Number 3 and its magic powers.
Does anybody know how the set of 1 3 9 27 81,... with two operations + and - can generate all integers, each member is used once. For example:32=27+9-3-1

Question

Number 3 and its magic powers.
Does anybody know how the set of 1 3 9 27 81,... with two operations + and - can generate all integers, each member is used once. For example:32=27+9-3-1

anonymous · Answer

1=1
2=3-1
3=3
4=3+1
5=9-3-1
6=9-3
7=9-3+1
8=9-1
9=9
10=9+1
11=9+3-1
12=9+3
13=9+3+1
so we can make 1-13 with 1,3,9 
14=27-13
15=27-12
16=27-11
....
28=27+1
29=27+2
...
40=27+13
It shows we could make all 1-40
now
41=81-40
42=81-39
.....
100=81+19
...
121=81+40
then we can continue and proof by induction method.