The Most Useful Pentominoes - An Experiment

The first strategy that I worked out for myself when just playing around with a set of pentominoes is to try and get certain shapes used up first, so you don't have to desperately find a home for them near the end of the construction. Pieces like the X pentomino especially I grew to hate with a vengeance. And after a while of solving I sort of developed a feel for which pieces needed to be used up pronto and which ones I didn't mind saving until near the end. This was generally something like:

This was all just based on gut feelings, on how much my blood boiled on picking up or even looking at a certain piece, and there was never anything approaching a sensible, methodical look into which pieces are most or least cooperative. Until now.

Whilst trying to do something else with the program mops (scroll to the bottom of the page for the various solvers) I discovered that if the rectangle size is set to something that doesn't use all the pieces, and you've got 'Determine Frequencies' ticked, it keeps counts of each time a piece is used in a solution.

So I set it to randomly solve 200,000 5x5 rectangles (since that's going to feel fairly similar to the last half of a 5x12 or 6x10 solution) and the frequencies of each pentomino are as follows:


Piece Count
L    149575
P    143160
Y    108150
I    107900
V    106884
C     91992
S     66828
F     55200
W     55017
T     55003
Z     50006
X     10285

Or, in hideous OpenOffice default graph form:

So my intuitions and this match up fairly well, for the most part - the L, P and I pentominoes are all near the top, and the X pentomino is right down there in a class of its own below useless.
Of course, whether these values have any bearing on how useful the piece is is up for debate. It's a rough guideline at the very least, I suppose.

And of course I want to go and do the same thing with hexominoes now, don't I?

Here's the raw values after running 125,000 6x8 boards in mops, using the hexomino naming scheme from here.

Hexomino      Count
A             37420
C             23250
D             33164
E              9351
HIGH F        18540
LOW F         19888
G             26659
H             21624
I             44063
J             37513
K             19738
L             63206
M             22871
LONG N        35834
SHORT N       27537
O             41949
P             61310
Q             30998
R             43327
LONG S        20482
TALL T        17613
SHORT T       27457
U             37957
V             57020
Wa            26479
Wb            14223
Wc            19354
X              6688
ITALIC X       6912
HIGH Y        47517
LOW Y         27467
SHORT Z       20379
TALL Z        16586
HIGH 4        15534
LOW 4         20090

So there's a similar kind of trend here as there was for the pentominoes - L-shaped pieces and 'blocky' pieces with 2x2 blocks in them are most 'useful', spiky/wiggle/cross-shaped bits, not so much.