6.28.06 Silhouette
To get an idea for what I mean by “silhouette”, take the following:
Suppose you wanted to remember the number 989, then you must remember that it is a three digit number, what the digits are (simply remembering the correct digits will make it unnecessary to remember that the number is a three digit number), and their order. But, you could also just remember that the space between (the silhouette) 989 and 1000, is 11; taking note that 11 is a two digit number. So really, you just have to remember a two digit number, and that it is framed by a larger four digit number. Of course this is more information to remember than the original three digit 989, but, and here is the important part, if there are a significant amount of three digit numbers that must me memorized, and they can all be truncated using this method, then it is more efficient to silhouette them.
Example 1:
There are 1000 three digit numbers that need to be memorized, and they all have random values. Because of how our number system works out, only the numbers with less space than 100 between themselves and 1000 could be silhouetted into a smaller two digit number. This means, only the numbers 901 through 999 could be silhouetted smaller against the 1000 background (everything else would just be silhouetted into a different three digit number). So, in the end, only 11% of three digit numbers could be truncated by the silhouette method, and they would only lose one third of their size (not forgetting that we still have to remember the four digit background they are related to, and that they each need to be un-silhouetted in order to get their original value). It would be necessary to create a set of numbers that had been silhouetted, so that we could remember what numbers to un-silhouette. And then we would have to associate that entire set with its appropriate background (in this case 1000). But still, this makes our original grouping of numbers smaller, digit wise, and should always cut off some amount each time.
To fix this problem, it is necessary to have multiple backgrounds. So for example, the proper background for the number range 901-999 is 1000, because the difference between any number in this range and 1000 is always a two or one digit number, but always less than three digits. However, the proper background for the number range 801-899 is 900, because the difference between any number in this range and 900 is always a two or one digit number. And, if the background was instead 1000, then these (801-899) would silhouette to just a different three digit number. Notice that for the number range 801-899, it also works if the background is 899, and in fact, this would allow the number 890 to have a one digit silhouette with the 899 background, rather then a two digit silhouette with the 900 background; the point being, that smaller backgrounds can help to create silhouettes with fewer digits.
Note: In order to un-silhouette a number, you must include it in a set that is related to it’s background through a function. Because of this need to remember the contents of the set, and the function that relates to the set, sets with few contents will be less efficient, memory wise.
Example 2:
Suppose there are 1000 randomly picked numbers, there range is 0-1000, there is no order to these numbers, and they must be memorized. Using the multiple background method, a good guess for some backgrounds could be the following:
100 600
200 700
300 800
400 900
500 1000
I picked these because it seems like there should be roughly an equal amount of numbers in any area between 0-1000 (there shouldn’t be any clusters of random numbers closer to any spot between 0-1000 because random number picking usually generates non-biased uniform spreads (by spreads, I mean that the numbers should be spread out uniformly, so that the average distance between a number and any other one is the same for any number in our range).
These backgrounds will silhouette all number from 0-1000, with the exception of 0-90 (these numbers will either be given more digits through silhouette, or they will remain with two digits, so they can not be made smaller with the given background). This means that up to about 9% of the numbers may not benefit from silhouette. But, the other 90% or so should be made one third or two thirds smaller, digit wise, with the majority of numbers being silhouetted with a reduction of only one third. But either way, it looks like the silhouette method will reduce a random assortment of numbers, with no order, by 33% digit wise, not counting for the extra set memorization, nor the functions that relate to them.
A. Dempsey