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In an infinite grid of regular triangles, Niels and Henrik are playing a game they made up. Every other time, Niels picks a triangle and writes × in it, and every other time, Henrik picks a triangle where he writes a ◦. If one of the players gets four in a row in some direction (see figure), he wins the game. Determine whether one of the players can force a victory, or if both players can stop the other from winning.

enter image description here

Source: The Niels Henrik Abel mathematics competition

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    $\begingroup$ Hi there! Please don't copy-and-paste puzzles without attribution here as it runs afoul of our policy for giving credit where credit is due. Feel free to write puzzles in your own words, but don't forget to link to wherever you got the idea. $\endgroup$
    – ffao
    May 21, 2017 at 21:07
  • $\begingroup$ @ffao what prove do you have that it isn't original? $\endgroup$
    – user64742
    May 22, 2017 at 1:03
  • $\begingroup$ @TheGreatDuck the edited link is undeniable proof that this post is not original. Whether or not that is where the OP took it from or if this link was the original or not is another matter though. $\endgroup$ May 22, 2017 at 1:16
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    $\begingroup$ @TheGreatDuck are you implying that the OP created that pdf in the link? Because the image and names are 100% identical to those in the PDF. $\endgroup$ May 22, 2017 at 1:45
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    $\begingroup$ Has a correct answer been given? If so, please don't forget to $\color{green}{\checkmark \small\text{Accept}}$ it. :) $\endgroup$
    – Rubio
    Jun 19, 2017 at 7:38

1 Answer 1

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The answer is no, no one can force a win. Suppose Niels goes first. Henrik can stop Niels from winning as follows: whenever Niels goes in a triangle, Henrik goes in the corresponding coloured triangle:

enter image description here

Any 4 in a row must go over a pair of coloured triangles, so Niels cannot win.

Now suppose Henrik can force a win. Then Niels steals Henrik's strategy and places his first move arbitrarily (see this). Then we know that Niels will win, by ignoring his first move and playing 'as the second player', which is a contradiction.

Since neither player can force a win, the game must be a draw.

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    $\begingroup$ Beaten by 18 seconds... :( $\endgroup$
    – ffao
    May 21, 2017 at 9:57
  • $\begingroup$ @ffao When I looked, it had already been up for 30 minutes so I thought 'it can't be that hard, can it?' $\endgroup$
    – Wen1now
    May 21, 2017 at 9:59
  • $\begingroup$ I don't see any squares, although I do see a lot of rhombi. Other than that pedantic note, nice answer. +1 $\endgroup$ May 21, 2017 at 11:56
  • $\begingroup$ Maybe I'm missing something - but the strategy of putting taking the corresponding triangle doesn't stop them from getting four in a row: i.imgur.com/aPkuvvo.png - Or am I missing something? $\endgroup$
    – Rob
    May 22, 2017 at 3:49
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    $\begingroup$ @Rob Those four crosses are not in a row - in a row means that the diagram can be reoriented to match the OP's question $\endgroup$
    – Wen1now
    May 22, 2017 at 4:08

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