It doesn't hold. Pick (0,1), (0,2), (0,3), (0,4), (0,5) on the plane z = 0
Well, you have five integer points on the same line.
Or is my assumption "more" means an integer point not in the set of five selected wrong?
Math Puzzle: Integer Points
41–49 of 49 posts
Re: Math Puzzle: Integer Points
#42Okay, we have a solution for the plane. The plane is in
two dimensions. Now, for any positive integer n, generalize the problem to n dimensions.
So, now the problem is, for any positive integer n, given any 2^n + 1 n-tuples of integers, at least two of these n-tuples if added have all even components.
Re: Math Puzzle: Integer Points
#43LOL. For any two integer points P(i, j), Q(l, m). We have the equation of a line:
y - j = ((m - j) / (l - i)) * (x - i)
Choose x = k * (l - i) + i y = k*(m - j) + j Since i, j, l, m, k are all integers we obtain another integer point. No need 5 points... I may misunderstand the question. :(
Re: Math Puzzle: Integer Points
#44Earlier quoted context omitted.
Well, you have five integer points on the same line.
"Suppose we arbitrarily choose 5 integer points in a plane. Show that we can always find 2 among these 5 integer points such that the line segment joining the 2 points contains at least 1 more integer point." I am not sure what you mean. I have 'arbitrarily' picked 5 integer points on a plane. Yes, they happen to lie on line. All their line segments contain only these integer points. Perhaps you mean that is line not…
In the case of {(0,0),(0,2)}, (0,1) counts as "1 more integer point" - the point has to be not one of the two points generating the line, not a point outside the set of 5 points on the plane.
Re: Math Puzzle: Integer Points
#45Earlier quoted context omitted.
Well, you have five integer points on the same line.
Or is my assumption "more" means an integer point not in the set of five selected wrong?
Yes, it's wrong. It means at least one integer point more than the two integer end points (which of course have to be on the line).
Re: Math Puzzle: Integer Points
#46I don't see any obvious problem with this counterexample:
(0,5), (3,0), (4,6), (7,1), (9,4)
http://farm9.staticflickr.com/8114/8661127874_f9269f0ee5_b.j...
Is there something I'm missing?
Re: Math Puzzle: Integer Points
#47I don't see any obvious problem with this counterexample: (0,5), (3,0), (4,6), (7,1), (9,4) http://farm9.staticflickr.com/8114/8661127874_f9269f0ee5_b.j... Is there something I'm missing?
(3,0) ... (6,2) ... (9,4)
So comprehensive testing is what you are missing :-)
Re: Math Puzzle: Integer Points
#48Technically, the problem doesn't specify that the points have to be distinct...
Yeah that bothered me too. If all the points are (0, 0), does that satisfy the conditions? Does the line segment (0, 0) to (0, 0) "contain" an integer point?
Re: Math Puzzle: Integer Points
#49This seems rather trivial. Take two arbitrary points, named A and B, where Ax Calculate the delta ∂ between the points, such that ∂x = Bx - Ax and ∂y = By - Ay Define a third point C such that Cx = Bx + ∂x and Cy = By + ∂y. C is integral, and is colinear with A and B.
C does not lie on the line segment AB.
Oh my, I didn't notice the word "segment" there. That does change the problem.