Conference rusure::math

One possible question is:

	Who believes the answer is 14 ?

Anyway, on a more serious note, could someone post some of the interesting
problems in this notes file so we don't have to buy the magazine ?
(probably a copyright infringement, but not a new precedent by any means)

This puzzle, that I also think is 14, goes roughly as follows ( I don't have 
the magazine here but I'll bring it in this week.  Others can please help with 
their own copies. I am a slow typist):
 
There is a 4x4 chess board.  What's the longest knight's tour from the 
northwest corner to the northeast corner?  On a knight's tour, only legal 
knight's moves are allowed, and no square is occupied more than once.  The 
final count should include the beginning and ending squares.  It does seem 
like two squares need to be left unoccupied, but I didn't prove it.

This issue of OMNI advertises the "World's Hardest IQ Test" on the cover.  
That test is where these puzzles come from. 


Another puzzle from OMNI: How many non-overlapping regions of space can three 
intersecting identically-sized cubes make?

Yet another from OMNI:  Into how many pieces can 5 planar slices cut a cube? 
(assuming no rearranging).

A question from me:  These last two puzzles seem to have the same answer. Are 
the two puzzles homologous in some way?  (or maybe they just don't have the 
same answer.)

Maybe if people type in other questions from OMNI they ought to make them 
separate notes, and this note can reference those other discussions.  OTW, 
this note might get too large to be easily usable.

-Jim Grant, Littleton MA.

> How many non-overlapping regions of space can three 
> intersecting identically-sized cubes make?

	Eight.  For four cubes, the answer is 16.  For five cubes...??

> Into how many pieces can 5 planar slices cut a cube? 
> (assuming no rearranging).

	Sixteen.  For n slices, the answer is:  1 + n(n+1)/2.

> Into how many pieces can 5 planar slices cut a cube?
> (assuming no rearranging).

>	Sixteen.  For n slices, the answer is:  1 + n(n+1)/2.

I believe the correct answer is actually twenty six.
For n slices this should be (n^3+5n+6)/6 
 (Steiner's division of space with planes).

The previous reply applies to dividing a square with lines...

What's the expression for n-space divided by hyperplanes?

I saw the issue on the newsstand... was the IQ test therein serious?

- Jim

>How many non-overlapping regions of space can three intersecting 
>identically-sized cubes make?

   I believe the answer is 26:
  
Let the three cubes by co-centric, so that all of the 24 corners (3 x 8) are 
tangent to a sphere.  Arrange the cubes so that none of the 24 corners are 
co-incident.  Then, each corner defines a region,  the center is another 
region, and "the rest of space" is another.  The answer "8" in .3 is for 
mutually "parallel" cubes.

On the "seriousness" of the OMNI IQ test:  The authors of the test, and three 
of the "high-IQ" societies take it seriously, but to me it seems very biased 
towards those with mathematical training.  On the other hand, that means that 
a lot of the readers of this file would probably get real good scores, so its 
fun for the ego.  NOTE: the rules specifically prohibit consulting with 
others, so if you want to take the test seriously, you should read no more 
notes about the OMNI puzzles.  I'm going to put some of the other puzzles as 
notes, marked as OMNI puzzles, so you can skip them if you want.