a) In each of these two grids of dots, 5 x 7 and 7 x 9, connect all of them so as to form polygons of 35 and 63 sides respectively (two consecutive segments can therefore not be collinear as they would then merge two sides of the polygon).

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b) Is such a connection possible in any grid of n x (n+2) dots for all n > 4?

  • 1
    $\begingroup$ The origin of this puzzle is the article in Mathematics Magazine (April, 2021) by Sam Chow, and Ayla and Pual Gafni: Connecting the Dots: Maximal Polygons on a Square Grid. There the matter is settled for square grids. $\endgroup$ Commented Apr 20, 2021 at 15:39

1 Answer 1


This answers b which includes a

Proof by example:
I hope the picture makes it clear how to get from N to N+4.

  • $\begingroup$ May I ask what software did you use to draw these graphs? $\endgroup$
    – Eric
    Commented Apr 24, 2021 at 13:40
  • $\begingroup$ @Eric Sure. python + numpy + matplotlib $\endgroup$
    – loopy walt
    Commented Apr 24, 2021 at 16:41

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