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Total lines possible in a grid.

Jemma and Gareth are playing a board game on a 7 X 7 square grid. They take turns to place a black counter in an empty square on the grid. The winner is the person who completes a line of three counters.

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In this game, Winning lines are horizontal, Vertical or Diagonal

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1. How many different horizontal winning lines are possible on the 7 X 7 grid?
2. Extend your investigation, making clear the rules and the methods that you use.

Any rules can be made for establishing how many lines are possible, i.e. You can use somebody else's counters to make lines. But you must stick to them.

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Solution: Given that a winning line is created by 3 black counters, which can be horizontal, vertical or diagonal. So we can consider different grids to find the sequence. Here we cannot use a 1*1 or 2*2 grid, since the winning line contains 3 counters.
So we can start with a 3*3 grid.
In this, considering a single row we have only one possibility as shown in the figure.

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Now consider a 4*4 grid. In this for a single row we have two possibilities as shown ...

Solution Summary

Here we can use the sequence method to solve the problem of how man total lines are possible in a grid.

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