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# Thoth's "Distance" Indicator

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While browsing through old posts I came across this gem of a post by Thoth.  The whole thread is well worth studying -- Thoth has done much in-depth analysis of the Pick 3 game and his posts are well worth some time.

I was particularly struck by his measure of the "distance" of a Pick 3 combination.

To find the distance, Thoth first arranges the combination from lowest to highest, so 327 becomes 237 and 996 becomes 699.  He then takes the delta of the first two numbers and the last two numbers.

For example, 327 ordered is 237.  We then subtract 3-2 to get 1 as our low-mid or front distance.  Similarly we subtract 7-3 to get 4 as our mid-high or back distance.   Thus the distance signature for this combination is 1-4.  For doubles, say 699, we subtract 9-6=3 for our front distance and 9-9=0 to get the back distance.  Thus, the distance signature of 699 is 3-0.

There are 55 of these signatures all together. 00 is the signature for all the triples.  There are 18 doubles signatures which all contain a zero.

A double signature will always be 01,02,03,04,05,06,07,08,09 or 10,20,30,40,50,60,70,80,90.  This neatly divides the doubles group exactly in half, based on whether the third digit is lower than the double digit (zero in back, eg.699 = 3-0) or the third digit is higher than the double digit (zero in front, e.g. 229 = 0-7).

The remaining signatures are assigned to the unmatched numbers and range from 1-1 to 8-1. The presence of a "1" in the signature always indicates a consecutive pair.  The unmatched numbers can also be evenly divided using the signatures with 60 combinations having a signature of 2-5 or less and the remaining 60 having a signature of 2-6 or higher.

100 of the 120 unmatched combination signatures will contain a '1' or a '2', so a combination with a distance signature such as 4-3 would be a rarity, only appearing about 16% of the time.  This is because there are few combinations that produce those signatures.

About a third of the doubles combinations will also contain a 1 or a 2 -- 17 from each group of 60 -- 01,02 or 10,20.

So far, the main way I have used this indicator is for doubles tracking.  Since the doubles are divided in half by the position of the zero in the signature, one can eliminate 50% of the doubles combinations for any game where you feel that the zero will fall in the front or back of the signature, and further elimination can be done if you want to include or exclude the '1' or '2' from the distance signature.

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