L+1+L+L-8+L-16
4L-24+1
4L-23
Formula should give for “L” number 12 total will be 80-23=57
20+21+12+4 = 57
My formula works for this grid size
This formula will not work for any other grid size.
I am going to try out my formula for 4L-23 for L number 13
L total = 4x13=52-23=39
1+7+13+14=35
I have found that this formula does not. I have only changed the grid size therefore the number that you will subtract must be related to the grid size. I am going to look at the algebra.
In a table of 6X6
L+1+L+L-6+L-12
4L-17
This formula will not work for any other grid size.
I am going to try to work out a formula that will work with any grid size:
I have noticed that you are taking away three times the number of the grid size each time and then adding one.
If G=grid size then the formula must be 4L-3G+1
This will work for any “L” shape that is three like the shape above.
I am now going to make a 8x8 box and see what happens to the total if I change the rotation of the shape by 180 degrees. The “L” number is in the bottom right of the shape. It is highlighted red
I am going to add 1+2+11+20
=This total equals 34
I am now going to add 2+3+12+21
This total equals 38
I am now going to add 3+4+13+22
This total equals 42
The formula for this is 4l+3g-1
I am now going to flip the “L” shape so that it is the other way around – this “L” shape will be coloured blue. The “L” number is the one in the top right. It is highlighted blue
I am now going to add 2+11+20+19
The total of this is 52
I am now going to add 3 +12+21+20
This total is 56
I am concluding that this is increases in sequences of four
The formula for this box is 4L+3g-1. This is the same as the sequence above. This means that their totals will be the same
I am now going to see what happens to the sequences if I increase the size of the table so that it is 9X9. This means that it will be nine squares across and 9 squares down
I am now going to add up the numbers in the “L” shape. The “L” number is the one highlighted in yellow
1+10+19+20 = 50
I am now going to add the next lot of numbers
2+11+20+21 = 54
I am now going to, again add the next sequence of numbers
3+12+21+22 = 58
I am going to make a table of results to show you what I mean
I am concluding that this sequence also goes up in fours. This surprised me however because I would expect it to go up in fives because I have added one more column of numbers onto the table. The total is always an even number.
The algebra of this sequence is:
This equals L+L+1+L-9+L-18
This is 4L-10
I am going to see what happens if you change the size of the table again so that it is 9 across but 8 down.
1+10+19+20 = 50
I am now going to add the next lot of numbers
2+11+20+21 = 54
I am now going to, again add the next sequence of numbers
3+12+21+22 = 58
These also go up in sequences of, yep’ you guessed it, four.
I am going to try and see what happens if you try and change the size of the “L”
I am going to make my “L” be 4 squares down and 4 across
I have worked out that this formula is:
12L-107+6
12L-101
12L-6G+6
I am going to try to work out the total of the numbers in the box
I have added 1+2+9+10+17+18+19+20+25+26+27+28
This total equals 202
I am now going to add 2+3+10+11+18+19+20+21+26+27+28+29
This total equals 214
I am now going to add 3+4+11+12+19+20+21+22+27+28+29+30
This total equals 226
I have concluded that it goes up in sequences of 12
I am now going to try to adjust the size of my “L” so that it is 5 down and 5 across
I am going to try to work out the total of the numbers in the box
I am going to add 1+2+9+10+17+18+25+26+27+28+29+33+34+35+36+37
The total of this is 367
I am now going to move the “L” shape along one space so now I am going to add 2+3+10+11+18+19+26+27+28+29+30+34+35+36+37+38
The total of this is 383
I am now going to move the “L” shape along one space so now I am going to add 3+4+11+12+19+20+27+28+29+30+31+35+36+37+38+39
The total of this is 399
I have concluded that the totals go up in sequences of 16
If I have an L, shape like this:
I have worked out a formula for this:
16L-168+4
16L-164
Or:
16L-8g+6
