I came across this number puzzle (pasted below) a few years ago, and
despite my best efforts have never been able to come up with the
solution.  I've shown it to many people and it has stumped all of
them.  I'm convinced that finding the solution involves looking at the
numbers in some sort of unconventional way (that is, not just using
standard mathematical operations).  This has bugged me for years, and
there must be someone out there who can see a pattern in these
numbers.  Please help!  Here is the puzzle with its original wording:

Fill in the blanks with the five one-digit numbers that will correctly
compete the series (note the semicolons, which divide the numbers into
groups):

3,6; 1,3,4,5,7; 1,3,4,6,7; 2,3,4,6; _,_,_,_,_; 1,2,4,5,6,7; 1,3,6

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Request for Question Clarification by rainbow-ga on 02 May 2003 07:02 PDT

Hi floridaguy,
I have located the answer (the 5 missing digits) but cannot explain
the reasoning. :)
Someone who understands Pascal and polish may be able to work out the
logic in it though. If giving you the missing digits and where I
located it is enough to post as an answer, let me know. Thanks.
Regards,
rainbow-ga

Hi, floridaguy-ga:

The missing group of digits is 1,2,4,6,7.

These represent the line segments of a "seven segment" display which
need to be on, in order for the digit five to be displayed.

If you think about digital clocks and similar instrumentation, the
display of numbers (e.g. in an elevator) is often reduced to seven
line segments (three horizontal and four vertical).  If all were
turned on, the pattern would look like this (numbering of segments to
one side):

 XXXX            1
X    X
X    X       2       3
X    X
 XXXX            4
X    X
X    X       5       6
X    X
 XXXX            7

i.e., a figure eight.  But selectively turning on these segments
allows for all ten digits to be shown.

The groupings given in the statement of your question are those for
digits one through seven, with the grouping for digit five missing.

The "clue" found by GA Researcher Rainbow-ga and independently by
Leli-ga (who first brought this problem to my attention) is a
newsgroup posting in Polish which has a Pascal code fragment for
displaying game scores.  The code given there sketches a way to break
a four digit number into its individual digits and then to "draw" the
line segments appropriate to displaying each:

http://groups.google.co.uk/groups?q=%223,6%3B+1,3,4,5,7%3B+1,3,4,%3B+%22&hl=en&lr=&ie=UTF-8&oe=UTF-8&selm=9invie%24lof%241%40news.tpi.pl&rnum=1

The solution to this puzzle was independently found by Eiffel-ga as
well.  Good researchers all!

regards, mathtalk-ga

That's it!  Thank you.

My first thought was thought the numbers are code for letters and the
sequence spells out a common phrase. (But it's not the letters and
numbers from a phone because 1 doesn't have any letters)

Very nice job, mathtalk-ga.

paf