Puzzler: The farmer and the 40 lb stone (10/22/11) (NO ANSWER GIVEN)

miter1 — At this juncture, the natural question to ask is "Why don't we use base-3 in computers, as base-3 can hold more information than base-2 in each digit?" The answer is that some extra information is also needed for base-3 to represent all non-negative integers. Consider 2 (base-10) is simply "10" in base-2. In base-3, it is "10"-"01". We need to take care of both "10" and "01", as well as the minus sign; so there is no saving at all. Sorry, there is no free lunch.
Actually, it is true that the optimal number system for integer bases is base 3, optimum in the sense that the number of number symbols times the number of 'places' needed to express any number is minimum. If the restriction that the base be integer is dropped, it is easy to show from first-year calculus that the optimal base is e. It is tempting to say that if the optimal non-integer base is e, then the optimal integer base is most likely 3, because 3 is closer to e than 2. That is about as far as I have ever gone with it. (But if you are familiar with linear and integer programming in operations research, you probably know that a linear programming solution rounded to the nearest integer is not necessarily the correct answer to an integer programming problem.)

However, I’m happy to have found a site on the web that shows for integer bases, 3 is the optimal base, 2 is the second most optimal base, and 4 the third most optimal base based on the minimal amount of symbols and places needed to express any number. What is the optimal numeric base?


I don’t think any respondent has explained in an easy-to-understand manner why a base 3 set of stone weights maximizes the number of unknown weights that can be measured on a balance scale when the stone weights can be placed on a scale on either side of the beam or not placed on a scale at all. Some of the explanations were correct, but a little bit muddy.
I did not think my original post would generate such a number of erudite comments.
On another topic, I’m somewhat miffed the show has degenerated into a bunch of snippets from the archives and re-runs of old puzzlers. I was a listener many, many years ago and only returned to the show over the last few months. All things change with time, I suppose.

I submitted to Tom & Ray two pretty-good (IMO) puzzlers four to six weeks ago. I’ve listened every Saturday morning since then, hoping my puzzlers will be aired on the show. However, I am discovering that the puzzlers they have been broadcasting are several years old.

For this reason I will give one puzzler on this board about 10 am CST on Saturday, November 12th 2011, after the show airs in Chicago (I still have hopes!). Tune back then for a tricky little algebraic puzzle. I’ll start a new topic with Puzzler #1.