Becoming Charlie

Poincaré died one hundred years ago, in July 1912. He was a French mathematician who also wrote essays on the nature of mathematics.  I happened to visit his Wikipedia page today, which would be his 158th birthday anniversary.  In the days before Wikipedia, this was a leading source for Poincaré's biography online in English. Also try this.

I wonder how many mathematicians there are in the world.
If "a mathematician is a person with an extensive knowledge of mathematics, a field that has been informally defined as being concerned with numbers, data, collection, quantity, structure, space, and change" (source) then I suspect that the number of people that I would consider to be mathematicians is 100,000 - 200,000.

You could think about membership of the American Mathematical Society (about 30,000)
Multiply by 3 (for counterparts in Europe/former SovietUnion and Other) = 90,000.
Double that number to reflect those outside pure mathematics but close enough = 180,000.

I also like this estimate of 150,000 people.

Other fun data: SIAM, Society for Industrial and Applied Mathematics has ~13,000 members, while Association of Women in Mathematics has ~2000 members. 

These days I'm watching season 5 of NUMB3RS again.  Charlie is a fictional math genius who teaches at CalSci but often assists his crime-busting brother at the FBI.  This TV show features imaginative and engaging stories of math life.  I find it inspiring.  

Improving all around in Mahjong Titans

Raising the score:
Two days ago I resumed playing Mahjong Titans.  I started applying my new knowledge of the rules: that you score bonus points for matching two pairs at once.  (You see, I don't even know the rules, I'd rather learn as I play.)  This is yielding higher scores (when I win) :
#1 top score in Spider (14th April; won 2 of 4 new games played since the 13th),
#1 (14th April) and #3 (13th April) in Crab (won 2 of 3 new games played)
and #3 in Fortress (this is 1 of 1 new game played, 13th April, possibly without the new rule.)
I expect that all, not just most of, my new wins will displace the old wins at the top, as soon as I've won about five more games in each.  

How to win at Turtle, modified: 
Then again, my previous posts were about winning at the game (which is not the same thing as scoring high, but related.)  I learned a new idea about winning at turtle too: ignore the two long rows (which I'd named "the feet") at the top and bottom of the layout.  Focus heavily on the middle (head, shell-top, to tail) since there are so many more tiles packed in this region than at the feet (even if the feet have long rows.)  I think it's working.  When I've tested it a bit more, I'll officially modify my rule book.

Improved or not?  Two more tests.
Criticizing Method One: 
In the previous post, I glanced at the plots of top 5 scores for each layout and declared that those that were going up and to the right were "improving" and so on.  I concluded that Turtle and Crab had a weak outlook, Cat and Spider were possibly stagnant, while Dragon and Fortress were strongly improving. But this was a messy and inconsistent measure.  Turtle was up and down and up again, making the slope rather meaningless.  Spider may have pointed down, not improving, but with such a slight slope (and small range; all the top five scores packed into three months)  that it's hard to tell. And so on.  Like I said, messy and inconsistent.

Method Two, are the wins recent or early? 
Think about what it means that only three of the top five scores in Turtle came in or after Oct 2011, with the other two long time ago in Dec 2010 and Jan 2011.  That means nearly all the winning scores in Turtle (28-3 of them) came in Sept 2010 - Aug 2011, the early period, while only 3 were in Sept 2011 - Jan 2012.  That is a lot of early wins; remember that I played more and won more Turtle than any other layout.
In fact, let's try to see a new win/early win breakdown for each layout.  Put September 2011 as the cut-off. 
WINS        New   +    Early      New % (of total wins)
Turtle         3        +        25           11%
Cat             4        +        9            31%
Crab           4       +       9          31%
Fortress      3       +      10          23%
Spider       5+       +      9-           36% or higher
Dragon       4       +       9            31%

This makes the point I wish to make better:  If play was for 16 months of which 11 are termed "early", then wins spread randomly will give 5/16 in the new period (or about 31%).    By this measure, Cat, Crab, and Dragon showed flat performance, a win distribution consistent with no improvement / no degradation.  Spider showed improvement, while performance in Fortress got poorer.  Turtle showed a marked degradation in performance.

Method 3: Let's look at this another way, when did the top 5 scores occur, on average?
Top 5 (Average of the dates in which the top five scores were attained in each layout)
Turtle 7/8/2011
Cat 8/7/2011

Fortress 9/23/2011
Crab 9/25/2011

Dragon 11/2/2011
Spider 12/5/2011

ALL : 9/21/2011

Answer: Early in Turtle and Cat, same as random in Fortress and Crab, recently in Dragon and Spider.
This suggests Turtle and Cat performance worsened, while Dragon and Spider improved.

TO SUMMARIZE an analysis of my progress in different layouts of Mahjong over 200 games, 6 layouts, and 1 1/2 years:

Above are linear regression lines for the top five scores in each layout.  The average date at top 5, or midpoint of the line, is more important than the slope of the line.  From bottom (early) to top (new) along the 3 mark, that gives clearly Turtle, Cat, Crab/Fortress , then Dragon, then Spider.  To decide the near-tie between Crab and Fortress, consider that Crab points down (degrading performance) while Fortress clearly points up (improving).  Hence Turtle, Cat, Crab, Fortress, Dragon, Spider. 
That's my answer and I'm sticking with it!

Coming up: An introduction to topology, a proof that women cheat, and more games, particularly my unsolvable computer solitaire.  Meanwhile, try Mahjong Titans and tell me your results.

Worsening performance in Turtle

I told you months ago about the sense I had that while I learned the tricks to play the most structured layouts in Mahjonng Titans e.g. Fortress, I seemed to lose the ability to win at Turtle as naturally as before.  Let me show you in numbers (the few numbers I have):

I think the scores are for about 18 months of play, from my first game in late 2010 to the last in January or February.
As you might expect, my best result is in 2012, with another two top-fives just the month before.  Basically my recent games won are good compared with my old games won.  This implies improvement.

Comment on using proxy measures: 
Of course the real data I would have liked to use is my wins and losses over time, to see if I won more (percentage wise) more recently than in the past.
I couldn't find such records in my game, but I found top-five "High Scores" with dates, which I am using as an alternative measure of performance trend over time.  Make sense? 

Now check out my scores for the other layouts:

Fortress seems to show a similar trend: top score in 2012, another two top-fives in Oct/Nov 2011.
Spider shows all five top scores come in Oct 2011 or later.  That is, better than what I played earlier --> improvement too.
Crab is somewhat similar; three top-fives since Oct 2011, just like Fortress.
Cat has four of the top scores in Sept-Oct 2011, and notably the number 2 top score from all the way in 2010.  Not as clear a picture of improvement with the Cat configuration.

Now look at the Turtle scores: of 28 games won in 16 months from say Oct 2010 to Jan 2012, two of the top five are in the early days: December 2010 and January 2011.  The remaining are in late 2011.  Compared to the other configurations, my top Turtle scores are not clustered in the most recent past. 

Or you could look at this picture I made by plotting the top 5 dates for each configuration in MS Excel:

Strongly improving: Fortress and Dragon
From rank 5 (or imagine the lowest rank wins in the blank space to the left of the chart) up to the top rank, the Purple Fortress appears to be marching up (improving).  To look at the best three scores in fortress and dragon, the message seems to be "the best is yet to come."  

Possibly stagnant: Spider and Cat
The performance in Spider is mildly falling, with best performances around October 2011 not looking likely to improve.  Cat is the same, if you ignore that the #2 performance in 2010.  Not getting better.

Weak outlook: Crab and Turtle
The green crab is almost certainly falling down, better past than future.  And the turtle?  It's a bit haphazard, but it's also sweeping down. 

It's not just my imagination, I used to play better Turtle before I learned all those specific tricks. 

Play to win: Mahjong Titans

1. The major constraint is that you can not see the lower tiles until you have peeled away the upper tiles. You can't move these 'covered' tiles.
2. There is an important secondary constraint: you can not move a tile when it is hemmed in on both sides by other tiles.  Think of these tiles as 'covered' from the side.

What should you do:
Of course, because of (1), you must try to get to the lowest level of tiles and eventually remove all the tiles. Dig vertically. 
Also, be strategic because of (2): when you see a long row of tiles with strong dependencies, focus on that row. Dig/Chisel/Peel horizontally too. 

Course of attack in Mahjong Titans

For example, in the Fortress layout, you will lose the game unless you focus almost absolutely on the two long rows of tiles.  If instead, you just pair tiles as they come, you will end up with several unpaired chunks/layers at the end of the game (i.e. loss of game).  Rule (2) is utterly important in Fortress.

I think it is also of the utmost importance in Spider, which has two super-long rows (focus on these), and then one other long row (the thin red arrow above the other two) where you should source for free tiles before considering other (largely freestanding) tiles.

(2) is also very important in Dragon, which is crowned with a super-long row of tiles.  You must focus on that row, and then also the two tricky bottom rows.  Secondarily, consider the two short half-rows.  Avoid the free-standing tiles as much as possible.

(2) is less important in the other layouts:

To solve Turtle: You can see why it's called a turtle: it has a hard shell on top - attack that, circled in red, very early or you may never win the game because the heart of the layout is buried layers deep.  It has a head and tail sticking out, cut those off early too.  First thing you should do in Mahjong Turtle, try to pair the top tile (circled), the and the three other tiles blocking progress (at the head and yansh).  There are also two long rows at top and bottom that you must peel back.  Turtle is a balancing act.  It was the first layout I played (previously I had a computer that had just this layout, back in the day).  I just matched tiles more-or-less randomly, although I noted the top two layers on the shell as well as the edge tiles.  I don't recall how much I won then. (Note: NEW Strategy for Turtle here, with very high win rate guaranteed) 

To solve Cat or Crab, it takes some Turtle skills (balance width with depth) and some rule (2) skills (don't leave long rows).  However, these are so easy that you could win without thinking about strategy.  Truly.

Anyhow,  in Crab, focus on two important long rows and the big belly, you should not focus on the small sections. Cat has a big region and a small region, so focus on the big region to some extent.  

I want to talk about other rules but they are less important and can wait till next time.  I am also dying to tell you why I think getting smarter at playing the "hard" layouts made me dumber at playing Turtle.
Footnote:  In a way, the gross strategy for playing a given game is to follow the rules of the game i.e. start pairing tiles in Mahjong. The finer strategy, then, is to discover what makes it nearly unsolvable (or nearly lost) and then combat that.  
For another example, I recently started playing chess: gross strategy - move in the allowed patterns, take hostages preferably big ones, take the king's head;  finer strategy (think what happens late in the game) : the queen is a vicious bitch so get your opponent's out, pawn promotion by your opponent is bad for you so frustrate the frisky pawns, etc.

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