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INVESTIGATION WINNING LINES
This investigation looks at the number of winning lines on a grid.
In a game you win by making a straight line of three O's on a 3 by 3 grid.
There are 8 winning lines of three O's on a 3 by 3 grid.
Show each winning line of three O's on the grids.
Three of the winning lines have been shown for you.
560
590
544
(a) Another grid is 4 by 4.
You now need four O's in a line to win.
Find the number of winning lines on a 4 by 4 grid.
You may use the grids to help you.
...................................................... [2]
(b) Another grid is 5 by 5.
You now need five O's in a line to win.
Find the number of winning lines on a 5 by 5 grid.
You may use the grids to help you.
...................................................... [2]
(c) Another grid is 6 by 6.
You now need six O's in a line to win.
Write down the number of winning lines on a 6 by 6 grid.
...................................................... [1]
509
586
556
(a) Complete this table using your answers to Question 1 and Question 2 and any patterns you notice.
[Table_1]
[3]
(b) A grid is $n$ by $n$.
You need $n$ Os in a line to win.
Find an expression, in terms of $n$, for the number of winning lines.
.............................................................. [2]
(c) Jibreel draws a very large square grid.
He thinks there will be 583 winning lines of Os on his grid.
Give a reason why he is wrong.
............................................................................................................................... [1]
(d) Harriet draws a square grid with 324 squares.
Find the number of winning lines of Os on this grid.
.............................................................. [3]
504
519
504
A grid is $n \text{ by } n$.
In a different game a winning line is one $\bigcirc$ less than $n$.
To make a line, the $\bigcirc$s must be in squares that are next to each other.
(a) In a $3 \text{ by } 3$ grid you need two $\bigcirc$s in a line to win.
These diagrams show some of the diagonal winning lines.
Complete the table to find the number of winning lines with two $\bigcirc$s.
You may use the grids below the table to help you.
[Table_1]
(b) In a $4 \text{ by } 4$ grid you need three $\bigcirc$s in a line to win.
Complete the table to find the number of winning lines with three $\bigcirc$s.
You may use the grids below the table to help you.
[Table_2]
(c) Copy your results from part (a) and part (b) into this table.
Complete the table.
You may use the grids below the table to help you.
[Table_3]
(d) In an $n \text{ by } n$ grid you need $(n-1)$ $\bigcirc$s in a line to win.
$n$ must be at least $3$.
In one grid the total number of winning lines is a square number less than $50$.
Find the grid size.
600
521
587
A rectangular grid has height 2 and width at least 2. You need two ⭕s in a line to win.
These diagrams show all the winning lines with two ⭕s on a 2 by 3 grid.
There are 11 winning lines
(a) Complete the table for the number of winning lines with two ⭕s. You may use the grid below the table to help you.
[Table_1]
(b) A 2 by $w$ grid has 111 winning lines with two ⭕s.
Find the width of the grid.
507
548
524
