Firstly we can identify certain rules the pattern is following by looking at which direction the line is moving with respect to its X and Y co-ordinates:
1. When the X co-ordinate is odd and the Y co-ordinate is even, then the next point will be (X+1, Y-1)
2. When the X co-ordinate is odd (and not 1) and the Y co-ordinate is odd, then the next point will be (X-1, Y+1)
3. When the X co-ordinate is even and the Y co-ordinate is even, then the next point will be (X-1, Y+1)
4. When the X co-ordinate is even and the Y co-ordinate is odd, then the next point will be (X+1, Y-1)
5. When the X co-ordinate is 1 and the Y co-ordinate is odd, then the next point is (X, Y+1)
(Please refer to the Excel spreadsheet for a more concise view of the above rules)
Using these rules we can answer the first question: 'If the pattern of arrows continues for ever, which point will the route visit immediately after (18,17)?'
18 is the X co-ordinate and is even. 17 is the Y co-ordinate and is odd. This means the next co-ordinate will be (X+1, Y-1), so it will be (19, 16)
For the second question, we need to identify an nth term of how many points have been visited for every Y value on the line X = a. In my case, I have decided to use the line X = 1.
(Please refer to the Excel spreadsheet to see the amount of points visited at every y value on the line X = 1)
You can immediately see that for every even Y value, the amount of points visited was just 1 higher than the previous odd Y value. This would make finding the nth term for the series very difficult, so I decided to consider the even Y values and odd Y values as 2 separate series.
I would firstly find the nth term for every odd Y value. This would enable me to add a '+ 1' to the nth term to find the nth term for even Y values.
I calculated the 'differences' and 'second differences' (please refer to my excel spreadsheet) and then used the following formula to find the nth term:
nth term = a + (n - 1)d + ½(n - 1)(n - 2)c
Definitions:
a = first number
n = undefined - used to locate value for the nth term
d = difference between 1st term and 2nd term
c = constant second difference
This gave me:
a = 1
n = undefined
d = 5
c = 4
nth term = 1 + ((n - 1) * 5) + (0.5 * (n - 1) * (n - 2) * 4)
nth term = 1 + 5n - 5 + (2 * (n - 1) * (n - 2))
nth term = 1 + 5n - 5 + 2n^2 - 6n + 4
nth term = 2n^2 - n
nth term for odd values of Y : 2n^2 - n
nth term for even values of Y: 2n^2 - n + 1
Now, with the use of both the rules and the nth terms, we can answer question 2: 'How many points will be visited before the route reaches the point (9,4)?'
9 is odd and 4 is even, which means the next point is (10, 3) and that the line is moving down-right with every iteration of x. Since we can figure out how many points have been visited at every Y value of the line X = 1, we need to figure out 2 things: the last Y value the line has visited AND the amount of points have been visited since then to get to (9, 4). We can then add these 2 values and find how many points have been visited.
Since we know the line is going down-right, we must travel up-left to get to the line X = 1 to find the Y value to input into one of our nth terms.
Here is the process of finding the Y value:
X = 9, Y = 4
To find point Y when X = 1, we must do 'Y = Y + (X - 1)' (which is effectively travelling up-left until we reach the line X = 1)
So Y = 4 + (9 - 1)
Hence Y = 12
The Y value is 12 and we have travelled 8 points since then.
Since the Y value is even, we must use the nth term '2n^2 - n + 1'
Inputting 12 as n gives: 2*12^2 - 12 + 1
The answer is: 277
However, we must now add 8 since we have visited 8 points since then.
The final answer is: 285 points