I will now try moving the stair upwards to see how this affects the sum of the encased numbers. I predict this will increase the total by 100, as moving up one square increases the sum by 10 moving ten squares up must increase the sum by 100.
This table shows that my prediction was correct and that as you move one roe up the sum of the encased numbers increases by 100.
As I have worked my way through the 4x4 stair I have come to realise that there are some rules that this stair also abides by.
I have investigated the relationship between the position and the sum of the encased numbers in a 4x4 stair I will now find the algebraic rule to support my investigating.
10n +110
I will now test the algebraic rule that I have come up with on 5 random numbers to see if the rule is correct.
This shows that my algebraic rule works, I have made a prediction, tested it, made a formula, tested it and have come to the conclusion that it works.
I will now investigate the relationship between the position of a 5x5 stair in a 10x10 grid I will try to find the formula and uncover some rules.
I have tested the 5x5 stair and I am now going to find a formula to explain it.
15n+220
I will now test my rule on 5 different and random square/corner numbers.
I have made a prediction, tested my prediction, made a formula, tested my formula and now I have finally come to the conclusion that it works.
Working through my work I have noticed a couple of pattern that have helped me make this investigation easier for myself.
I have also noticed that the algebraic rules have patterns.
Amount of Algebraic Additional
Steps Rule Part
1 n 11x0
2 3n+11 11x1
3 6n+44 11x4
4 10n+110 11x10
5 15n+220 11x20
From looking at this I can see a relationship between the amount of steps and the first part of the rule for each of the algebraic rules. I can see that if you do
(Step number(step number +1)
2 n
If you wanted to find the n part of the algebraic rule for a 5 step stair then you would do
(5(5+1)
2 n
I will now investigate the relationship between the position of the stairs and the sum of the encased numbers in a 9x9 grid. Investigating different sized grids will help me achieve my goal of finding a super rule.
As the number increases by 6 when you move the stair to the right the number will be 6n+40
1+2+9+10+18=40
I will now look at a 4x4 stair in a 9x9 grid to investigate the relationship between the sum and the position.
As the stair moves to the right 10 is added on to the sum this means than the algebraic rule will begin 10n, the second part of the rule is determined by this.
1+2+3+9+10+11+18+19+27=100 So the rule is 10n+100
I will now investigate the relationship between the position and the sum of a 5x5 stair in a 9x9 grid.
My results shows me that the algebraic rule will be 15n + 200
1+2+3+4+9+10+11+12+18+19+20+27+28+36=200
I will now investigate the relationship between the position and the sum of the encased numbers inside an 11x11 grid to gather as much research to help me find the super rule.
I predict that with a 3x3 stair in an 11x11 grid moving the stair to the right will increase the sum by 6, I also predict that moving the stair upwards will increase the sum by 66 because 11x6=66.
Looking at the table I can see that the algebraic rule for this is
6n+48
1+2+11+12+22=48
I will now look at a 4x4 stair in an 11x11 grid, I predict that as I move the step to the right the sum will increase by 10, I also predict that as I move the stair up the sum will increase by 110 because 11x10=110
Looking at the table I can see that the algebraic rule will be
10n+120
1+2+3+11+12+13+22+23+33=120
I will now look at a 5x5 stair in an 11x11 grid, I predict that as I move the step to the right the sum will increase by 115, I also predict that as I move the stair up the sum will increase by 165 because 11x15=165
I can tell from looking at the table that the algebraic rule will be
15n+240
1+2+3+4+11+12+13+14+22+23+24+33+34+44=240
Here is a table that incorporates all the algebraic rules I have come across so far this may help point out any ways that I might come across the super rule.
If I piece together what I have found I can build up the super rule
I need to test my super rule to make sure it is correct.
Imagine the corner number is 45 the step size is 3 and in a 11x11 grid
You would work out the total number in this way
(3(3+1) + (11+1) x
2 n
6 n + 12 x 1(1+1) + 2(2+1)
2 2
6 n + 12 x 1 + 3
6 n + 12 x 4
6 n + 48
6x45 + 48
270 + 48 = 318
45+46+47+56+57+67=318
I have came up with a rule and tested it and it works. Woop
I hope you have enjoyed reading and marking my coursework as much as I have enjoyed doing it.