[deleted]
Math Puzzle: Integer Points
11–20 of 49 posts
Re: Math Puzzle: Integer Points
#12There are four pairs of values modulo two, so if you have five points some two of them must be the same modulo two. Say these points are A and B. Then A + B = (even, even), and so (A + B) / 2 has integer coordinates.
That was tough to follow, let me restate,
Let the two points be (x1, y1) and (x2, y2) such that x1 % 2 == x2 % 2 and y1 % 2 == y2 % 2, i.e., have the same parity.
Then x1 + x2 = 2 * x3 (is even) and y1 + y2 = 2 * y3 (is even), and (x3, y3) is an integer point.
Given five points, two such points must exist because of the pigeon hole principle.
Re: Math Puzzle: Integer Points
#13[deleted]
Re: Math Puzzle: Integer Points
#14Re: Math Puzzle: Integer Points
#15back of the napkin:
take the first 4 points to be a unit box with the lower left point at the origin. You can draw any of 4 lines to the 5th point (x,y). These 4 lines will have slopes x:y, x-1:y, x:y-1, x-1,y-1. You can always arrive at a slope ratio that is evenly divisible through some combination of -1 and reducing the ratio to simplest form. An evenly divisible slope ratio will pass through another point.
example: 5th point (11,4). Choose line with slope 10:4. Divisible by 2, this line passes through the point (5,2). A ratio divisible by 3 would pass through 2 additional points etc.
Re: Math Puzzle: Integer Points
#16back of the napkin: take the first 4 points to be a unit box with the lower left point at the origin. You can draw any of 4 lines to the 5th point (x,y). These 4 lines will have slopes x:y, x-1:y, x:y-1, x-1,y-1. You can always arrive at a slope ratio that is evenly divisible through some combination of -1 and reducing the ratio to simplest form. An evenly divisible slope ratio will pass through another point. exampl…
A real proof would work for an arbitrary first four points (with any segment between two not containing a lattice point), but I think nikln already provided this :)
Re: Math Puzzle: Integer Points
#17back of the napkin: take the first 4 points to be a unit box with the lower left point at the origin. You can draw any of 4 lines to the 5th point (x,y). These 4 lines will have slopes x:y, x-1:y, x:y-1, x-1,y-1. You can always arrive at a slope ratio that is evenly divisible through some combination of -1 and reducing the ratio to simplest form. An evenly divisible slope ratio will pass through another point. exampl…
Why would 4 of the points form a square?
Re: Math Puzzle: Integer Points
#18There are four pairs of values modulo two, so if you have five points some two of them must be the same modulo two. Say these points are A and B. Then A + B = (even, even), and so (A + B) / 2 has integer coordinates.
So that means that there will be at least on line which the center is an integer point.
Re: Math Puzzle: Integer Points
#19There are four pairs of values modulo two, so if you have five points some two of them must be the same modulo two. Say these points are A and B. Then A + B = (even, even), and so (A + B) / 2 has integer coordinates.
So that means that there will be at least on line which the center is an integer point.
There are four possible coordinate parities
(even, even), (even, odd), (odd, even), (odd, odd)
Between two points of same parity the difference has the parity (even, even).
As (even, even) is divisible by two there's an integer midpoint.
Re: Math Puzzle: Integer Points
#20There are four pairs of values modulo two, so if you have five points some two of them must be the same modulo two. Say these points are A and B. Then A + B = (even, even), and so (A + B) / 2 has integer coordinates.
So that means that there will be at least on line which the center is an integer point.
true, you can always make one of the 4 lines have a slope that is even/even. the point composed of x/2,y/2 will lie on this line.