Can someone please help me to come up with a winning strategy for a 4 x 4 grid game with the use of 16 counters.
Players take turns to place either 1,2 or 3 counters on the grid (one counter per square). If 2 or 3 counters are put on any one turn, then they must form an unbroken line (horizontally, vertically or diagonally). The loser is the player who places the last counter on the grid.
2 players
Well, the added rule of the unbroken line could make it more difficult. Normally, you would always have an advantage of going first because you will always be able to force taking four in total, and then win in the end (if you take three in the beginning, there are 13 squares left, so no matter how the opponent plays you can always sum up to four-> e.j., 1+3, 2+2, 3+1). If the board is laid out like so:
You can clearly see there are at most 5 configurations for the 3-in-a-row, so it would be very difficult for your opponent to take them all. The only problem is in the worst case scenario: if the opponent tries to seal you off, you might not be able to complete three in a row. Example: You play first:
For ease of operation opponent will always play 3 except the last round (he is "Z")
your turn again [take corners to prevent being forced to play only 1]
Z tries to prevent you from winning [But it's obvious at this point who's going to win: you]
As you can see the "worst case scenario" won't happen because you're taking the corners. When going first, you can take one corner from the get start, and even though Z can attempt to "seal off" a corner like so (your turn):
He won't succeed because there are 7 squares left and you can force Z into a situation where he can only play two or one (and you can thus always force 5 with this situation):
So in conclusion: -You win if you go first -Play 3 on the get go -Always take the corners if possible -Sum up with opponent to 4 (unless you can only play 2 or 1) -If Z tries to force you into being only able to play 2 or 1, it won't change a thing since you can still force a sum of 3 and win with the rest. (He can't force you into this situation if he only plays one at a time)