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Dominion Articles / Re: Castles
« on: April 09, 2018, 09:59:54 am »
I feel like this might be due to a certain lack of precision in the Castle Sniper... I could not implement something that checks the number of Castles left in supply, so I did "lowest supply pile". But in a game where the opponent hardly ever buys Castles (and your "Castle" strategies may easily lead to games where they never buy Castles), the lowest supply pile will be Province or Duchy, leading to suboptimal decision making.
Either you misread my strategy or I misunderstood how the simulator works.  I put Castles above Provinces in the buy rules, and always gave them the same money cut-off (except when I gave a smaller one for castles).  My understanding is that it should always buy castles first, then provinces if it can't afford a castle (or the castles are out).
Geronimoo's simulator can check castles left in I went ahead and implemented that the updated rules for the sniper are:

Castles if count in supply Castles < 3
Provinces if total $ in deck > $18
Castles if count in supply Castles < 7 and > 3
Duchy if count in supply Provinces < 5
Estate if count in supply Provinces < 3
Duchy if count in supply Province < 7
This does about the same (57-40 vs buying castles-then provinces at $19+).
In a sample game, the Castles strategy picked up, 2 Estates, 3 Duchies, 4 Provinces, and 5 of the Castles, while the Sniper got 2 Estates, 2 Duchies, 4 Provinces, and 2 of the Castles (Grand and King's...he bought Small as well, but trashed it for King's).  So I'm pretty sure these buy rules are accurate.  The Sniper in the sample only got 1 province before the castles strategy started in on them, but he got both the really big Castles.

Dominion Articles / Re: Castles
« on: April 06, 2018, 03:10:37 pm »
I'm pretty sure in a BM game, mostly-Castles-and-some-Provinces beats Provinces-first-and-Castle-sniping-late. This should be reasonably simulatable because it's Big Money, I'd be curious to see the numbers.
Well I have done some simulation, though it's not optimized.

The Castle-countering strategy was fairly simple. It would always compete for Grand and King's Castle, and competes for Opulent, Haunted and Small Castle if it cannot afford a Province.

For the Castles-centric strategy, I have tried to implement a strategy that goes for Castles earlier than Provinces (BMU goes for Provinces with $19+ in deck). I fiddled around with the number of $ needed to start buying Castles, but I got the best results leaving it at $19+ as well, and then it still loses to Castle-sniping 33-64.

Another way to do Castles is to dive straight in. Surpringly, a strategy that always prefers Castles over Silver (i.e. that would open Small/Crumbling on 3/4) performs better! But it still loses to the Castle Sniper 41-56.

Feel free to come up with better bots!
I'm not sure what you did for the Castles-centric strategy...when I tried it, I got castles winning over the sniper you posted 57-40.  (I just added a Castle buy-rule above Provinces with the same $19+ requirement).

I wondered from discussion in this thead, if waiting longer would be better.  Increasing the wait time for provinces/castles beyond $19 improves things a bit, but hurts the mirror match.

So if we take the buy rules to be:
Castles if total money in deck > $X
Provinces if total money in deck >$18 and >$X
Duchies if Provinces < 5
Estate if Provinces < 3
Duchy if Province < 7

Increasing just the province start-money has minimal impact as far as I can tell.  I haven't played much with decreasing it below castles.

then for X=$18 I get: 57-40 over sniper (70-27 over BMU)
for X=$20 I get: 59-38, so it does just a bit better.  (but loses 48-50 vs X=$18)
for X=$22 I get: 61-36 vs sniper, but loses 47-50 vs X=$18.
Dropping X below 18 doesn't improve things much though, unless you drop it to 0, but even then it's barely squeaking out against X=$18 at 50-47.  And loses to the Castle Sniping (as noted by faust, 41-56).

General Discussion / Re: STAR WARS
« on: January 07, 2018, 09:16:59 pm »
R comes before T. 4 > R1 > 2.
Attack of the Clones doesn't have a "the" (unlike 5-8, 1, and 3).

General Discussion / Re: Maths thread.
« on: May 05, 2017, 09:55:31 pm »
I have a second problem (the same stupid lecture). I need to prove something, but it really appears to be false. That probably means I messed up, but idk how, so...

Let (E^2, d) be an Euclidean space and [P, Q] and [P', Q'] two line segments of equal length. I am supposed to prove that there are exactly two possible isometries T :: E^2 -> E^2 which map P onto P' and Q onto Q'; T(P) = P' and T(Q) = Q'. Also an isometry is defined as a function which preserves distance, so d(P, Q) = d(T[p], T[Q]).

Except this isn't true. I think.

Let f be a function which just shifts all points by (P' - P), so that P lands exactly on P' and all other points land wherever.

Let g be a rotation with fixed point P' that rotates exactly far enough so that f(g(Q)) = Q'.

Let h be a reflection around the axis that goes through P' and through the middle point of f(Q) and Q', so that h(f(Q)) = Q'

Then g ∘ f and h ∘ f both map P onto P' and Q onto Q' (f already maps P onto P' and g and h both don't change the point P').

Similarly, by moving Q to Q' first and then doing the same maneuvers to the point where P lands to P' without changing Q', I get two more functions.

All of them preserve distance, and they don't seem to be identical. You can get 4 more by first swapping/mirroring and then shifting I believe (and there are probably infinity other ways), but well just having 4 proves that there aren't just 2. Am I wrong or is the hypothesis wrong?

They may not seem identical, but they are.
It might help to think of a concrete example, here's a spoilered one.
You're translating (f), rotating (g), and reflecting(h).  Take a piece of paper, move it around, flip it over rotate it.  If you want to move 2 holes onto 2 nails you can do it in lots of ways, but there are only 2 different end results.
Anyways, I don't think that will directly help you prove it, but it should improve your intuition about it.

General Discussion / Re: Random Stuff Part III
« on: March 21, 2017, 02:27:34 pm »
you forgot the constant...
it's a (natural) log cabin by the sea!

General Discussion / Re: Maths thread.
« on: February 24, 2017, 04:14:52 pm »
I don't think sudgy's proof works without the fundamental theorem of arithmetic.

In particular, I'm not sure that
Because n only contains one of each of its factors, and a2 contains each of its factors at least twice, n|a
holds without fundamental theorem of arithmetic. 

I think with non-unique factorizations, you can't require n to have one of each factor and a2 to have 2 of each factor and every factor of n to be one of those duplicated factors.  You can only require any 2 of them.

General Discussion / Re: Random Stuff Part III
« on: December 07, 2016, 09:06:11 am »
Is there any instance where you'd rather have 1D8+1 over a D10, or vice versa?

Obviously when rolling a one sided die. But what about with a D4? Would you maybe want the D8+1 instead of the D10?

For D4, you'd want D8+1. For D4+5, you'd want D10. That's because the numbers 1-10 are symmetric in such a way that D10 has the advantage for the higher numbers and D8+1 for the lower numbers (and it all adds up so that they're both just as good against anything that rolls numbers between 1 and 10 symmetrically).
To generalize:
d10 is more likely to roll less than 5, and more likely to roll greater than 6.

So considering static numbers: (define "win" as roll better and "lose" as not "win")
d8+1 "wins" against 1, 2, 3, and 4 more frequently.
d10 "wins" against 6, 7, 8, and 9 more frequently.
They win equally often against 5.
And they both always win for 0 or less, and always lose for 10 or more.

Then for any given probability distribution including numbers from 1-9, you can see how it matches up against those values.  They aren't all equivalent probability wins, but they're symmetric about 5.  Basically, if comparing against other symmetric distributions, you'll just have to consider if one side is more likely than the other.  Against a uniform distribution that's just counting:
Here are some examples.
1d4, 1-4 are lots more likely (only possible) 1d8+1 wins more.
1d4+5 6-9 lots more likely (only possible) 1d10 wins more.
1d6 more 1-4 counts than 6-9 counts, 1d8+1 wins more.
1d6+2 has 3, 4, 6, 7, 8...more 6-9 counts so 1d10 wins more.

How about, 2d4? It has values 2-8 symmetric about the mean (5)...and so, it compares equally to d8+1 (winning more frequently on 2, 3, and 4), and d10 (winning more frequently on 6, 7, and 8).

Aside: What if we include ties?  well, if we adjust our definition of "win" to be "at least tie" then:
d8+1 "wins" against 2, 3, 4, and 5 more frequently.
d10 "wins" against 7, 8, 9, and 10 more frequently.

So, d8+1 will at least tie against 2d4 much more frequently than d10.  Since they're equal for "win", we see that if getting a tie has any advantage then d8+1 is better in this case.

What about 2d6?  2d6 has a symmetric probability about 7.  If you know much about rolling 2d6, you know 6, 7, 8 are more likely than 2,3,4...and it doesn't include 1, so clearly d10 is better here.

What about 2d6-1 though?  this includes all the numbers in both ranges, but it's still more likely to roll in the 6-9 range than the 1-4 range. (4 and 8 have the same likelihood and it goes down as you get farther from 6).

Finally, looking at 2d6-2.  Mean is 5, probability goes down as you get away from it.  So probability of 4 is same as 6, etc...makes them equal for win again.
(and again if we consider ties, 1d8+1 wins again)

Rules Questions / Re: The One-card Shuffle
« on: May 19, 2016, 04:04:51 pm »
fwiw, IMO a very small minority reads the poll question if the title of the thread is a question (I didn't as a single example).  If you had not put a question in your thread title, I would believe the results more strongly.  (Thread title saying "Please give your vote: can you shuffle one card?" means that people are voting on that question, not reading the more nuanced version)

As-is I am surprised by the number of bgg users that say that you can't shuffle one card.  I would simply say:
The question: "do most people believe that you can shuffle a single card?" probably has answer: "no".  As I would say that BGG is still more specialized than general population.

I don't think that's a particularly reasonable position in the abstract-world of game rules, but there are plenty of people that play games without having a complete understanding of the rules.  Many people miss intricacies of rules, and rule-interactions, and sometimes it has a negative effect sometimes not.  It's not persuasive, but the majority viewpoint doesn't need to be persuasive it simply is.

For this particular issue, I don't think that misinterpreting this rule would have a negative impact.  I would be inclined to say you can shuffle a 0-or-1 card deck because it's in line with pedantic game-rules (i.e. MtG), and I think it's better to be in-line with pedantic rules than general intuition.  But what I think on this particular issue isn't necessarily relevant.

As far as interesting side-cases:
Card commands you to shuffle your empty (or single-card) deck (I would count it).
Card commands you to draw and you have empty deck and single-card discard pile (I would count it).
Card commands you to draw and you have empty deck and empty discard pile (I would not count it, because I think the inability to draw a card takes precedence over the shuffling of discard into draw deck...i.e. you can't draw even after you shuffle your discard into draw, so you skip it.  Though, I could be convinced I'm wrong).

FWIW: I'd expect (based on current poll-results) good questioning to return results along the lines of:  (left being people that think shuffling a single card counts as "shuffling")
among mathematicians: 95-5
among fds people: 80-20
among gamers: 60-40
among general population: 30-70

Hearthstone / Re: List of Hearthstone IDs
« on: July 18, 2014, 11:10:10 am »
Theorel #1470 NA

Treasure Hunts / Re: More google maps challenges?
« on: April 29, 2014, 11:15:58 am »
Looking back, I think I did all the others...except 8 since there's no image.

Power Grid General Discussion / Re: Map Comparison Article
« on: June 12, 2013, 10:12:48 am »
So, there's several things we could potentially analyze here...and several ways to achieve that analysis.  We should probably have a bit more discussion on the matter of how to achieve the analysis we're looking for.

Note on including city costs:
If we're building 68 cities, then every map has $680 added in.  This results in maps appearing more similar than they actually are.  I personally think that this is bad.  Unless you're trying to figure out how much money a player is expected to spend on cities for a given map, it's not useful to include it.  (Note: that is potentially useful).  So, the question is: are we analyzing the maps relative to each other, or as absolute quantities.  If relative to each other, then I believe we should reduce the commonalities in the maps.  This will highlight their differences which is what we're interested in.

(Note: regardless all the analyses will be relative, so it's more a question of if we're comparing say 1.55 and 1.3 or comparing 1.11 and 1.06)

So, that's all I have to say about the city costs, I prefer leaving them out.
Let's talk a bit about analyzing the maps:
1. Relative Cost:
We could use the true MST here.  It gives nice consistent numbers, and it gives generally smaller numbers.  They're easy to compute, and "cleaner" to compare.
Disadvantage: you only build each city once, and must build each city once.  It doesn't take into account maps where some city might not get built.
That takes us to the 68-city "MST".  This does not give consistent numbers, so we should technically compute it for all starting cities and find the minimum in order to introduce consistency.  The numbers are bigger.  The result seems less "clean".

I don't think that the disadvantage on the MST is really there.  I think the 68-city MST will build every city at least once.  A "cheap" area getting built 3 times isn't that different from 2 moderately expensive areas being built twice.  I like the clean truth of a true MST over the simulated mess of a 68-city quasi-MST.

2. The impact of competition:
This is where the 68-city MST is really useful.  If we take it as an approximation of a "cooperative" game.  Then we can use the 4x17-city MST as an approximation of a "competitive" game.  I think that's a fairly accurate prediction of the impact of competition.

Currently we're doing 1 city at a time (I think that's what AdamH did?).  If we did 1 player at a time that might give information about value of turn order (how big is the difference between player 1 and player 4).  This could give a feel for impact of board position.

We could also analyze the actual board positions, seeing where the optimal starting positions are.  We don't do this when we minimize total cost, but if we minimize individual costs we can determine "ideal" starting positions for each player (note: this would still assume either completely equal or completely unequal development, which is not true in game and could certainly impact the value of starting positions.  It also ignores whether certain bits of expansion are affordable).  Anyways, it's not perfect but I think it could be an interesting basis for some discussion.

Power Grid General Discussion / Re: Map Comparison Article
« on: June 11, 2013, 01:20:54 pm »
Oh, and here's the code if you want it.

It should parse all of the files that you posted.  If you add a name line above each line, it will then name all the cities.  If you add a "Name, Color" line it will parse that, and then you can enable/disable the various sections of the map using the enum.

Power Grid General Discussion / Re: Map Comparison Article
« on: June 11, 2013, 12:35:40 pm »
Okay, I've got it all in place now.  So, let's see:
1. For the quasi-competitive implementation I was allowing a single player to build a city multiple times.  Removing that results in things similar to your results.

2. For USA-Max, non-competitive.  If I choose index 14 to start with I get your result.  However, choosing something like San Diego (index 27) results in the costs I saw before.  I noted this already in my write-up earlier.  Starting city matters if you're not doing a straight MST.  (P.S. that means that your max for USA should be 1343, not 1406)
(here's the full transcript for the 1343 build: text file)

3. For competitive, I'm still getting better results than you though.  I realized I wasn't allowing to build through cities (which gave better results, because competition was lessened).  But with that in place, I'm getting the following results for USA.
Santa Fe
San Francisco
Kansas City

New Orleans

Power Grid General Discussion / Re: Map Comparison Article
« on: June 11, 2013, 10:39:48 am »
Okay, I'm sick again and not at I just implemented it at home.

I found an error in your min-cost map.  You have the New Orleans-Jacksonville connection at cost 6 instead of 16.

Power Grid General Discussion / Re: Map Comparison Article
« on: June 07, 2013, 08:44:24 am »
I coded stuff at the office during down-time.  And worked from home all week (due to illnesses), so I still don't have the code.

I'm trying to figure out the differences in our US results though.

For starters, I assume you're actually charging for cities, which I didn't do.  So, you have to add 680 to my costs, straightforward enough.
For min-cost map that takes it to: 1209 for me vs. 1207 for you.  Okay, that's reasonably close, starting city could have an impact there, I didn't try all of them, maybe even which route you took in ties could have had an impact.

For the max-cost US map though, you've got 1406, while I got 1343.  That's substantial.  So, I'm wondering where the bugs are.  I can chart you the connections used once I have the code again to help find them...could even just be in the connections matrix you've done.  Or it could be in my connections stuff somewhere.

For the competitive version, what did you minimize?  Total cost, or maximum cost?  Total cost is still collusion in start position.  Even maximum cost is to some extent.

If you minimized over total cost though, then I'm confused again as to why my min-cost is some 50-points lower here.  Again, I can get all the connections used to verify when I'm back in the office, hopefully next week sometime.

Power Grid General Discussion / Re: Map Comparison Article
« on: June 03, 2013, 06:55:23 am »
Um...when I open that zip file it contains grid.lnk.  Maybe some sort of shortcut to the folder that you intended to zip?  Maybe you used windows' Recent Files thing somehow in a file browser?

Power Grid General Discussion / Re: Map Comparison Article
« on: June 03, 2013, 06:52:14 am »
I'll try to get the code posted sometime this week.

Power Grid General Discussion / Re: Map Comparison Article
« on: May 29, 2013, 10:16:57 am »
Okay, so I tried to do some stuff.
I made a tree in java, and did some simple hueristic MST-type stuff.

I've only done the US Map so far...but I might code up some more maps later.  I can't see the attachments AdamH put up, so I didn't bother to make a text parser...if I can get them visible, I could try to build a parser for it.

Anyways, the interesting bits are the results:
So, US Map including/excluding various regions.  I did a simple heuristic MST for it (just take the cheapest connection from the current MST and add that city).  I'm pretty sure this works for us because we're directionless, but I don't feel like proving it.

Here's the total costs for those trees for each possible 4-region set:
Green, Brown, Blue, Purple: ILLEGAL
Purple, Blue, Red, Brown: ILLEGAL
Green, Brown, Yellow, Blue: ILLEGAL

Green, Brown, Yellow, Red: 134
Green, Brown, Red, Purple: 147
Green, Brown, Yellow, Purple: 154
Green, Red, Yellow, Purple: 159
Brown, Red, Yellow, Purple: 163
Green, Brown, Red, Blue: 181
Green, Red, Blue, Purple: 191
Green, Red, Yellow, Blue: 193
Brown, Red, Yellow, Blue: 197
Brown, Yellow, Purple, Blue: 200
Green, Yellow, Purple, Blue: 201
Yellow, Red, Blue, Purple: 208

So, then I allowed you to visit each node 3 times, and built 68 cities.  There are several problems with this as an approximation of total costs but based on the later stuff it's not too bad.
Just cheapest and most expensive this time:
Green, Brown, Red, Yellow: 529
Red, Yellow, Blue, Purple: 663 (starting from phoenix)

Note: here starting city matters, and it's clearly not optimal.  I think the issue is probably that it builds some double-connections before single connections that open up cheaper regions.  In particular it's cheaper to build the SouthWest out before you triple-up in the fargo area.  Even though that first connection is more expensive.

Then I just had it do 4 of these simultaneously.  This goes 1-city at a time and builds out to 17 cities per "player" 
This is maybe the most interesting for analyzing starting positions although it's a far cry from a simulation.  Interestingly the total is cheaper than above...I'm supposing it's due to the 4 "free" connections for starting cities.
Green, Brown, Red, Yellow: 125, 130, 139, 128: 522
(Savannah, NewYork, OklahomaCity, Duluth)
Red, Yellow, Blue, Purple: 208, 144, 152, 154: 658
(LA, Fargo, Houston, Cheyenne)

For the cheap map, it's pretty stable regardless of where you start (though if you clump everyone together cost goes up a bit).  For the expensive map it is highly unstable.  It's really interesting to see how starting positions effect one another.  Especially the LA-player.  His starting position is obviously terrible using this method, but if you move him elsewhere he's too close to another player and only goes down to ~ 190 while driving them up to similar values.

Anyways, enjoy.

Treasure Hunts / Re: Ozle Google Maps Challenge 1
« on: April 15, 2013, 09:39:05 am »
Hey, I finally solved it! yay...

Treasure Hunts / Re: Ozle Google Maps Challenge 6
« on: April 02, 2013, 08:32:20 am »
I'm stuck...anybody want to talk about this one?
Land of Abraham:
Canaan?  Illinois?  Other?

Connection between 4 musicians/musical groups...I have no idea.  I didn't even know one of them existed, one of them I've only heard of, and the remaining 2 I'm only passingly familiar with some of their music.

Treasure Hunts / Re: Google Image Challenge 5
« on: March 28, 2013, 10:13:31 am »
Well, I'm not british...
Just checking the "exit"s QVist mentioned, it's not hard to find a road that looks like above, which is required for the image to be found.

Treasure Hunts / Re: Ozle Google Maps Challenge 4
« on: March 28, 2013, 09:12:27 am »
yay, found it.

Treasure Hunts / Re: Google Image Challenge 5
« on: March 28, 2013, 08:59:50 am »
Found it.

Treasure Hunts / Re: Ozle Google Maps Challenge 4
« on: March 26, 2013, 09:05:15 pm »
Okay, I'm pretty sure I've figured out the last bit, but I'm trying to figure out the first half.

Wild guess here: Considering the latter clues strongly suggest a golf course near the palace of Versailles in France (More than likely the 12th hole of the Golf Des Yvelines, which was the first golf course in France), I'm guessing the "country's prime" refers to a king, probably King Louis the XIII, since 13 is a prime number. Additionally, Louis the XIII was the king in Alexander Dumas book "The Three Musketeers". The "detective" line could refer to the book's sequel "Man in the Iron Mask", wherein the titular man is a son of Louis the XIII, hidden in a prison.

"Sweet Spot" refers to a manner hitting a golf ball in just the perfect manner.

Anyway, my guesses.

Although cool, I think you might have the same misunderstanding as ashersky had.  The goal is not to solve the riddle and guess where the image is, the goal is to find the image in google maps.  Looking over the Golf Des Yvelines, I cannot find this image...thus it isn't the correct answer (or I just didn't find it, which is possible)

However, this is a golf course, and likely the 12th is intended to guide us to that.  (I don't know how golf courses are laid out, maybe 12th will direct us to the very image for someone who knows it).

It appears to me that gold courses look the same the world over, so I think we really need to solve the first bit to get anywhere.

It used to take a detective to find this country's prime.

My wife suggested that it could be the prime minister.
A google search for Prime Minister Watson turned up Australia.  But I don't see any "sweet spot"s there.  And a search for antionette didn't turn up anything useful.  Any other ideas?

I assume the "queen who stored a lot of cake" is Marie Antoinette, and so likely we're looking for something named after her.  i.e. antoinette street or some such thing...but probably something less specific.  i.e. in the right area maybe a maryborough, or some such would get us close enough (although it doesn't appear to be the maryborough golf course in Australia).

Treasure Hunts / Re: Ozle Google Maps Challenge 3
« on: March 26, 2013, 08:29:19 pm »
Also, not sure what more clues I could give to this one without deliberately SPELLING it out.

You have 'a cop went' around, once you work out what that means the rest is pretty simples. And to give any more clues on that one would be just akin to telling the answer. its not as complicated as it sounds.
By specific "hints" I just meant solutions to the various parts.  i.e. what the first line means, second line, etc.

Yes, it's essentially the same as giving the full answer, but it lets people work out the later bits instead of just receiving a map location and trying to reverse understand the clue.

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