It's not at all impossible. Points can be on the line. If
M and N are both odd, the line will go through at least one
red and at least one blue point.
Of course, that still wouldn't address the problem with infinite sets.
In fact, it does work with infinite sets, and it works in
more dimensions as well. This theorem, saying that in d dimensions, given d sets there is a
hyperplane of dimension d - 1 that divides all
sets in two parts of equal size, is also known as the
"ham-cheese sandwich cut", because the theorem implies that
you can always divide a ham-cheese sandwich in two equal
parts (both parts having the same amount of ham, cheese and
bread) with a single cut, even if you leave the cheese in
the fridge.
Abigail | [reply] |
Oh, duh. Points on the line. Right... my mistake.
Regarding infinite sets, I simply meant that defining the problem in terms of floor() was problematic. You avoided that in your reply by using the phrases "two parts of equal size" and "two equal parts".
-sauoq
"My two cents aren't worth a dime.";
| [reply] |
Regarding infinite sets, I simply meant that defining the problem in terms of floor() was problematic. You avoided that in your reply by using the phrases "two parts of equal size" and "two equal parts".
In my original statement of the problem, I was describing,
finite, discrete sets. After all, I was saying one set had
N points, the other M points. I would not
be able to give a number of points if any of the sets was
infinite. In my later reply, I was refering to the continuous
form of the problem, in which we have continuous sets (but
not necessarely connected). Then it's right to talk about
two parts of equal size.
Abigail
| [reply] |