6x + 1 + 2 + 3 + 10 + 11 + 20 =
6x + 44
=11+12+13+21+22+31=110
The stairs total for this three-step stair is 110
I can say the total for square x is, “x + x + 1 + x + 2 + x + 3 + x +10 + x + 11 + x + 20”, because where ever x is on the grid, the number of the square:
· One place right of it will be x +1
· Two places right of it will be x +2
· One place above it will be x +10
· One place above it then one place to the right will be x + 10 + 1 = x + 11
· Two places above it will be x + 20
To get the total you multiply ‘x’ by six then add 1, 2, 10, 11 and 20 which gives you a general formula of 6x + 44.
I can test my theory by making doing the following:
First I need to make a table giving a sample of results found by adding each stair up, square by square:
I then need find the difference between the totals:
Because there is an instant pattern in the first difference – the totals go up in 6, I need find out what 6n would come to:
6x doesn’t give me the correct answer so I add 44 to 6x and get the Stair Total:
Testing the Formula
To ensure the formula works in every case, I tested it in five situations where
‘x’ was different each time.
Using the formula 6x + 44
If n = 1 then
· 6 x 1 + 44 = 6 + 44 = 50
If x = 12 then
· 6 x 12 + 44 = 72 + 44 = 116
If x = 23 then
· 6 x 23 + 44 = 138 + 44 = 182
If x = 34 then
· 6 x 45 + 44 = 270 + 44 = 314
All the results using the formula are correct, so I can come to the conclusion that the formula for the total of a stair:
· On a 10 by 10 grid
· Which travels downwards from left to right
· With a height of 3 squares
Is: 6n + 44
In every case of a three step, the difference in position has a different total of 6.
Part Two (Extension)
Investigate further the relationship between the stair total and other step
stairs on the number grids.
For past of this investigation I am going to find out the totals for each step, starting from 2 step, to a 10step. As I found a formula for part one, I will continue to use that, in this part of this investigation, to see if there is any initial relationship. To see if there are any relationships between the number of stairs and there total, I will then put the results into a table, to make it clearer.
Two and Three Step
Two Step
‘x’ =43
X + (x + 1) + (x + 10) =
3x+11
=43+44+53=140
The stairs total for this two-step stair is 140
Three Step
‘X’=17
X + (X + 1) + (X + 2) (X + 3) (X + 10) (X + 11) + (X + 20) =
6x+44
=17+18+19+27+28+37 = 146
The stairs total for this two-step stair is 146
Four and Five Step
Four Step
‘X’=33
X + (X + 1) + (X + 2) (X + 3) (X + 10) (X + 11) + (X + 12) + (X + 20) + (X + 21) + (X + 30) =
10x+110
=33+34+35+36+43+44+45+53+54+63= 440
The stairs total for this two-step stair is 440
Step Five
‘X’=56
X + (X + 1) + (X + 2) (X + 3) (X + 4) (X + 10) + (X + 11) + (X + 12) + (X + 20) + (X + 21) + (X + 22) + (X + 30) + (X + 31) + (X + 40) =
15x+220
=56+57+58+59+60+66+67+68+69+76+77+78+86+87+96= 1060
The stairs total for this two-step stair is 1060
Six Step
‘X’=45
X + (X + 1) + (X + 2) (X + 3) (X + 4) (X + 5) + (X + 10) + (X + 11) + (X + 12) + (X + 13) + (X + 14) + (X + 20) + (X + 21) + (X + 22) + (X + 23) + (X + 30) + (X + 31) + (X + 32) + (X + 40) + (X + 41) + (X + 50)=
21X+385
=45+46+47+48+49+50+55+56+57+58+59+65+66+67+68+75+76+77+85+86+95= 1330
The stairs total for this two-step stair is 1330
Seven Step
‘X’=12
X + (X + 1) + (X + 2) (X + 3) (X + 4) (X + 5) + (X + 6) + (X + 10) + (X + 11) + (X + 12) + (X + 13) + (X + 14) + (X + 15) + (X + 20) + (X + 21) + (X + 22) + (X + 23) + (X +24) + (X + 30) + (X + 31) + (X + 32) + (X + 33) + (X + 40) + (X + 41) + (X + 42) + (X + 50) + (X + 51) + (X + 60) =
28x+616
=12+13+14+15+16+17+18+22+23+24+25+26+27+32+33+34+35+36+42+43+44+45+52+53+54+62+63+72=952
The stairs total for this two-step stair is 952
Eight Step
‘X’=23
X + (X + 1) + (X + 2) (X + 3) (X + 4) (X + 5) + (X + 6) + (X + 7) + (X + 10) + (X + 11) + (X + 12) + (X + 13) + (X + 13) + (X + 14) + (X + 15) + (X + 16) + (X + 20) + (X +21) + (X + 22) + (X + 23) + (X + 24) + (X + 25) + (X + 30) + (X + 31) + (X + 32) + (X + 33) + (X + 34) + (X + 41) + (X + 42) + (X + 43) + (X + 50+ (X +51) + (X + 52 + (X + 60) + (X + 61) + (X + 70)=
36x+924
=23+24+25+26+27+28+29+30+33+34+35+36+37+38+39+43+44+45+46+47+48+53+54+55+56+57+63+64+65+66+73+74+75+83+84+93=1752
The stairs total for this two-step stair is 1752
Nine Step
‘X’=11
X + (X + 1) + (X + 2) (X + 3) (X + 4) (X + 5) + (X + 6) + (X + 7) + (X + 8) + (X + 10) + (X + 11) + (X + 12) + (X + 13) + (X + 14) + (X + 15) + (X + 16) + (X + 17) + (X +20) + (X + 21) + (X + 22) + (X + 23) + (X + 24) + (X + 25) + (X + 26) + (X + 30) + (X + 31) + (X + 32) + (X + 33) + (X + 34) + (X + 35) + (X + 40+ (X +41) + (X + 52) + (X + 43) + (X + 44) + (X + 50) + (X + 51) + (X + 52) + (X + 53) + (X + 60) + (X + 61) + (X + 62) + (X + 70) + (X + 71+ (X + 80)=
45x+1320
=11+12+13+14+15+16+17+18+19+21+22+23+24+25+26+27+28+31+32+33+34+35+36+36+41+42+43+44+45+46+51+52+53+54+55+61+62+63+64+71+72+73+81+82+91=1814
The stairs total for this two-step stair is 1814
Ten Step
‘X’=1
X + (X + 1) + (X + 2) (X + 3) (X + 4) (X + 5) + (X + 6) + (X + 7) + (X + 8) + (X + 9) + (X + 10) + (X + 11) + (X + 12) + (X + 13) + (X + 14) + (X + 15) + (X + 16) + (X +17) + (X + 18) + (X + 20) + (X + 21) + (X + 22) + (X + 23) + (X + 24) + (X + 25) + (X + 26) + (X + 27) + (X + 30) + (X + 31) + (X + 32) + (X + 33+ (X +34) + (X + 35) + (X + 36) + (X + 40) + (X + 41) + (X + 42) + (X + 43) + (X + 44) + (X + 45) + (X + 50) + (X + 51) + (X + 52) + (X + 53+ (X + 54) + (X + 60) + (X + 61) + (X + 62) + (X + 63) + (X + 70) + (X + 71)+ (X + 72+ (X + 80) + (X + 81)+ (X + 90)=
55x+1815
=1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+21+22+23+24+25+26+27+28+31+32+33+34+35+36+36+37+41+42+43+44+45+46+51+52+53+54+55+61+62+63+64+71+72+73+81+82+91=1906
The stairs total for this two-step stair is 1906
Having found out the stairs totals and the formula’s for the steps of 2-10, I am then going to put my findings into a table, as I am not able to see any instant relationship between the group of results as they are scattered all about.
Having interpreted the results of the formulas and stairs totals into a table, I am now able to see a relationship between the formulas for working out each step. This being that the formulae’s for each step are triangular numbers.
The first 10 triangle numbers are: 3,6,10,15,21,28,36,45,55
As you can see, they all are in the formulas.
Starting from the basic two stairs, the formula (relationship) is the first triangular number, and continues through out the table, and beyond ten stairs.
As you carry on with the number stairs, for example working out the 11th stair formula, you will notice that it will be the 11th triangular number, this will be for all of the formulas.
The table below states this:
