I last week picked up one of the same octagonal cubes (Wonderful Puzzler
brand) as Alan R. Katz in his message. I might point out that it shares the same
workmanship and twistability (lack thereof) which Katz attributes to their brand
of standard cube.

After some experimenting, I found that the "parity error" involved is always a
pair of edges in reverse locations on the bottom. My solution algorithm is top,
equator, bottom [corner locations, edge locations, edge orientations, corner
orientations]. On an ordinary cube, one pair of edges reversed never happens
when you go to position edges. I suspect that if your algorithm does edge
locations before corners, then two corners in wrong locations would be the parity
error. Incidentally, the parity of Katz's cube is the reverse of mine, though the
center cubie colors are in the same relation to each other.

Edge orientation (I assume that is what Katz meant by "edges must be fliped")
parity errors are irrelevant to getting the cut-edge-colors rightly paritied. Edge
orientation parity errors happen because one of the cut-edges may have been
oriented wrongly. Or more precisely, an odd number of them reversed from the
way they were virginally (but they look right either direction). This is easily
cured by any of the "flip a pair of edges" macros.

The change in cube shape is actually helpful in solving, except for being
difficult to grab sometimes. Corners that need orientation really stand out. The
Greenwich meridian edges (assuming you solve it from the top down with the
octagonal faces left and right), being different shape, are instantly located.

The best part of the Wonderful Puzzler is the instructions, which I quote here:

THE CHALLENGE:
"ORIGINAL PUZZLER" presents not only a unique challenge but offers the
possibility of countless hours of relaxation. Your mental ingenuity may be tested
for a few hours -- many days -- several weeks -- or even a period of much
longer duration. If you can determine the key to unlock the knack for solving
the PUZZLER, the final trimph can be the psychological turning point in your
life. Mathematicians may be tested to the limit and cry over this one -- and you
may, too! You will gain a measure of satisfaction when you align one plane.
You will be delighted with the completion of two. You will be elated with the
completion of three or four! The completion of the fifth plane will quicken your
pulse!! -- and you will have scaled the peak once the last unit of the sixth plane
falls into place!!!!

PREPARATION & CAUTION:
*Spin the PUZZLER several times, as indicated on the cover, until all color units
are randomly distributed on each of six planes.
*Do NOT remove any color unit in the process of play.
*Initially, activate the random distribution GENTLY. With little use, this can be
accomplished easily and smoothly.
Patience and persistence will beat the Puzzler! Good Luck!

Challenge
Can you contend with more than 18'000'000'000 combination to reach a solution?
*end of quote*

I think the errors remaining above are all theirs. It is full of little gems: turning
point in your life, solving five planes (on the standard cube, which I believe
has the same instructions), 18 billion (better than Ideal's guess). The GENTLY
caution is valid; a neighbor kid blew mine apart, including a center cubie, by
ignoring this.