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Articles and essays Articles and essays
the first click
Where to start the game?
 
 
 

Article written by Manu (1-12-53), on August 3rd 2003

contents :

introduction
probabilities
meaning
comments

Introduction

You often wonder if there is a particularly favorable place to make the first click on. In order to clear the minefield without leaving some uncleared areas on which it is necessary to return, some sweepers prefer beginning on a corner and trying to evolve in diagonal towards the opposite corner. They claim to have a hole more frequently by beginning with an extremity … Why not?
The small calculations which follow explain why you observe effectively more openings on the corners than on the edges, and more on the edges than right in the middle.


Probabilities

The calculations are based on the following hypotheses :

1. the mines distribution on the board is unpredictable. All the possible boards for a certain level have an equal probability to come out. As far as we know, that it is completely false, it will maybe be the object of another article … However it is the model which allows to make simple calculations and which gives very realistic results when we compare them with the obtained results by beginning several hundreds of games. In addition, this model of "perfectly unpredictable Minesweeper" corresponds to what we are entitled to expect from an update of the game. Moreover, many players ignore that the Microsoft Minesweeper's boards are not randomly generated …


2. The first click is never a mine. This is an undebatable truth, because that it is the way that it was made by Microsoft.

With these hypotheses, here are the obtained results:

BEGINNER (WIN 98) (10 mines on 8 × 8 = 64 squares)
 
1st click...
2nd click if the 1st one didn't open anything…
No mine on the square you’ve chosen. There are 10 mines on the 63 other squares, so 53 chances out of 63 that there is no mine on each square surrounding the one you’ve clicked on
After the first click, there are 10 mines left on 63 squares ( 53 empty squares ). To open an area clicking in the middle, you need 9 of those empty squares. Not just 8, like when it was the first click
Middle
Edge
Corner


The same thing can be done for inter and expert…

INTERMEDIATE (40 mines on 16 × 16 = 256 squares)
 
1st click…
2nd click 1st one didn't open anything…
Middle
Edge
Corner

 

EXPERT (99 mines on 16 × 30 = 480 squares)
 
1st click…
2nd click 1st one didn't open anything…
Middle
Edge
Corner

 

Meaning

I clarify the way it is necessary to understand the obtained results by taking an example …


At the expert, you have 24.93 % chance to have an opening by making the second click on an edge:

-> ...whatever is the place of the first click : if this first click is in corner, on the edge or in the middle, the second click is 24.93 % of chances of opening if it is made on the edge.

-> ...if the first click didn't open anything. Indeed, if the first click made a hole of 45 squares for example, there is only 480-45 = 435 undiscovered squares, which only 435-99 = 336 are empty. It gives a probability of only = 21.24 % to have a second opening, what is sharply lower than the announced 24.93 %. It is easily understood by thinking that if the first click made a hole, the density of mines on the remaining area increased, what decreases the chances of opening. Moreover, who would want to try to make a second hole if he has in the first click of a big opening easy to exploit ?

-> and if the 2nd click is independent from the first one. Indeed, if the second click is made on a square touching the discovered square by the first click, it is then necessary to take into account the indication given by the square as in the example below :

In this expert's situation, the first click gave a 1. If the second click is made on the ? the probability of opening is not 24.93 % as calculated previously. Indeed, to have an opening in this precise situation, following both conditions have to be respected :

1. The mine of the 1 is under one of the 2 crosses ; corresponding probability : 2/5 = 0.40 = 40 %.
2. Both squares under points are empty ; corresponding probability: = 62.94 %


On the whole, the probability of opening by clicking on the ? is 0.4×0.6294 = in 0.25.17 = 25.17%. You notice that the presence of the 1 vary this probability of opening from 24.93 % to 25.17 %. The variation would have been sharply more significant (and in the other way) if this 1 had been a 2. If that had been a 3 or a 4, the ? would have been surrounded with at least one mine and the probability of opening would then have been 0%. There would have been even an important probability to find one of these mines under the ? and also to lose the game. If the 1 had been a 5, the game was necessarily lost by clicking on the ?.

You also understand that the probability given at the beginning of this article concerning the 2nd clicks are valid only if this 2nd click is not made next to the 1st. Should the opposite occur, the calculations are different and often more complex.


Comments


• It is funny to notice that the probability of opening are slightly bigger for the intermediate than for the beginner, in spite of the fact that the density of mines is the same for these two levels : 10/64 = 40/256 = 0,15625. It is due to the fact that the first click is never a mine and that the real densities to be taken into account are thus 10/63 = 0.15873 for the beginner and 40/255 = 0.15686 for the intermediary.

• It is very important to understand one thing : it is easier to obtain an opening in corner than in the middle because you "ask" at less squares to be empty. However, openings obtained in the middle are in bigger, given that they are made in 8 directions, vs only 5 for an opening on the edge and 3 for an opening on a corner. In summary, clicking in the middle opens less often but opens bigger ! Everything is thus question of choice : you can click in corner to have frequently holes from the beginning, even if it means having holes difficult to exploit (even impossible, what arrives sometimes also), or you can click in the middle and you prefer to begin numerous games by waiting for a hole presenting numerous zones of easy work. To cut definitively the debate, it could be useful to calculate the average size (in number of squares) of a corner-opening of corner, an edge-opening and a middle-opening. Math student could do that…

• I compared these theoretical results with the results obtained by Paul Kerry (the USA, 2-14-54) in an experimental way. Here are the results :

Place of the first click
Milieu
Bord
Coin
Theoretical percentage of opening by considering a perfectly unpredictable distribution of mines
15.69
31.42
49.93
Percentage of opening noticed by Paul Kerry, on a trial sample of 424 clicks in the middle, 196 in the edge and 135 in corner
14
31
44


The comparison brings us to notice that even if we know that the distribution of mines is not perfectly unpredictable, the results are generally close to the theoretical model. I call to the volunteers to experience on more boards (1000 clicks of every case would be good) to clarify a little these first results.

 

• The last version of Windows XP proposes a railing(bars) of 9*9 squares instead of 8*8 compartments for the beginner level. Here is the influence of this change on the probability of opening in the first two clicks

Probability of opening at the first click
 
DEBUTANT WIN 98
DEBUTANT WIN XP
Middle
Edge
Corner

 

Probability of opening at 2nd click if th e1st one didn't open anything...
 
DEBUTANT WIN 98
DEBUTANT WIN XP
Middle
Edge
Corner


These results on 2nd clicks are valid with the same conditions as previously.

The observed enormous variations are owed to the fact that in this new version, the density of mines was returned from 0.15625 to 0.125 : a fall of 20 %! In these conditions, it is evident that Windows XP favors enormously the records. Moreover a player as Damien Moore (excellent Canadian player: 1-11-46 and n°3 world) managed to make three times 2 sec and once 1 sec on these boards 9*9 in his first 20 trial minutes ! It is to say that this board must in my humble opinion be maintained as unofficial as regards the ratification of records. 8*8 board comes already with too important probabilities of sub 3sec records, that I don't like beginner.

That's it. I hope to have been clear and informative. Do not hesitate to make me known your reactions, remarks and corrections of possible errors on the forum.

Good minesweeping to you all,

Manu
 
Last update: 6 August 2006 Grégoire Duffez 2001-2006 - Contact - Sitemap - Rules