The grape codes looked like they were in base 2. The only exception was that there could be digits higher than 0 and 1. Another thing I noticed is that when you shift it to the left it multiplies by 2 and when shifting to the right it divides by 2. This is the same for all bases except you divide or multiply by the base number. An example is when you shift a number in base 3 to the left. It multiplies by 3.

The left/right shift is a really interesting observation.

If you can create 10 with two grapes. What's the relationship between

2 x 10 and minimum # grapes?

Similarly, what's the relationship between 1/2 x 10 and minimum # grapes?

The relationship is that when you multiply/divide by 2^n, then you just move the numbers to the left/right n times. -aj

If I gave you a larger number, for example 320, could you quickly figure out the minimum # of grapes needed to represent it?

Well, for 320 yes, because I know 320=256+64=2^8+2^6 so 2 but for a different/larger number probably no.

I was trying to get at the idea: both 320 and 10 can be represented by the same number of grapes because 320 = 10 x 2^5.

Another example of this concept: let's take 112. We know it's divisible by 2, so we start halving the number until we can't anymore.

112

56

28

14

7

We arrive at 7; we know that 7 and 112 can both be represented by the same # of grapes. We observe that 7 can be represented by three grapes (4 + 2 + 1), therefore 112 can also be represented by three grapes.

We can also use this idea to figure out the placement of the grapes. To go from 7 to 112 we shift to the left 4 times.

Yes that is what I had meant.