Box 2
33x55=1815
→ 1855-1815=40
35x53=1855
Box 3
14x36=504
→ 544-504=40
16x34=544
I have noticed the pattern here. Whenever the numbers are diagonally multiplied and then the difference is found, you always end up with 40. The hypothesis I am going to make is that if I was to work out the difference of another box taken from a 10x10 in a 3x3 snapshot, the difference I will find will be 40.
Box 4
17x39=663
→ 703-633=40
19x37=703
My predictions that I made earlier about ‘Box 4’ have turned out to be correct as when I multiplied the numbers in the 3x3 box and worked out the difference, I was left with 40.
I should now try the same method but with boxes of 4x4 dimensions.
4x4 Boxes
Box 1
2x35=70
→ 160-70=90
5x32=160
Box 2
7x40=280
→ 370-280=90
10x37=370
Box 3
67x100=6700
→ 6790-6700=90
70x97=6790
I have noticed the pattern here. Whenever the numbers are diagonally multiplied and then the difference is found, you always end up with 90. The hypothesis I am going to make is that if I was to work out the difference of another box taken from a 10x10 in a 4x4 snapshot, the difference I will find will be 90.
Box 4
47x80=3760
→ 3850-3760=90
50x77=3850
My predictions that I made earlier about ‘Box 4’ have turned out to be correct as when I multiplied the numbers in the 4x4 box and worked out the difference, I was left with 90.
I should now try the same method but with boxes of 5x5 dimensions.
5x5 Boxes
Box 1
1x45=45
→ 205-45=160
5x41=160
Box 2
6x50=300
→ 460-300=160
10x46=460
I have noticed the pattern here. Whenever the numbers are diagonally multiplied and then the difference is found, you always end up with 160. The hypothesis I am going to make is that if I was to work out the difference of another box taken from a 10x10 in a 5x5 snapshot, the difference I will find will be 160.
Box 3
23x67=1541
→ 1701-1541=160
27x63=1701
My predictions that I made earlier about ‘Box 3’ have turned out to be correct as when I multiplied the numbers in the 5x5 box and worked out the difference, I was left with 160.
I should now try the same method but with rectangles.
Rectangles
Rectangle 1
1x24=24
→ 84-24=60
4x21=84
Rectangle 2
7x30=210
→ 270-210=60
10x17=270
I have noticed the pattern here. Whenever the numbers are diagonally multiplied and then the difference is found, you always end up with 60. The hypothesis I am going to make is that if I was to work out the difference of another box taken from a 10x10 in a 4x3 snapshot, the difference I will find will be 60.
Rectangle 3
22x45=990
→ 1050-990=60
25x42=1050
My predictions that I made earlier about ‘Box 3’ have turned out to be correct as when I multiplied the numbers in the 4x3 box and worked out the difference, I was left with 60.
Algebra Variables
For this specific investigation, I have decided to use algebra to represent and then make formulae for each snapshot I have taken from the 1-100 grid. I will use the letter H to represent the height of each snapshot and use the letter W for the width of each snapshot, no matter of size and dimensions. In this, S is the top left corner in each snapshot. I will now prove how 2x2 boxes always leave you with a difference of 10 through algebra.
Equation
S(S+11)=S2+11
(S+1)(S+10)=S2+10S+S+10
=S2+11S +10
Difference Finding
S2+11S+10-S2-11S
=10
I have noticed that the difference of the boxes is continuously changing. I will next show what happens when you do the same thing with algebra but using a 5x5 box.
5x5 Box
S(S+44)=S2+44S
(S+4)(S+40)=S2+40S+4S+160
=S2+44S+160
Finding The Difference
S2+44S+160-S2-44S=160
I will now show you how to find the formula for any box no matter of the sizes and dimensions.
Finding the Formula That Can Be Used On All Boxes
(S+10W-10)(S+W-1)=
S210WS-10S+(W-1)S+10(W-1) 2
xS+(W-1)+(10W-10)=
SxS+10W-10+(W-1)=
S2+10WS-10S+(W-1)S
S2+10WS-10S+(W-1)S+10(W-1)2-S2+10WS
-10S+(W-1)S=10(W-1)2
= S2-20W-11S+11WS+10W2+10
Next, I will show how to find the formula for any Rectangle.
Finding the Formula Which Can Be Used On Any Rectangle
S+10L-10xS+(W-1)=
S2+10LS-10S(W-1)S+10(L1)(W-1)
SxS+10(L-1)+(W-1)=
S2+10LS-10S(W-1)S+10(L-1)(W-1)-(S2+10LS-10S+(W-1)S)
=10(L-1)(W-1)
I have now found the general rule to work out the formula for any rectangle on a 10x10 grid. The rule is 10(L-1)(W-1).
Expanding the Brackets
s(s+11)
s2+11s
(s+10)(s+1)
s2+11s+10-(s2+11s)
s2+11s+10-(s2-11s)
=+10
Algebra
2x2 Box
s(s+11)(s+1)-s(s+11)
3x3 Box
s(s+10+10+1+1)-(s+1+1)
(s+10+10)
s+20
4x4 Box
s(s+33)-(s+30)(s+3)
5x5 Box
s(s+44)-(s+40)(s+4)
DIFFERENCE IS 10 WHEN YOU GO DOWN A BOX.
