Newsgroups: rec.puzzles
From: hoey@AIC.NRL.Navy.Mil (Dan Hoey)
Date: 4 Aug 92 12:39:34 GMT
Subject: sod.! (was a series problem and answer)

In response to the series question

>> >3 6 9 27 ?

dwr2...@zeus.tamu.edu (Dave Ring) gave:

>> 3^3^3, whatever that means.

explaining

> the first term '3' is a little ambiguous, but is reasonable.

I have to take exception to that:  One of the rules of the sequence of
arithmetic operations is that

  op[k](n,n)=op[k+1](n,2)

so whatever op[0] is that comes before op[1]="+", it must satisfy
op[0](3,3)=3+2=5, not 3.  Otherwise, I have to grant that Dave made
a valiant try at providing interesting answers to a couple of pretty
mediocre puzzles (and if ``there are no bad questions'' then the
``good'' questions that are bad puzzles belong in rec.questions, not
here.  (No, it doesn't exist.  BFD, CFD.))

Series puzzles are generally pretty lousy, partly because they are so
ambiguous (and for all the people who have asked for the ``simple''
rule I have yet to see a proof of optimum simplicity, or even a decent
attempt to frame a useful measure) and partly because they are usually
followed by a couple of dozen reposts of the audioactive sequence
(series.07 in the FAQ).  But this problem, before it was melted down
into a series puzzle, was actually about a fairly interesting
function, sod.!, where sod.!(n) is the sum of the digits of n!.

So tell me, for how many pairs n > m+1 is sod.!(n) = sod.!(m), or if
infinite, how big can n-m get, or if infinite, how big can n/m get?
Are there any interesting results beyond the fact that for n>5,
9|sod.!(n)?  Is there even a proof that sod.!(n)>9 whenever n>8?

I wish this were a better puzzle, but it's probably just some
questions.  I hope at least they are not quite so trivial, ambiguous,
and trite as the other ones I saw today.

Dan Hoey
Hoey@AIC.NRL.Navy.Mil