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(a) This is a 3 by 3 diagram.
The diagram shows:
• 6 horizontal connectors
• 4 up diagonal connectors.
Each connector joins 2 dots.
Complete the diagram by drawing the 6 vertical connectors and the 4 down diagonal connectors that join 2 dots.
(b) This is a 4 by 4 diagram.
On this 4 by 4 diagram,
(i) draw the horizontal connectors and the vertical connectors that join 2 dots, [1]
(ii) draw the up diagonal connectors and the down diagonal connectors that join 2 dots. [1]
(c) Complete the table for the numbers of connectors that join 2 dots. Use part (b) and any patterns you notice.
You may use the square dotty paper on page 2 for diagrams.
| Size of diagram (n by n) | Horizontal | Vertical | Up diagonal | Down diagonal | Total |
|---|---|---|---|---|---|
| 1 by 1 | 0 | 0 | 0 | 0 | 0 |
| 2 by 2 | 2 | 2 | 1 | 1 | 6 |
| 3 by 3 | 6 | 6 | 4 | 4 | 20 |
| 4 by 4 | |||||
| 5 by 5 | 20 | 16 | |||
| 6 by 6 | 110 |
(d) In an $n$ by $n$ diagram there are $n$ rows and $n$ columns.
(i) Find an expression, in terms of $n$, for the number of up diagonal connectors that join 2 dots on an $n$ by $n$ diagram.
.......................................................... [2]
(ii) Find an expression, in terms of $n$, for the number of horizontal connectors that join 2 dots on an $n$ by $n$ diagram.
.......................................................... [3]
(e) Use your answers to part (d) to find the total number of connectors that join 2 dots on a 15 by 15 diagram.
.......................................................... [3]
577
555
589
(a) This is a 4 by 4 diagram.
Find the number of horizontal, vertical, up diagonal and down diagonal connectors that join 3 dots. Two horizontal connectors have been drawn for you.
Horizontal .......
Vertical .......
Up diagonal .......
Down diagonal .......
(b) Complete the table for the numbers of connectors that join 3 dots.
Use your answers to part (a) and any patterns you notice.
You may use the square dotty paper on page 2 for diagrams.
[Table]
Numbers of connectors that join 3 dots
| Size of diagram (n by n) | Horizontal | Vertical | Up diagonal | Down diagonal | Total |
|-------------------------|------------|----------|-------------|--------------|-------|
| 2 by 2 | 0 | 0 | 0 | 0 | 0 |
| 3 by 3 | 3 | 3 | 1 | 1 | 8 |
| 4 by 4 | | | | | |
| 5 by 5 | 15 | | | | 80 |
| 6 by 6 | | | | | |
(c) (i) This is an expression for the number of up diagonal connectors that join 3 dots on an $n \times n$ diagram.
$(n-2)^2$
Work out the number of up diagonal connectors that join 3 dots on a 20 by 20 diagram.
.......
(c) (ii) This is an expression for the number of horizontal connectors that join 3 dots on an $n \times n$ diagram.
$n^2 + an$
Find the value of $a$ and write down the expression.
.......
502
580
517
(a) Complete the table for the numbers of connectors that join 4 dots.
[Table_1]
| Size of diagram (n by n) | Horizontal | Vertical | Up diagonal | Down diagonal | Total |
|---|---|---|---|---|---|
| 3 by 3 | 0 | 0 | 0 | 0 | 0 |
| 4 by 4 | 10 | ||||
| 5 by 5 | 10 | ||||
| 6 by 6 | 18 | 18 | 9 | 9 | 54 |
[2]
(b) (i) Write down an expression, in terms of n, for the number of up diagonal connectors that join 4 dots on an n by n diagram.
$\text{...............................}$ [1]
(ii) Find an expression, in terms of n, for the number of horizontal connectors that join 4 dots on an n by n diagram.
$\text{...............................}$ [2]
(c) Show that the total number of connectors that join 4 dots on an n by n diagram is
$$4n^2 - 18n + 18.$$
[2]
(d) Find the size of the diagram which has a total of 180 connectors that join 4 dots.
$\text{...............................}$ [2]
588
512
543
