Code: Select all
Binary digits: 0, 1
Ternary digits: 0, 1, 2
Signed-ternary digits: 0, +, -
Binary digits Ternary Signed-ternary
16g 8g 4g 2g 1g Decimal 27g 9g 3g 1g 27g 9g 3g 1g
0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 1 0 0 0 1 0 0 0 +
0 0 0 1 0 2 0 0 0 2 0 0 + -
0 0 0 1 1 3 0 0 1 0 0 0 + 0
0 0 1 0 0 4 0 0 1 1 0 0 + +
0 0 1 0 1 5 0 0 1 2 0 + - -
0 0 1 1 0 6 0 0 2 0 0 + - 0
0 0 1 1 1 7 0 0 2 1 0 + - +
0 1 0 0 0 8 0 0 2 2 0 + 0 -
0 1 0 0 1 9 0 1 0 0 0 + 0 0
0 1 0 1 0 10 0 1 0 1 0 + 0 +
0 1 0 1 1 11 0 1 0 2 0 + + -
0 1 1 0 0 12 0 1 1 0 0 + + 0
0 1 1 0 1 13 0 1 1 1 0 + + +
0 1 1 1 0 14 0 1 1 2 + - - -
0 1 1 1 1 15 0 1 2 0 + - - 0
1 0 0 0 0 16 0 1 2 1 + - - +
1 0 0 0 1 17 0 1 2 2 + - 0 -
1 0 0 1 0 18 0 2 0 0 + - 0 0
1 0 0 1 1 19 0 2 0 1 + - 0 +
1 0 1 0 0 20 0 2 0 2 + - + -
1 0 1 0 1 21 0 2 1 0 + - + 0
1 0 1 1 0 22 0 2 1 1 + - + +
1 0 1 1 1 23 0 2 1 2 + 0 - -
1 1 0 0 0 24 0 2 2 0 + 0 - 0
1 1 0 0 1 25 0 2 2 1 + 0 - +
1 1 0 1 0 26 0 2 2 2 + 0 0 -
1 1 0 1 1 27 1 0 0 0 + 0 0 0
On the left, you see the representation using the binary weight set. Note that as you read down the columns, adding one to the total weight with each row, you cycle through the binary digits, duplicating them in proportion to the value of the weight, that is, the 1g column uses each digit once before going on to the next (or back to the first if you prefer) (0, 1, 0, 1, ..) while the 2g column uses each digit twice (0, 0, 1, 1, 0, 0, 1, 1, ...). Just to the right of the decimal column, you see a scheme using power-of-3 weights on only one pan, where you would need two of each size. Again, note the cycling of the digits in each column. Finally on the far right, you see the scheme I have termed signed-ternary. The tricky part here is the conception of the cycling. In the 1g column, we start with 0, then add the 1g weight (the +). Now to add another 1g to the total, we cannot add another 1g weight. So we remove it (which reduces our total by 1g), then add it to the other side (which reduces our effective total by another 1g). So now we are down 2g, and we add the 3g weight for a net gain of 1g. It all resembles the operations of carrying and borrowing that we learned when we started adding and subtracting numbers of more than one digit. It also carries an echo of Roman numerals, where we count i, ii, iii, then jump to iv, which is v minus i, before proceeding to v.
All of this gets me to thinking about the way computers represent positive and negative numbers. There was a time when some computers kept integers in sign-and-magnitude format, where one bit determines whether the number is positive or negative, and the rest determine the actual value of the number. For computational reasons, this representation is less common now than what is known as two's-complement format, where the most significant bit of the number has a negative weight. This would correspond to adding a 1024g weight to our binary set, but only being allowed to put it on the pan with the sample, never across from it. Then a -1g weight (remember the balloons?) would be balanced by placing the 1024g weight with it on one side and the original nine weights (total 1023g) on the other. This signed-ternary representation distributes the sign of the number across all the digits, so that makes it unusual. Of course, ternary systems don't fit well with binary things like transistors, but sometimes, physicists come up with new mechanisms which have more than two states, and theorists dream of strange computers.
I hope you puzzle fans have found all this interesting.
