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Can you arrange 7 trees so that there are 6 rows of 3 trees? It is entirely possible.

Note: A tree can be part of more then one row, for example a if you arranged a 3x3 square (9 trees) the tree in the center is part of 4 rows.

Rows are usually horizontal but in this case they can be horizontal, vertical, or diagonal.

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4 Answers 4

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7 trees - 6 lines

(Joel Rondeau's solution would look like this:

    1
   2 3
    4
 5  6  7

With lines: 125, 137, 146, 247, 345, 567. A more symmetric solution than mine)

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  • $\begingroup$ That's slightly off... $\endgroup$
    – warspyking
    Commented Oct 26, 2014 at 21:41
  • $\begingroup$ How? Please show how this solution is wrong? Could it be there are more solutions that you are not aware of? $\endgroup$
    – Ole Tange
    Commented Oct 26, 2014 at 21:55
  • $\begingroup$ Now it looks good. Your solution with numbers didn't line up properly. $\endgroup$
    – warspyking
    Commented Oct 26, 2014 at 22:50
  • $\begingroup$ @OleTange it was possibly a bit confusing because in the diagram you've drawn your cross section has been lined up correctly. $\endgroup$
    – Nick Coad
    Commented Nov 10, 2014 at 5:50
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An equilateral triangle with 3 trees at the corners, 3 trees at the midpoints and one tree in the center.

Each side has 3 trees. Each altitude has 3 trees.

enter image description here

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  • $\begingroup$ Could you please provide a picture? $\endgroup$
    – warspyking
    Commented Oct 26, 2014 at 21:34
  • $\begingroup$ Picture added, seems a bit too late. $\endgroup$
    – Joel Rondeau
    Commented Oct 27, 2014 at 3:23
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this is my solution...

and this is done by euler line which i studied in high school.

euler line

enter image description here

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  • $\begingroup$ Tangentially related: the Fano plane, the Transylvanian lottery problem, and the board game fire and ice (played on a Fano plane of Fano planes). $\endgroup$
    – user2322
    Commented Oct 27, 2014 at 1:17
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North pole, south pole and 5 along the equator...10 rows :-)

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    $\begingroup$ Hello and welcome to PSE :) Congratulations on your very first answer but it is a bit too unclear. Perhaps giving a picture of your answer will be better. Nevertheless, I encourage you to write and answer puzzles in this site. Don't forget to visit the Help Center (puzzling.stackexchange.com/help) and check out other puzzles. Happy Puzzling :D $\endgroup$
    – Kevin L
    Commented Sep 3, 2018 at 2:33
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    $\begingroup$ That's definitely an interesting answer, as it uses the third dimension (and the original question didn't specify the trees must be on a plane). What does confuse me is that with 5 trees on the equator, there are only 5 lines that join each equatorial tree with the poles (N. Pole -> Equator -> S.Pole), so I'm a bit lost as to how you get 10 lines, unless you've double counted something. One way around this would be to arrange 4 of the 5 equatorial trees such that you can make a straight line from Equator -> N. Pole -> Opposite Equator, and mirror it with the other pole, giving 9 lines in all. $\endgroup$
    – Phylyp
    Commented Sep 3, 2018 at 3:06
  • $\begingroup$ there are 5 trees on the equator..1,2,3,4,5..so 123, 234, 345, 451 and 512 are 5 more lines..similarly 1NP2 is a line, 1NP3 is another, so is 1NP4..also 1SP2, and so on (any longitude is a straight line).. i think this will mean there are a total of 30 such lines $\endgroup$
    – sriram sathyamoorthy
    Commented Sep 26, 2018 at 8:22
  • $\begingroup$ While clever, I'm not sure if spherical geometry is in the spirit of the answer $\endgroup$
    – qwr
    Commented Jun 2, 2022 at 5:14
  • $\begingroup$ I know it's not in the spirit...but I did follow the letter, or rather the line of the puzzle :-) Sriram Sathyamoorthy $\endgroup$
    – sriram sathyamoorthy
    Commented Aug 29, 2022 at 2:58

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