Introduction
I was looking at the Mac Pro memory chart, and I asked myself,
how can we be sure it is complete? So I set out to calculate the
total number of possible configurations and I compared that
against the number shown in the chart. It turns out that the chart
is not complete.
Here is the chart:
http://homepage.mac.com/tribe3/.pictures/MemConfig.jpg
Preliminaries
In the chart, each choice of memory modules (e.g., 2x2GB + 1x1GB)
is laid out in only one way: one memory choice -> one configuration.
Each configuration consists of a number of filled memory slot pairs,
and each pair is filled with chips of the same capacity. So we only
need consider slot pairs.
Counting configurations
For n filled slot pairs we can choose to install 2*x*512MB, 2*y*1GB
and 2*z*2GB FB-DIMM memory modules, such that the following hold:
The number of configurations are thus:
There are 4 fillable slot pairs in total, so the total number of possible
configurations is:
And the memory chart does indeed show 34 configurations. However,
1 of these is invalid: the 15GB configuration showing all 8 slots unfilled.
So one configuration is missing. Which one?
That I leave as an exercise for the reader
I was looking at the Mac Pro memory chart, and I asked myself,
how can we be sure it is complete? So I set out to calculate the
total number of possible configurations and I compared that
against the number shown in the chart. It turns out that the chart
is not complete.
Here is the chart:
http://homepage.mac.com/tribe3/.pictures/MemConfig.jpg
Preliminaries
In the chart, each choice of memory modules (e.g., 2x2GB + 1x1GB)
is laid out in only one way: one memory choice -> one configuration.
Each configuration consists of a number of filled memory slot pairs,
and each pair is filled with chips of the same capacity. So we only
need consider slot pairs.
Counting configurations
For n filled slot pairs we can choose to install 2*x*512MB, 2*y*1GB
and 2*z*2GB FB-DIMM memory modules, such that the following hold:
Code:
0 <= x <= n
0 <= y <= n - x
z = n - (x + y)The number of configurations are thus:
Code:
1 filled slot pair: 1 + 2
2 filled slot pairs: 1 + 2 + 3
n filled slot pairs: (n+1)(n+2)/2There are 4 fillable slot pairs in total, so the total number of possible
configurations is:
Code:
3 + 6 + 10 + 15 = 34And the memory chart does indeed show 34 configurations. However,
1 of these is invalid: the 15GB configuration showing all 8 slots unfilled.
So one configuration is missing. Which one?
That I leave as an exercise for the reader