Algebra.Com's Answer #787110 by ikleyn(52803)![\"\"](\"/images/stars/stars-13.gif\")  
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N lattice points in coordinate 8-space are chosen at random. The line segments joining all pairs of these N points are drawn.
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document.write( "What is the smallest value of N that guarantees that the midpoint of at least one of the resulting line segments is itself a lattice point?
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document.write( "Each lattice point x = ( ,  ,  , . . . ,  ) is the set of 8 (eight) integer numbers, - its coordinates, \r\n" );
document.write( "that are positive, zero or negative integer numbers.\r\n" );
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document.write( "The line segments, joining all pairs of these lattice points, are vectors, whose components are the differences of the \r\n" );
document.write( "lattice points' coordinates.\r\n" );
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document.write( "For each such a joining segment/vector, its midpoint is a lattice point if and only if ALL 8 components of this segment/vector \r\n" );
document.write( "are EVEN numbers.\r\n" );
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document.write( "As I just said above, each component of the joint segment/vector is the difference of the coordinates of the original \r\n" );
document.write( "lattice points.\r\n" );
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document.write( "Therefore, for every joining segment/vector, its midpoint is a lattice point if and only if for the respective pair of the lattice points \r\n" );
document.write( "all 8 their coordinates have the same parity as integer numbers, i.e. in each of 8 positions they are either EVEN or ODD integer \r\n" );
document.write( "numbers simultaneously.\r\n" );
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document.write( "Next, it is clear, that in any set of  vectors with integer components in 8-space  , at least two vectors have identical parity \r\n" );
document.write( "of integer components in all 8 positions.\r\n" );
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document.write( " | It follows from the \"pigeonhole principle\". |\r\n" );
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document.write( "It leads us to the conclusion that if  (the number of all joining segments) is greater than = 257,\r\n" );
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document.write( "then there are at least two vectors that have identical parity of integer components in all 8 positions.\r\n" );
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document.write( "Therefore, the minimal number N the problem asks for, is N satisfying inequality\r\n" );
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document.write( " >= 257, or\r\n" );
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document.write( " N*(N-1) >= 2*257 = 514.\r\n" );
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document.write( "And since  = 22.67, the final inequality and the ANSWER is N >= 24.\r\n" );
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document.write( "ANSWER. The smallest value of N that guarantees that the midpoint of at least one of the resulting line segments is itself a lattice point, is 24.\r\n" );
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document.write( "Solved.\r
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document.write( "The post-solution notice:\r
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document.write( " This value of 24 is not in your post answers list, meaning that the post is/was DEFECTIVE.\r
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document.write( "Thanks for submitting this remarkable problem.\r
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