Patterns in Rolling 3 Dice

Date: 09/16/97 at 15:22:42
From: Jeremy Schaper
Subject: Probability-Outcomes/Triangular numbers

Dr. Math,

I have listed all the outcomes of rolling 3 dice. I have come out 
with 216 outcomes. I found the simple pattern: 1 die 6 outcomes or 
6^1, 2 dice 36 outcomes or 6^2, 3 dice 216 outcomes 6^3, 4 dice 1296 
outcomes or 6^4, and so on.

While I was listing all the outcomes for rolling 3 dice, I began to 
see a pattern: 

  number     number of outcomes
    3               1
    4               3
    5               6
    6              10
    7              15
    8              21
    9              25
   10              27
   11              27
   12              25
   13              21
   14              15
   15              10
   16               6
   17               3
   18               1

The pattern for number of outcomes looks like the triangular numbers, 
that is until the number 9, when the number of outcomes is 25 and the 
number 10 has 27 outcomes.

I fully expected the number of outcomes to ascend until the middle 
values and descend after.

Why does the pattern change after number 8 and go back to the pattern 
after number 13?

Thanks,

Jeremy Schaper

Date: 09/29/97 at 17:29:28
From: Doctor Pipe
Subject: Re: Probability-Outcomes/Triangular numbers

Jeremy,

You are very observant to have noticed the connection to triangular 
numbers - congratulations!  There is a reason for this connection. 

Consider the problem of determining how many ways the positive integer 
N can be written as a sum of three positive integers.  The answer is a 
triangular number.

In your dice problem, "number of outcomes" is a sequence of triangular 
numbers for dice totals of 3 through 8 because for these numbers, no 
positive integer greater than six can be summed with two other 
positive integers and result in a sum less than or equal to 8. In 
other words, for sums up to 8, the dice problem is the same as the 
problem in the previous paragraph of finding three positive numbers 
which add up to N with no condition on how high the three numbers can 
be. So the dice problem follows the general case, up to this point.

For dice totals greater than 8, the "number of outcomes" is not a 
triangular number because we are not including the ways the number, 
say 9, can be written as the sum of three positive integers where one 
of the integers is greater than 6 - where you expected "number of 
outcomes" to be 28 it is 25 because you can't roll 1,1,7 or 1,7,1 or 
7,1,1 with your three dice. From the dice totals 9 and up the dice 
problem does not follow the general case.

So you see, your dice problem (How many ways can the numbers 3, 4, 5,  
, 18 be written as the sum of three numbers from {1, 2, 3, 4, 5, 6}?) 
is a modified form of the general case (How many ways can a positive 
integer N be written as a sum of three positive integers?)

Now, why is the "number of outcomes" sequence, when descending for the 
numbers 13 through 18, a sequence of descending triangular numbers?  
This is a result of the symmetry that exists between the dice totals 
3 and 18, 4 and 17, 5 and 16, etc. There is only one way to throw a 3 
with three dice, just as there is only one way to throw 18 with three 
dice (1+1+1; 6+6+6); there are three ways to throw a 4 with three dice 
just as there are three ways to throw 17 with three dice; etc. To make 
this observation precise, note that if the three dice read x, y, and z 
and add up to N, then the three numbers 7-x, 7-y, and 7-z give a 
solution to finding three dice rolls that add up to 21-(x+y+z)=21-N, 
and every such solution arises in this manner. (That's why the whole 
list of numbers you wrote form a symmetric sequence around 21/2=10.5)  
Since the "number of outcomes" for 3 through 8 is a sequence of 
triangular numbers, so is the "number of outcomes" for 13 through 18.

-Doctor Pipe,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 09/30/97 at 11:08:49
From: Jeremy Schaper
Subject: Re: Probability-Outcomes/Triangular numbers

Doctor Pipe,

Thanks for your help. My students were anticipating your answer. The 
change in the pattern seems obvious to me now.

I fully understood the symmetry that exists between the dice totals 3 
and 18, 4 and 17, 5 and 16, etc.  My question was solely on the 
pattern change, which you answered exceptionally.

Thanks again,

Jeremy Schaper
Math/Computer Teacher
Timber Ridge Magnet School
Skokie, IL