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{\Large CONCERNING AN INVESTIGATION ON DICE}
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{\Large (SOPRA LE SCOPERTE DEI DADI)}
(\textsc{Galileo Galilei}, \textit{Opere}, Firenze, Barbera,
\textbf{8} (1898), pp.\ 591--594)
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\noindent
The fact that in a dice-game certain numbers are more advantageous than
others has a very obvious reason, i.e.\ that some are more easily and
more frequently made than others, which depends on their being able to
be made up with more variety of numbers.  Thus a 3 and an 18, which are
throws which can only be made in one way with 3 numbers (that is, the
latter with 6.6.6 and the former with 1.1.1, and in no other way), are
more difficult to make than, e.g.\ 6 or 7, which can be made up in
several ways, that is, a 6 with 1.2.3 and with 2.2.2 and with 1.1.4,
and a 7 with 1.1.5, 1.2.4, 1.3.3, and 2.2.3.  Nevertheless, although 9
and 12 can be made up in as many ways as 10 and 11, and therefore they
should be considered as being of equal utility to these, yet it is known
that long observation has made dice-players consider 10 and 11 to be
more advantageous than 9 and 12.  And it is clear that 9 and 10 can be
made up by an equal diversity of numbers (and this is also true of 12
and 11): since 9 is made up of 1.2.6, 1.3.5, 1.4.4, 2.2.5, 2.3.4,
3.3.3, which are six triple numbers, and 10 of 1.3.6, 1.4.5, 2.2.6,
2.3.5, 2.4.4, 3.3.4, and in no other ways, and these also are six
combinations. Now I, to oblige him who has ordered me to produce
whatever occurs to me about such a problem, will expound my ideas, in
the hope not only of solving this problem but of opening the way to a
precise understanding of the reasons for which all the details of the
game have been with great care and judgment arranged and adjusted.
And to achieve my end with the greatest clarity of which I am capable, I
will begin by considering how, since a die has six faces, and when
thrown it can equally well fall on any one of these, only 6 throws can
be made with it, each different from all the others.  But if together
with the first die we throw a second, which also has six faces, we can
make 36 throws each different from all the others, since each face of
the first die can be combined with each of the second, and in
consequence can make 6 different throws, whence it is clear that such
combinations are 6 times 6, i.e.\ 36.  And if we add a third die, since
each one of its six faces can be combined with each one of the 36
combinations of the other two dice, we shall find that the combinations
of three dice are 6 times 36, i.e.\ 216, each different from the others.
But because the numbers in the combinations in three-dice throws are
only 16, that is, 3.4.5, etc.\ up to,18, among which one must divide the
said 216 throws, it is necessary that to some of these numbers many
throws must belong; and if we can find how many belong to each, we shall
have prepared the way to find out what we want to know, and it will be
enough to make such an investigation from 3 to 10, because what pertains
to one of these numbers, will also pertain to that which is the one
immediately greater.
Three special points must be noted for a clear understanding of what
follows.  The first is that that sum of the points of 3 dice, which is
composed of 3 equal numbers, can only be produced by one single throw of
the dice: and thus a 3 can only be produced by the three ace-faces, and
a 6, if it is to be made up of 3 twos, can only be made by a single
throw.  Secondly: the sum which is made up of 3 numbers, of which two
are the same and the third different, can be produced by three throws:
as e.g., a 4 which is made up of a 2 and of two aces, can be produced by
three different throws; that is, when the first die shows 2 and the
second and third show the ace, or the second die a 2 and the first and
third the ace; or the third a 2 and the first and second the ace.  And
so e.g., an 8, when it is made up of 3.3.2, can be produced also in
three ways: i.e.\ when the first die shows 2 and the others 3 each, or
when the second die shows 2 and the first and third 3, or finally when
the third shows 2 and the first and second 3.  Thirdly the sum of points
which is made up of three different numbers. can be produced in six
ways.  As for example, an 8 which is made up of 1.3.4. can be made with
six different throws: first, when the first die shows 1, the second 3
and the third 4; second, when the first die still shows 1, but the
second 4 and the third 3; third, when the second die shows 1, and the
first 3 and the third 4; fourth, when the second still shows 1, and the
first 4 and the third 3; fifth, when the third die shows 1, the first
3, and the second 4; sixth, when the third shows 1, the first 4 and the
second 3.  Therefore, we have so far declared these three fundamental
points; first, that the triples, that is the sum of three-dice throws,
which are made up of three equal numbers, can only be produced in one way;
second, that the triples which are made up of two equal numbers and the
third different, are produced in three ways; third, that those triples
which are made up of three different numbers are produced in six ways.
From these fundamental points we can easily deduce in how many ways, or
rather in how many different throws, all the numbers of the three dice may
be formed, which will easily be understood from the following table
{\footnotesize
\begin{tabular}{r||r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|}
1& \\
6& 10& &  9& & 8 &   &   7 &   &   6 &   &   5 &   &   4 &   &   3 &   \\
\hline
10&631&6&621&6&611& 3 & 511 & 3 & 411 & 3 & 311 & 3 & 211 & 3 & 111 & 1 \\
15&622&3&531&6&521& 6 & 421 & 6 & 321 & 6 & 221 & 3 &     &   &     &   \\
21&541&6&522&3&431& 6 & 331 & 3 & 222 & 1 &     &   &     &   &     &   \\
25&532&6&441&3&422& 3 & 322 & 3 &     &   &     &   &     &   &     &   \\
27&442&3&432&6&332& 3 &     &   &     &   &     &   &     &   &     &   \\
\cline{1-1}
108&433&3&333&1&   &   &     &   &     &   &     &   &     &   &     &   \\
\cline{2-17}
108 && 27 && 25 && 21 && 15 && 10 && 6 && 3 && 1 \\
216 &&    &&    &&    &&    &&    &&   &&   &&   \\
\end{tabular}
on top of which are noted the points of the throws from 10 down to 3,
and beneath these the different triples from which each of these can
result; next to which are placed the number of ways in which each triple
can be produced, and under these is finally shown the sum of all the
possible ways of producing these throws.  So, e.g., in the first column
we have the sum of points 10, and beneath it 6 triples of numbers with
which it can be made up, which are 6.3.1, 6.2.2, 5.4.1, 5.3.2, 4.4.2,
4.3.3. And since the first triple 6.3.1 is made up of three different
numbers, it can (as is declared above) be made by six different
dice-throws, therefore next to this triple 6.3.1 a 6 is noted: and since
the second triple 6.2.2 is made up of two equal numbers and a third
which is different, it can only be produced by 3 different throws, and
therefore a 3 is noted next to it: the third triple 5.4.1, being made up
of three different numbers, can be produced by 6 throws, therefore a 6
is noted next to it: and so on with all the other triples.  And finally
at the bottom of the little column of numbers of throws these are all
added up: there one can see that the sum of points 10 can be made up by
27 different dice-throws but the sum of points 9 by 25 only, the 8 by
21, the 7 by 15, the 6 by 10, the 5 by 6, the 4 by 3, and finally the 3
by 1: which all added together amount to 108. And there being a similar
number of throws for the higher sums of points, that is, for the points
11, 12, 13, 14, 15, 16, 17, 18, one arrives at the sum of all the
possible throws which can be made with the faces of the three dice,
which is 216.  And from this table anyone who understands the game can
very accurately measure all the advantages, however small they may be,
of the \textit{zare},\footnote{\textit{Zara} was a game played with 3
dice, the rules of which were possibly those of the game known to us as
Hazard.  \textit{Incontri} is used here probably as a technical term in
\textit{Zara}.} the \textit{incontri},$^1$ and of any other special rule
and term observed in this game.
\bigskip\bigskip
\textit{Translated by E. H. Thorne and published in} Games, Gods and
Gambling \textit{by F~N~David.}