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T-Totals 10x10 Grid

Extracts from this document...

Introduction

T-Totals – 10x10 Grid

image02.png

image03.pngimage14.png

T-Number=22                        T-Number=45

T-Total= 40                        T-Total=145

1+2+3+12+22                        24+25+26+35+45

T-Number=82                        T-Number=99image25.pngimage35.png

T-Total=340                        T-Total=425

61+62+63+72+82                78+79+80+89+99

  • When the T-Number is even, so to is the T-Total.
  • When the T-Number is odd, so to is the T-Total.

I will now find the rule which links the T-Number with the T-Total.

image00.pngimage36.png

n+n-10+n-19+n-20+n21= 5n-70image37.png

Example: When n=59        (5x59)-70=225

T-Total= 225        38+39+40+49+59

T-Totals – 9x9 Grid

image38.png

image39.pngimage40.png

T-Number=20                        T-Number=41

T-Total= 37                        T-Total=142

1+2+3+11+20                        22+23+24+32+41

T-Number=65                        T-Number=80

T-Total=262                        T-Total=337image04.pngimage05.png

46+47+48+56+65                61+62+63+71+80

  • When the T-Number is even, the T-Total is odd.
  • When the T-Number is odd, the T-Total is even.

...read more.

Middle

image08.pngimage09.png

T-Number=18                T-Number=37

T-Total=34                        T-Total=129

1+2+3+10+18                20+21+22+29+37

T-Number=        58                T-Number=63

T-Total=234                T-Total=259image12.pngimage11.png

41+42+43+50+58        46+47+48+55+63

  • When the T-Number is odd, the T-Total is odd.
  • When the T-Number is even, the T-Total is even.

I will now find the rule which links the T-Number with the T-Total.

n+n-8+n-16+n-17+n-18= 5n-56image13.png

image15.png

Example: When n=36        (5x36)-56=124

T-Total=157        19+20+21+28+36=124

image00.png

T-Totals – Any sized Grid

I will now find the general rule for any sized grid, which links the T-Number with the T-Total.image01.png

image16.png

n+n-G+n-2G-1+n-2G+n-2G+1= 5n-7G

Example: When n=65, and G=10

image00.png

T-Number= 65        

T-Total= 255         (5x65)-(7x10)

T-Total= 255         44+45+46+55+65


T-Totals - Translatingimage17.png

image18.png

If I translate the T three vectors right it will become:image19.png

        T-Number=43                        T-Number=46

        T-Total=145                        T-Total=160

22+23+24+33+43                25+26+27+36+46

  • The T-Total has increased by 15.
  • This is because there are 5 sequences in the T-Total, which have all increased by 3. 5x3=15.
  • Consequently, if the vector is (image21.png) the formula for moving the T across would be: T-Total+5aimage20.png
...read more.

Conclusion

To do this I must combine the rules for rotation with the ones for translations:

90*=5n+5c-5dG-5d-5cG+7+5a-5bG

180*=5n+10c-10dG+7G+5a-5bG

270*= 5n+5c-5dG+5d+5cG-7+5a-5bG


T-Totals –Conclusion

I have found that you can rotate and then translate a T (combination) using the formulae:

90*=5n+5c-5dG-5d-5cG+7+5a-5bG

This allows you to rotate the T by 90* and then move it by any vector within the Grid limitations.

180*=5n+10c-10dG+7G+5a-5bG

This allows you to rotate the T by 180* and then move it by any vector within the Grid limitations.

270*= 5n+5c-5dG+5d+5cG-7+5a-5bG

This allows you to rotate the T by 270* and then move it by any vector within the Grid limitations.

Example: Move the T by the vector (image33.png)

image34.png

The new T does not fit within the limitations.

...read more.

This student written piece of work is one of many that can be found in our GCSE T-Total section.

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