take a beautiful base, and destroy as many of its good properties as possible
by combining the various techniques found on this formum. Here's my entry
in the contest.
As you can tell by my moniker, I'm a big fan of base 6. I'm also a big fan
of twistaff notation, and ballanced bases. But what happens when you put them
all together? Do you get easier divisibility tests, nice rounding, and
straightforward conversion between positive and negative numbers? Uh, no.
We'll do a 3on2 version of twistaff senary. But for an extra twist,
we'll use ballanced ternary for the upper stave:
decimal  balanced ternary 
1  I 
0  0 
1  1 
and we'll use straight binary for the lower stave. Standard tables
don't look too bad....
Code: Select all
Addition:

Upper staff Lower staff
 
 I 0 1  0 1
+ +
I  I0 I 0 0  0 1
 
0  I 0 1 1  1 10

1  0 1 10
Multiplication

Upper staff Lower staff
 
 I 0 1  0 1
+ +
I  1 0 I 0  0 0
 
0  0 0 0 1  0 1

1  I 0 1
... but here's how the numbers 12 through 12 decimal come out in this scheme:
Code: Select all
dec formula twistaf notation

12 (2*1 + 1) * 6^0 I0.00
(2*I + 0)
11 (2*1 + 1) * 6^0 I0.01
(2*I + 0)
10 (2*1 + 1) * 6^0 I0.10
(2*I + 0)
9 (2*1 + 1) * 6^0 I0.11
(2*I + 0)
8 (2*1 + 0) * 6^0 I1.I0
(2*I + 0)
7 (2*1 + 1) * 6^0 I1.I1
(2*I + 0)
6 (2*0 + 0) * 6^0 I1.00
(2*I + 1)
5 (2*0 + 1) * 6^0 I1.01
(2*I + 1)
4 (2*1 + 0) * 6^0 I1.10
(2*I + 1)
3 (2*1 + 1) * 6^0 I1.11
(2*I + 1) * 6^1
2 (2*I + 1) * 6^0 00.I0
1 (2*I + 1) * 6^0 00.I1
0 (2*0 + 0) * 6^0 00.00
1 (2*0 + 1) * 6^0 00.01
2 (2*1 + 0) * 6^0 00.10
3 (2*1 + 1) * 6^0 00.11
4 (2*I + 0) * 6^0 01.I0
(0*0 + 1) * 6^1
5 (2*I + 1) * 6^0 01.I1
(2*0 + 1) * 6^1
6 (2*0 + 0) * 6^0 01.00
(2*0 + 1) * 6^1
7 (2*0 + 1) * 6^0 01.01
(2*0 + 1) * 6^1
8 (2*1 + 0) * 6^0 01.10
(2*0 + 1) * 6^1
9 (2*1 + 1) * 6^0 01.11
(2*0 + 1) * 6^1
10 (2*I + 0) * 6^0 10.I0
(2*1 + 0) * 6^1
11 (2*I + 1) * 6^0 10.I1
(2*1 + 0) * 6^1
12 (2*0 + 0) * 6^0 10.00
(2*1 + 0) * 6^1
MY EYES!!! MY EYES!!! its so ugly it burns. How ugly is it? Let us
count the ways:
1. Yeah, counting is super hard. My favorite is going from 2 to 3.
Which implies we've taken senary's legendary prolixity and put it
Which implies we've taken senary's legendary prolixity and put it
on steroids. 3 takes 4 staves to represent, and parsing it to
find the actual value is a challange in itself: What is the value
of I1.11? lets see, that would be 12 + 6 + 2 + 1 = 3.
2. Which implies we've taken senary's legendary prolixity and put it
on steroids. 3 takes 4 staves to represent, and parsing it to
find the actual value is a challange in itself: What is the value
of I1.11? lets see, that would be 12 + 6 + 2 + 1 = 3.
3. We've managed to destroy the ballanced base property that there are
multiple representations for each number. Sounds like a good thing;
but we have also thereby destroyed cleverly exploiting that ambiguity
to be able to give ourselves banker's rounding by truncation. Rounding
in this notation mindbendingly difficult, at least for me.
4. We've also managed to destroy the balanced base property that
the negation of a number is a simple swap of positive for negative
staves. 1 is 01, but 1 is I1. Hideous!
5. Is crisscross multiplication at least any easier? How
hard can it be to just be multiplying by zero or plus/minus
one all the time? Well, the "pairtriple" table, or whatever
you call it, which is analogous to the dickerdozen table
in twelfty, actually has an entry with a carry to overflow.
So no, criss cross multiplication is totally screwed as well.
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"Dicker Dozen"

 I 0 1
+
0  0I 00 01

1  10 11 100 <ugly!!!
6. We have, however, managed to preserve all of the difficulties
inherant in doing long division in ballanced bases.
Well, my work here is done. I can't think of any more ways to
to make senary harder to work with. Can you?