Thinking outside the box


1) 3×3*
Try to draw 1 line consisting of 4 straight segments through all 9 dots. It is not allowed to lift your pen from the paper, to backtrack over a line, or to go through a dot twice.
square with 3x3 dots

2) 4×4**
With the conditions from the previous puzzle, how many straight line segments do you need to connect all 16 dots in the figure below?
square with 4x4 dots

3) 5×5**
With the conditions from the first puzzle, how many straight line segments do you need to connect all 25 dots in the figure below?
square with 5x5 dots

4) 3×3**
The first problem is often used as an example of the need to think “out of the box”. “Out of the box” does not mean “as shipped”, but rather “outside the box”, a term which is also used and more accurate. Then intention is that to solve the problem, the problem solver has to shake off his unconscious borders and step over these borders in order to solve the problem.

To practice some real “outside the box” thinking, try to solve the first problem with 1 (yes, one) straight line, every dot crossed exactly once, no backtracking, etc. There are at least 4 solutions.

5) 3×4**
There is no reason to limit ourselves to squares. Can you draw 5 straight lines through these dost, connecting them all as in problem 1, and end where you started?
3x4 dots

6) nxn dots****
There seems to be a clear pattern
3×3 dots: 4 straight lines
4×4 dots: 6 straight lines
5×5 dots: 8 straight lines
This suggests an infinite series:
nxn dots: 2*(n-1) straight lines.
As far as I know, this problem is unsolved. It is trivial that there is an upper limit of 2n-1 lines.

You can check your solutions:
solution 3×3 dots
solution 3×4 dots
solution 4×4 dots
solution 5×5 dots
solution 5 ways with 1 line

Did you know…
Exercising your brain may help reverse the effects of brain damage?

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3 thoughts on “Thinking outside the box

  1. This is the proof of your “clear pattern”:

    The best,
    Marco

    1. Thank you :)
      Here you can read the 3D generalization and some considerations about the nxnx…xn puzzle.

      M

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