Given a cube, how many cubes (all identical to the original) can you place around it so they all touch it?

This problem was invented by Martin Gardner, who wrote the column
"Mathematical Games" in Scientific American for 20-odd years.

When I began receiving Scientific American by subscription, the very
first issue I got was the July 1971 issue.  Gardner's column in this
issue (pages 106-109) gives a list of 36 "quickie" problems, including
some joke problems.  Number 33 on the list is the touching cubes problem.

In his August column (pages 102-105), he gives the answers, and the answer
given is 20 cubes.  But wait -- there's more!

In his November column (pages 114-121), he reports on many letters from
readers relating to the 36 quickie problems.  For almost half of them,
readers produced quibbles or improved answers.  The cube touching problem
was one of these.  The November column illustrates *two different*
solutions with 22 cubes, and mentions that "while I was recovering" from
the arrival of these, some of the people who had produced them came up
with even better ones -- one with 23 and one with 24 cubes!

In his December column (pages 96-99), he explains the 24-cube solution,
which was invented by physicist Robert S. Holmes.  It requires explanation
because some of the overlaps are too small to clearly diagram.  He also
conjectures that this is the best possible, and that his original 20-cube
solution is best if (as the original problem did *not* require) the whole
surface of the original cube must be covered by the touching cubes.
None of the 22-to-24-cube solutions cover all of the surface.  Rather
aptly, the cover of the issue features an optical illusion with cubes.

I presume that all this material has appeared in one of Gardner's books
collecting his columns, but I couldn't say which.

Incidentally, the July column also mentions, in connection with a question
asked in an earlier column, the relationship between the fictional computer
name Hal and the corporate acronym IBM.  Gardner says that he assumed it
was intentional until Clarke assured him otherwise.

-- Mark Brader <msb@sq.com>

[ Editor: Brader's remarks on problem 25 of the same column are under
geometry/manhole.cover. ]

[ Editor: further material on this puzzle, edited separately from other collected postings (batch 20); it may repeat some of the above. ]

[ Editor: problem as stated there: ]
What is the largest number of identical cubes that can touch a given
cube of the same size?

[ Editor: problem statement reconstructed from the postings (the
  thread "Quickie #6"); the original wording was not preserved. ]


Spoilers for a guess follow:

90tms@eng.cam.ac.uk (T.M. Speight) writes:
>
>                       ______
>                      |      |
>                ______|      |______
>               |      |      |      |
>               |      |______|      |
>               |      |      |      |
>               |______|  \/  |______|
>               |      |  /\  |      |
>               |      |______|      |
>               |      |      |      |
>               |______|      |______|
>                      |      |
>                      |______|

Take three instances of this configuration of seven white cubes.
Replace one white center cube with the red cube.
Rotate that instance 45 degrees about the X.
Overlay that on a white centered instance of the configuration.  (X over
X.)
Overlay the third white centered instance on the red one.  (X over the
X's.)

So far, that's 7+7+6 = 20 white blocks touching the red one...

-- David Blume <blume@twg.com>

There is a long discussion in Gardner's book about this "quickie."  Let
me quote a passage:

     The problem of touching cubes was one I thought of several years
     ago and had answered with 20 cubes.  I was staggered to receive
     two different solutions, each with 22 cubes.

He then has a [non-ASCII-reproducible] diagram of a two 22-cube
solution.  He has a little more discussion, then he writes:

     While I was still recovering from the 22-cube solution, [Robert
     S.] Holmes (who is working for his doctorate in particle physics
     at the University of Rochester) delivered the knockout punch:
     a 24-cube solution!

This time, there are THREE diagrams to show how the construction is
completed.  In involves some minuscule overlaps in some places, and is
really very tricky.  (In other words, I don't believe the person who
said his "quick" answer was 24 really had the solution.  If he did,
I suggest that he send it to Gardner.)

Anyway, sorry that I can't provide the solution, but you know where to
look it up if you are interested.  I would congratulate those people
who got 20, which means you are at least as good as Gardner! 8-)

-- Timothy Vaughan <tvaughan@buphyk.bu.edu>

In article <1993Aug18.162234.2965@ericsson.se>, etljhan@etlxd30.ericsson.se
(john allen xd/d) writes:
|> If anybody can manage to get 8 cubes on the table arranged so they
|> all touch the top one, I should like to know about it...

One can, in fact, get 8 nonintersecting unit squares to overlap one
unit square.  This is the basis of one of the 22-cube solutions shown
in Gardner's book, and it is attributed to Rudolf Bergan, Rudolph
Krutar, and Robert Holmes.  I continue to be frustated by the fact that
these diagrams are impossible to draw in ASCII, so I just encourage
people to look at Gardner's book _Wheels, Life, and Other Mathematical
Amusements_.

-- Timothy Vaughan <tvaughan@buphyk.bu.edu>

Ok, here's a description of one I found after reading the above hint.

Consider this arrangement:     ABC
                                0D
                                 E
where 0 is the original square.  C just overlaps the UR corner.  AB are
glued together and are together tilted so that A can barely overlap the
UL corner.  Similarly for DE.  Put a piece directly on top of 0, but
rotated 45 degrees.  Now two pieces can touch the LL corner.  8 total.

I think it's possible to move A slightly downward, tilt B so that it
touches a different face of 0, and fit an extra cube in the middle
layer.  If this can be done without intruding into the bottom layer
then the same trick can be done again on the other side for a 24 cube
solution.

-- David Wayne Ring <Cdude@phys.tamu.edu>