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Math Puzzle: Integer Points

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Re: Math Puzzle: Integer Points

#4

Post by aeflash »

Scaevolus wrote: post #2

I don't understand. Choose points (0,0) (0,1) (1,0) (1,2) (2,1) (2,2). What line contains three points? Diagram: _OO O_O OO_ e: oh, not a point in the original set, any integer point. thanks!

The line segment described by (0,0) (2,2) goes through the integer point (1,1).

Re: Math Puzzle: Integer Points

#5

Post by anonymoushn »

Scaevolus wrote: post #2

I don't understand. Choose points (0,0) (0,1) (1,0) (1,2) (2,1) (2,2). What line contains three points? Diagram: _OO O_O OO_ e: oh, not a point in the original set, any integer point. thanks!

You chose 6 points, and there are 3 line segments among them containing (1,1).

Re: Math Puzzle: Integer Points

#6

Post by shib71 »

Scaevolus wrote: post #2

I don't understand. Choose points (0,0) (0,1) (1,0) (1,2) (2,1) (2,2). What line contains three points? Diagram: _OO O_O OO_ e: oh, not a point in the original set, any integer point. thanks!

I think the question might be better put this way:

Pick 5 random points on an infinite plane. Draw lines between all of them. One of those lines will contain another integer point. Prove this is true for every set of 5 random points.

Re: Math Puzzle: Integer Points

#9

Post by nilkn »

There 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.
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