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Is it possible to place 4 players on a football field in such a way that the 6 distances between every pair of them are 1, 2, 3, 4, 5, 6 meters?

Source: Moscow Math Olympiad 2001 (Look Inside to Page 8, see Problem 1 under Level B)

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    $\begingroup$ In case anyone is wondering, as I was, just how big a football field is (and which sort of football we are working with here), the answer is "big enough, for all sports called football, that the exact dimensions don't matter". $\endgroup$
    – Gareth McCaughan
    Jan 5 '21 at 13:58
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    $\begingroup$ Do all 6 distances need to be present, that is, each one appearing once? $\endgroup$
    – xnor
    Jan 5 '21 at 14:02
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(I'm understanding the problem to mean that each if the six distances appears exactly once.)

Yes, it's possible

Explanation:

Put the players in a line, at positions 0,1,4,6.

This is the only solution, up to symmetries. For the distance of 1, any triangle containing it must be degenerate because the other side lengths are distinct whole numbers, so the triangle inequality allows equality at best. This forces all points to lie on the same line as the distance-1 segment.
Put the distance-6 pair at 0 and 6 on a number line. The other two points must lie on whole numbers within this interval. The distance 3 segment can't use either endpoint or there would be a second distance of 3. So, it must go from 1 to 4, giving the above solution, or symmetrically from 2 to 5.

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[The following is completely wrong because I misunderstood the question to be asking "can you make all the distances be taken from {1,2,3,4,5,6}?" rather than the apparently intended "can you make the set of distances be {1,2,3,4,5,6}?". The question has since been edited to make its meaning harder to misunderstand. I'm leaving this wrong answer here instead of deleting because I don't believe in hiding my errors.]

The answer is

yes

because of

the existence of the 3-4-5 right-angled triangle.

So

put two players 4m apart, and the other two also 4m apart offset from them perpendicularly by 3m. Done.

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  • $\begingroup$ Could you pleas explain how they are gonna be put on the pitch? $\endgroup$
    – Rahim
    Jan 5 '21 at 14:04
  • $\begingroup$ What about 1 , 2, 6 ? $\endgroup$
    – Rahim
    Jan 5 '21 at 14:10
  • $\begingroup$ Sounds like you're describing i.stack.imgur.com/3Aijp.png , which doesn't seem like a correct solution. $\endgroup$
    – msh210
    Jan 5 '21 at 14:12
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    $\begingroup$ ... Oh, I think I completely misinterpreted the question. $\endgroup$
    – Gareth McCaughan
    Jan 5 '21 at 14:14

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