each of the 2 as-yet unguessed options. You can either guess them with 2 of the Y's, or with the R and one of the Y's. If there are no unguessed options, you will get both Red, and eliminate them from all Spirits, so it won't have mattered where you guessed them (also, you are guaranteed that at least one will be Red). If one of the Spirits wants one of the unguessed ones, you will either match it or not. So, is it better to try matching with a Y, or with the R?

Hmmm... There are 530 arrangements with an unguessed options, and 420 with only Y options. 265 possible with either of the unguessed. 53 possible with each of the Spirits. But, is that 53 possible with the Spirit that gave an R? No. There are 90 possible arrangements where an unguessed is in the R position, 440 where either is in the Y position (and 420 where it is in neither position). This is where my reasoning runs out. I need to adjust my spreadsheet for concrete info.

 

[Deleted User - 1759805]

so technically it is possible to build a spire calculator that gives the most optimal spire options based on.
I would say an awesome AI based project for a college graduate to build a system that can play the spire in the most optimal fashion.

You don't really need an AI for this game, it is simple enough for the usual minimax approach. I have built such engine in the beginning, but haven't used it in a long time as with only 2 turns (1st turn is always the same) it is much faster just to eyeball it. There is really not much added value in the calculator as we're not looking for the fastest solution, we're looking for a solution in 3 moves, and we're only really selecting 2.

 

[AstralSoul]

The Spire (and Mastermind) comes nowhere close to being a complicated game. There are a fixed number of arrangements, a fixed number of guesses, and while these numbers are larger than can be encompassed at once, they come nowhere near overwhelming. This is not Go, or Chess, or Poker, or Blackjack. When you eliminate the stupidity of random guessing that allows repeated guesses of the same option for the same Spirit, a 7 option game resolves in 7 rounds. While you could, it's probably overkill to set an AI to optimizing the strategy.

Agree.

It only really gets tough in the 3rd and 4th sections of the 3rd floor. The second floor with logic you might need no diamonds. You can auto-fight the first whole. Especially if you have booster buildings from the tourney.

Have five enchanted magic was (from 2nd boss), and 3 of each booster menus, and you will do fine. With sentient goods it will get trickier, but the nature is still the same. It is easy to get the whole 1st, and 2nd floor. Even a fragment of the third, and if you want to push it one week, only the 3rd fight, and the beefed up boss in the third floor will be really insane (variety, and quantity asked.)