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
