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tersums are like Nim-sums but base 3 is used instead of base 2, <br>
see sequence A004489, 'write m and n in base 3 and add mod 3 with no
carries'<br>
<a href="http://www.research.att.com/~njas/sequences/A004489" eudora="autourl">http://www.research.att.com/~njas/sequences/A004489<br><br>
</a>Below are images based on a table of tersums for n and m from 0 to
26. <br><br>
<img src="cid:.0" width=300 height=304 alt="1d4608fe.jpg"><br><br>
PARI code used to calculate the values:<br>
M=matrix(27,27); for(i=1,27,for(j=1,27,A = vector(4); B = vector(4); C =
vector(4); a = i-1; b = j-1; <br>
for(n=1,3,A[5-n]=floor(a/3^(4-n));a=a-A[5-n]*3^(4-n));A[1]=a; <br>
for(n=1,3,B[5-n]=floor(b/3^(4-n));b=b-B[5-n]*3^(4-n));B[1]=b; <br>
for(n=1,4,C[n]=(A[n]+B[n])%3); <br>
M[i,j]=sum(n=1,4,C[n]*3^(n-1))))<br><br>
I would think that ter-multiplication etc. is or could be defined
similarly to <br>
the way Nim-multiplication is defined. Is that correct? If so, is there
some <br>
practical significance?<br><br>
The tersum values:<br><br>
<font face="Courier New, Courier" size=2> 0 1 2
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
18 19 20 21 22 23 24 25 26<br>
1 2 0 4 5 3 7 8 6
10 11 9 13 14 12 16 17 15 19 20 18 22 23 21 25 26 24<br>
2 0 1 5 3 4 8 6 7
11 9 10 14 12 13 17 15 16 20 18 19 23 21 22 26 24 25<br>
3 4 5 6 7 8 0 1 2
12 13 14 15 16 17 9 10 11 21 22 23 24 25 26 18 19 20<br>
4 5 3 7 8 6 1 2 0
13 14 12 16 17 15 10 11 9 22 23 21 25 26 24 19 20 18<br>
5 3 4 8 6 7 2 0 1
14 12 13 17 15 16 11 9 10 23 21 22 26 24 25 20 18 19<br>
6 7 8 0 1 2 3 4 5
15 16 17 9 10 11 12 13 14 24 25 26 18 19 20 21 22 23<br>
7 8 6 1 2 0 4 5 3
16 17 15 10 11 9 13 14 12 25 26 24 19 20 18 22 23 21<br>
8 6 7 2 0 1 5 3 4
17 15 16 11 9 10 14 12 13 26 24 25 20 18 19 23 21 22<br>
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 0
1 2 3 4 5 6 7 8<br>
10 11 9 13 14 12 16 17 15 19 20 18 22 23 21 25 26 24 1
2 0 4 5 3 7 8 6<br>
11 9 10 14 12 13 17 15 16 20 18 19 23 21 22 26 24 25 2
0 1 5 3 4 8 6 7<br>
12 13 14 15 16 17 9 10 11 21 22 23 24 25 26 18 19 20 3
4 5 6 7 8 0 1 2<br>
13 14 12 16 17 15 10 11 9 22 23 21 25 26 24 19 20 18 4
5 3 7 8 6 1 2 0<br>
14 12 13 17 15 16 11 9 10 23 21 22 26 24 25 20 18 19 5
3 4 8 6 7 2 0 1<br>
15 16 17 9 10 11 12 13 14 24 25 26 18 19 20 21 22 23 6
7 8 0 1 2 3 4 5<br>
16 17 15 10 11 9 13 14 12 25 26 24 19 20 18 22 23 21 7
8 6 1 2 0 4 5 3<br>
17 15 16 11 9 10 14 12 13 26 24 25 20 18 19 23 21 22 8
6 7 2 0 1 5 3 4<br>
18 19 20 21 22 23 24 25 26 0 1 2 3 4
5 6 7 8 9 10 11 12 13 14 15 16 17<br>
19 20 18 22 23 21 25 26 24 1 2 0 4 5
3 7 8 6 10 11 9 13 14 12 16 17 15<br>
20 18 19 23 21 22 26 24 25 2 0 1 5 3
4 8 6 7 11 9 10 14 12 13 17 15 16<br>
21 22 23 24 25 26 18 19 20 3 4 5 6 7
8 0 1 2 12 13 14 15 16 17 9 10 11<br>
22 23 21 25 26 24 19 20 18 4 5 3 7 8
6 1 2 0 13 14 12 16 17 15 10 11 9<br>
23 21 22 26 24 25 20 18 19 5 3 4 8 6
7 2 0 1 14 12 13 17 15 16 11 9 10<br>
24 25 26 18 19 20 21 22 23 6 7 8 0 1
2 3 4 5 15 16 17 9 10 11 12 13 14<br>
25 26 24 19 20 18 22 23 21 7 8 6 1 2
0 4 5 3 16 17 15 10 11 9 13 14 12<br>
26 24 25 20 18 19 23 21 22 8 6 7 2 0
1 5 3 4 17 15 16 11 9 10 14 12 13<br><br>
</font>I've got tables and images of Nim-multiplication and Nim-powers
for n and m <br>
from 0 to 255. They are fascinating but a bit large for an email and need
some<br>
more checking. I'll try to put them on a web site. I'm working on a table
of <br>
Nim-multiplication for powers of 2, so far for 1 to 2^8:<br>
<font face="Courier New, Courier" size=2> 1
2 4 8 16
32 64 128 256<br>
2 3 8 12
32 48 128 192 512<br>
4 8 6 11
64 128 96 176 1024<br>
8 12 11 13 128
192 176 208 2048<br>
16 32 64 128 24
44 75 141 4096<br>
32 48 128 192 44
52 141 198 8192<br>
64 128 96 176 75
141 103 185 16384<br>
128 192 176 208 141 198
185 222 32768<br>
256 512 1024 2048 4096 8192 16384 32768 384<br><br>
</font>Happy New Year!<br>
Gerald<br>
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