From cube-lovers-errors@mc.lcs.mit.edu  Fri Sep  3 13:16:50 1999
Return-Path: <cube-lovers-errors@mc.lcs.mit.edu>
Received: from sun28.aic.nrl.navy.mil (sun28.aic.nrl.navy.mil [132.250.84.38])
        by mc.lcs.mit.edu (8.9.1a/8.9.1-mod) with SMTP id NAA19489
        for <cube-lovers-outbound@mc.lcs.mit.edu>; Fri, 3 Sep 1999 13:16:49 -0400 (EDT)
Precedence: bulk
Errors-To: cube-lovers-errors@mc.lcs.mit.edu
Message-Id: <002d01bee8fc$928efe80$74c4b0c2@home>
From: roger.broadie@iclweb.com (Roger Broadie)
To: "Cube-Lovers" <Cube-Lovers@ai.mit.edu>
Subject: Re: Rubik's Cube Perpetual Calendar
Date: Tue, 17 Aug 1999 23:05:08 +0100

A calendar cube has just sold on eBay.  Quite possibly it comes from
the same source as the cube Chris Pelley mentioned.  The description
includes a picture, from which I see that it is not the same cube as
the one I described on 30 July, which was bought in the UK.  The eBay
version uses a different type-face and uses upper and lower case for
the week-day names, unlike the British version, which uses all
capitals.   It also has logos, whereas the British version has blanks
on all faces that are not used for date markings. The British version
was nonetheless a genuine Ideal cube and came in a cardboard drum
bearing the title Rubik's Calendar and the copyright notice
"[c-in-a-circle] MMLXXXI Ideal Toy Co Ltd, Wokingham, Berks".  There
are other European calendar cubes pictured on eBay which are like the
British one

There is something very strange about the cube in the eBay photo.  In
fact, I am convinced that is does not work for all dates, which drives
me to the conclusion that the stickers have been rearranged.  I'll
explain my reasons and see if others disagree.

The photo shows only one view, so we have only three faces to work on.
They look like this:

-------------------
|     |     |     |
|  2  |  M  |  5  |
|  V  |  >  |  <  |
-------------------
|     |     |     |
|  J  |  P  |  1  |
|     |  V  |  <  |
-------------------
|     |     |     |
|  Mon|  P  |  6  |
|  <  |  >  |     |
-------------------

-------------------   -------------------
|     |     |     |   |     |     |     |
|Satur|day  |     |   |  7  |  A  |  0  |
|     |     |     |   |  V  |  >  |  <  |
-------------------   -------------------
|     |     |     |   |     |     |     |
|  F  |  E  |  B  |   |     |  C  |  3  |
|     |     |     |   |     |  V  |  <  |
-------------------   -------------------
|Rbk's|     |     |   |     |     |     |
|Cube |     |  8  |   |Thurs|Ideal|  9  |
|     |     |     |   |     |     |     |
-------------------   -------------------
The orientation is shown by the Vs, which point to the local upright
for those markings that are not upright as shown - I've followed Dan
in this.

Once the markings on the various pieces of a calendar cube have been
fixed, the way the cube is assembled does not matter, since to show
the date we need solve only one face, and that can be done from any
starting position, any necessary counterbalancing twists or
permutations taking place in the other layers.  Let's assume the face
used to show the date is the front face.

Therefore we only need to worry about what markings each piece will
bear.  The straightforward approach in designing a cube of this sort,
it seems to me, is as far as possible to keep all the markings of the
same type - that is, destined for the same position on the front
face - together on the pieces carrying them.  So the week-day names
would be on one set of corner pieces, destined for the top left of the
front face, and the numbers forming the units digit of the day of the
month would be on another set, destined for the bottom right of the
front face.  In that way, clashes in which the piece would be needed
in two places at once are avoided.  If this approach is not followed,
then either there must be no clash, or markings must be duplicated.
An example of the first would be an edge piece that combined the F or
B of FEB with the 3 of the tens digit of the day of the month, a
combination that is possible because there is no FEB 30.  The picture
on the container of the British calendar cube, though not the cube
itself, illustrates the other possibility, since one edge-piece
combines J and 2.  That means that JAN 20 and similar dates cannot be
shown unless either the J or the 2 is duplicated.  As it happens,
there is one spare edge-piece face on that cube, so one duplication
could be managed, but no more, but it is impossible to see from the
picture if there is any duplication.  Probably not, because there are
some other impossibilities and inconsistencies in the pictures which
suggest they show non-functioning mock-ups.

The different markings that need to be accommodated on the edge
pieces, defined by their position on the front face, are:

Top:  DAY

Left: the eight initial letters of the month, J F M A S O N D

Right: the ten letters completing the abbreviation for the month, N B
R Y L G P T V C

Bottom: the four numbers for the tens digit of the day of the month, 0
(or blank) 1 2 3.

The edge-piece at the top will always stay there, since it is needed
to show DAY (assuming no duplicates).  So the other face of this piece
will never show on the front face and cannot carry a useful marking.
It can be blank, or carry a logo.

In total, including the face backing DAY, we now have 24 faces, and
that is exactly the number we have available if we have one face for
each of the markings above.  As it happens, the one letter that occurs
at both the start and the end of a month, N, is symmetrical in the
sans-serif typeface used, and in the British cube is made to double as
a starting and an ending letter, since it can be either way up. That
frees up one face to permit one duplication.  It is impossible to see
if the same approach is followed in the eBay cube.

As a matter of interest, the British cube mostly (but not in all
cases) puts an initial and a final letter together on the same
edge-piece.  That is possible without too much juggling to avoid the
clash of having the start and end of a month on the same edge piece,
but is not necessary - the principle of segregating the different
types of marking would lead to four edge-pieces with the initial
letters and five with final letters.

If we turn back to the eBay cube, we find that the visible edge-pieces
are as follows.

Day/P, J?, M?, 1A, F?, 3?, B/blank, blank/? logo/?, 3x??

Since Day is combined with P, we need a duplicate of one or other if
dates in SEP or APR are to be showable.  But even then we would not be
able to show APR 10 unless a duplicate of 1 or A was included.  Thus
even if one N is used for both the start of NOV and the end of JAN or
JUN, we would already have one face too many.  Yet on top of that
there are two blanks and a logo.  One blank is usable as the blank for
the tens digit in dates like JAN 1, and one among the month letters is
there is only one N.  But that still means that there is an extra
blank even if there are no duplicates.

If extra blank faces are included, then other needed markings must be
omitted and dates involving those markings could not be shown.  In
fact the only reasonable explanation I can see is that some stickers
have been removed and put back wrongly.

Roger Broadie

From cube-lovers-errors@mc.lcs.mit.edu  Fri Sep  3 19:04:22 1999