As a Chinese version of this post, http://saolei.net/BBS/Title.asp?Id=12924is too rough and too much to read. I'll try to rearrange & simplify all the issue in a different way.

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0. About the work on board difficulty, 3bv is the most wide-spread benchmark even in smartphone platform, it means the minimum of leftclicks-only needed to solve the given board. You can easily find 3bv solved per second meaningful. So the 1st Speed-Efficiency Model is:

*3bv*=*Time***3bv/s*Since

*Time*is the goal and you can't determine the

*3bv*,

*3bv/s*is naturally being another target of this game, the efficiency to approaching your goal. Somebody call it the speed of playing.

0'. Now that we have

*3bv/s*as a start of statistical analyzing, the work following is to study our operation. The most obvious indicator is how many clicks used in the game, and how fast your click is. So the 2nd Speed-Efficiency Model is:

*Click*=*Time***Click/s*Till now everything is easy. Suppose that every game needs same click, if you can click faster in the same solving, you will improve easily. But what I suppose here is not true.

1. If you are just clicking faster, some clicks won't make sense. What click makes sense is the click which is used to solve more 3bv. We come to the concept of

**Index of Efficiency (IOE)**now,

*IOE*:=

*3bv*/

*Click*(":=" is stand for "be define as" in math). Compared Model 1 and Model 2, we now have the 3rd Speed-Efficiency Model:

*3bv/s*=*Click/s***IOE*Remind that

*3bv/s*is the target, at the very beginning you don't know the 3bv of board, what you can do is to get as higher

*3bv/s*as possible, so that you are possible to break your time or 3bv/s as your new record after several games. If you improve your Click/s or IOE, or both, you will be a better player.

2. Click-Efficient (

*C-eff*) benchmark makes refined model of Model 3 possible, which counts the clicks that make difference on the board.

(copied from http://www.minesweeper.info/forum/viewt ... f=26&t=273 )All clicks are efficient, except:

- left clicks which open no square

- chord clicks which open no square

- right clicks which do not set or remove a flag

We define:

*Corr*(Correctness):=100%*

*C-eff*/

*Clicks*It's the click efficiency of player, no more than 100% always.

*Thrp*(ThroughPut):=

*3bv*/

*C-eff*

*CeS*(C-eff per second):=

*C-eff*/

*Time*

*Thrp*is the refined ver. of

*IOE*,

*CeS*is the refined ver. of

*Click/s*, you can understand them as

*Thrp*=

*IOE*/

*Corr*(IOE if there're all eff clicks),

*CeS*=

*Corr**

*Click/s*(eff click rate)

Here comes the 4th Speed-Efficiency Model(Refined Model 3):

*3bv/s*=*CeS***Thrp*In fact, Minesweeper Arbiter breaks IOE into Corr*Thrp, so if you write down Model 3 like

*3bv/s*=

*Click/s**

*Corr**

*Thrp*, Model 4 is shown if you combine the first two factors. Model 4 is obviously better than Model 3. For FL player, concentrate on improving

*Thrp*please. For NF player, Thrp is limit by the board, at most 92% at normal speed, so the only way to improve is higher

*CeS*.

3. Think in a different way. The operation of mouse is not only the clicking, but the moving. If one player awared larger pattern than others and was able to move his cursor faster to the easy part, he must be a better player. This indicator must rely on the "Path" benchmark, it has the dimension Pixel (every cell has the size of 16pixel*16pixel). Define

*Path-Sq*:=

*Path*/16pix.

Define

*Move-Sq*(MoveSpeed by Square):=

*Path-Sq*/

*Time*.

Then what's the efficiency of moving? Define

*IOME*(Index of Moving Efficiency):=

*3bv*/(

*Path-Sq*). It's almost impossible to achieve IOME>1, though I have find out 2 beginner games in saolei.net with 0.97+ IOME. A player who achieved IOME>0.5 on expert must have very good grasp of patterns.

Now 5th Speed-Efficiency Model is:

*3bv/s*=*Move-Sq***IOME*This model has a problem that

*Move-Sq*can be cheated, say, leave 1 cell unsolved and moving very fast in circles But

*IOME*is a good benchmark, considering

*Move-Sq*when

*IOME*>0.3 is enough. Cheating

*Move-Sq*is useless because there's no record for that

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0''. What influence NF player most is Openings (Op), they need 2~6 clicks to solved an op, but an experienced FL player need 1~4 only. A relative concept is Islands (Is), remove all mines and all opening squares(with opening's edge), then a set of continuous (including diagonal) number cells is an island. Both Op and Is do something into the board difficulty. Read this for more about difficulty!http://members.ziggo.nl/ronny_de_winter ... 0trial.pdf

0'''. ZiNi stands for the minimum of clicks needed to solve the given board, when 3bv aimed at difficulty of NF, ZiNi aimed at difficulty of FL (definition is the same except of "left-only"). There're several ways to count ZiNi but none of them is theoretical ZiNi, they are Human ZiNi (H-Zini), Greedy ZiNi (G-ZiNi, which is mostly the best algorithm, so called ZiNi for short), Random ZiNi. Notice that solving all op by L-click at the beginning is not always most efficient if there's certain pattern on a board, see the statistic herehttp://www.minesweeper.info/forum/viewt ... f=17&t=274

Every indicator in 1st part has a ZiNi version, 3bv/s vs ZiNi/s, IOE vs ZNE, Thrp vs ZNT...just place new difficulty benchmark at where 3bv is. You can even generate more benchmarks using H-ZiNi, ZiNi+Op, 3bv+Op, 3bv+2*Op, OBV...as difficulty if you want. But what I used most are only two of them, that is,

**for FL skill,**

*ZiNi/s***as limit of IOE (which is not always true).**

*3bv/ZiNi*