Addition Chart Possibilities

V+N=Z

1+1=2
W=1

1+2=3
W=1

1+3=4
2+2=4
W=2

1+4=5
2+3=5
W=2

1+5=6
2+4=6
3+3=6
W=3

1+6=7
2+5=7
3+4=7
W=3

1+7=8
2+6=8
3+5=8
4+4=8
W=4

1+8=9
2+7=9
3+6=9
4+5=9
W=4

1+9=10
2+8=10
3+7=10
4+6=10
5+5=10
W=5

1+10=11
2+9=11
3+8=11
4+7=11
5+6=11
W=5

1+11=12
2+10=12
3+9=12
4+8=12
5+7=12
6+6=12
W=6
     So it seems that with each even
number and continuing on to the odd
number after it the number of sum
possibilities in the addition chart
increments by one. I will attempt to
form a proof for this if it is true
or not true or something else.
     An application for this could be for
determining cross domain relations for
alternative route mathematics such as for
logic or spoken language such as for
finding how many different ways there are
to form a statement or to describe an
idea or question and what that could be
given an exact methodology.
     Notice that small numbers have fewer
addition chart possibilities and that the
larger numbers get the more addition
chart possibilities there are in sequential
pairs.

1+12=13
2+11=13
3+10=13
4+9=13
5+8=13
6+7=13
W=6

1+13=14
2+12=14
3+11=14
4+10=14
5+9=14
6+8=14
7+7=14
W=7

1+14=15
2+13=15
3+12=15
4+11=15
5+10=15
6+9=15
7+8=15
W=7

1+15=16
2+14=16
3+13=16
4+12=16
5+11=16
6+10=16
7+9=16
8+8=16
W=8

1+16=17
2+15=17
3+14=17
4+13=17
5+12=17
6+11=17
7+10=17
8+9=17
W=8

1+17=18
2+16=18
3+15=18
4+14=18
5+13=18
6+12=18
7+11=18
8+10=18
9+9=18
W=9

1+18=19
2+17=19
3+16=19
4+15=19
5+14=19
6+13=19
7+12=19
8+11=19
9+10=19
W=9

1+19=20
2+18=20
3+17=20
4+16=20
5+15=20
6+14=20
7+13=20
8+12=20
9+11=20
10+10=20
W=10

1+20=21
2+19=21
3+18=21
4+17=21
5+16=21
6+15=21
7+14=21
8+13=21
9+12=21
10+11=21
W=10

1+21=22
2+20=22
3+19=22
4+18=22
5+17=22
6+16=22
7+15=22
8+14=22
9+13=22
10+12=22
11+11=22
W=11

Hypothesis:
For All Z not equal to 2
If V+N=Z
Then (V+1)+(N-1)=Z

       Given: V,N,Z,W are in the Natural Number System
       Proof: When Z is even then V+N / Z = 2W
       and when Z is odd then V+1 + N-1 = Z  such that Z-1 = 2W

Wednesday, November Fourth, Two-Thousand and Nine.
11/04/2009
Justin M Coslor
I give my thanks to Research Scientist Wilfried Sieg
at Carnegie Mellon University for training and inspiration in the
Fall 1997 Humanities Philosophy Department Math in Context course.
11/13/2009 I also am grateful to Andrew J. Dougherty of the
Formalized Research Database: Cluster, Study, Apply (FRDCSA)
for encouragement in formalizing my experimental mathematical
endeavors and to Seth Casana for proofreading this idea, and
also I am thankful to Daniel Belcher for asking good questions
about this idea and its possible applications and for verifying it.
I also want to thank my parents for moral support.