# Tic-Tac-Toe problem

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.

Source: The Niels Henrik Abel mathematics competition

• 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. – ffao May 21 '17 at 21:07
• @ffao what prove do you have that it isn't original? – user64742 May 22 '17 at 1:03
• @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. – stack reader May 22 '17 at 1:16
• @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. – stack reader May 22 '17 at 1:45
• Has a correct answer been given? If so, please don't forget to $\color{green}{\checkmark \small\text{Accept}}$ it. :) – Rubio Jun 19 '17 at 7:38

## 1 Answer

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:

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.

• Beaten by 18 seconds... :( – ffao May 21 '17 at 9:57
• @ffao When I looked, it had already been up for 30 minutes so I thought 'it can't be that hard, can it?' – Wen1now May 21 '17 at 9:59
• I don't see any squares, although I do see a lot of rhombi. Other than that pedantic note, nice answer. +1 – John Coleman May 21 '17 at 11:56
• 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? – Rob May 22 '17 at 3:49
• @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 – Wen1now May 22 '17 at 4:08