Determine X in each of the following sequences (two from the Titan test, which appeared in Omni magazine,...

[ Editor: from a sci.math thread, December 1993 - January 1994.  The
discussion of whether such problems have well-defined answers is in
series/series.00. ]

Well some of it was hard.  But some of it was easy.  Like the
following:

Determine X in the following sequence:

2,  15,  1001,  215441,  X

Now I thought that this couldn't be solved by a person with an I.Q.
under 170 in less than a week, but I was proved wrong by a person with
an I.Q. of 155 who solved it in 36 minutes or so he claimed.

To revenge himself on me for having frittered away so much of his time
he sent me the following which he concocted while trying an
unsuccessful approach to the first.

Determine X in the following sequence:

99,  120,  143,  201,  X

To which I came up with an ugly solution, which although correct was
not as pretty as the solution that he had intended and involved a
lateral thinking leap.

Most of this Titan test was in fact little but mental gymnastics where
concentration mattered more than intelligence.  But the problems were
amusing and it was good exercise.

However there was one Titan problem which I had great difficulty with,
in fact it took me two years to find a solution to it.  It is stated as
follows:

Determine X in the following sequence:

7,  8,  5,  3,  9,  8,  1,  6,  3,  X

Now after much deliberation I found a solution to it, and the solution
is correct I am sure.

But the problem was that the solution was rather ugly and not
mathematically pleasing in the way that all the other solutions to the
Titan test numerical problems were mathematically pleasing.

-- Árni Thoroddsen <arnith@ismennt.is>

----------

>2,  15,  1001,  215441,  X

Easy.  X = 95041567.  This took 2 minutes; I used the unix program
'factor' but only to confirm my suspicions.

>99,  120,  143,  201,  X

I'll guess 243, but the pattern is not so nice.  99, 120, 143, 168
would make much more sense if the answer is along the lines of what
I'm thinking.

>7,  8,  5,  3,  9,  8,  1,  6,  3,  X

X = 3.  I admit I used 'bc' to find this one.  Here's a counter-
puzzle: 7 8 5 5 3 4 4 6 9 7 8 X.

-- Daniel S. Rice <rice@enterprise.Berkeley.EDU>

----------

I just solved it in under 30 seconds.  What does that make my IQ?

        2               = 2
        15              = 3 * 5
        1001            = 7 * 11 * 13
        215441          = 17 * 19 * 23 * 29
        95041567        = 31 * 37 * 41 * 43 * 47

OK, OK, you caught me, I used "primes" and "factor".  So, lower my IQ
20 points for this.

-- Gene Stark <stark!gene@newsserv.cs.sunysb.edu>

----------

rice@enterprise.Berkeley.EDU (Daniel S. Rice) writes:
>       Easy.  X = 95041567.  This took 2 minutes; I used the unix
>program 'factor' but only to confirm my suspicions.

Ok, but a lot of people get lost going down wrong deductive paths.

>       I'll guess 243, but the pattern is not so nice.  99, 120, 143,
>168 would make much more sense if the answer is along the lines of
>what I'm thinking.

Wait a minute.  Do we have a guesser here?

Illuminate us as to the deduction you used.

>       X = 3.  I admit I used 'bc' to find this one.

What is 'bc'?  Pardon my ignorance.  But what I am really interested in
is the deductive path.  I considered this kind of solution but rejected
it on the ground that there are so many interesting real numbers, that
the testers could not have meant for the search space to be so large.

Actually this is quite interesting since my solution calculated the
sixth digit from the first five numbers, the seventh number from the
next five and so on.

Does this mean that there is a memnomonic formula for PI/4?

-- Árni Thoroddsen <arnith@ismennt.is>

[ Editor: pi/4 = 0.7853981633..., whose digits give the sequence and
X = 3. ]

----------

In article <31608@rvik.ismennt.is> arnith@ismennt.is (Arni Thoroddsen)
writes:
>But what I am really interested in is the deductive path. I considered
>this kind of solution but rejected it on the ground that there are so
>many interesting real numbers, that the testers could not have meant
>for the search space to be so large.

I disagree with this.  There are not all that many "interesting" real
numbers.  The kind of real numbers that show up on a test are almost
always going to be related in some simple way to pi, e, phi, or
rational numbers, e.g., sqrt(2), e^(1/e), pi*e.  You're not going to be
expected to recognize the decimal expansion of
sin(e*tan(gamma)-27/97).  On the other hand, this does illustrate how
solving puzzles like this depends a lot on your having the same kind
of experience with mathematics as the testers.

>Actually this is quite interesting since my solution calculated the
>sixth digit from the first five numbers, the seventh number from the
>next five and so on.
>Does this mean that there is a memnomonic formula for PI/4?

Please post!  I suspect that your rule will eventually generate a digit
different from the expansion of PI/4, but I'm sure many people would be
interested in seeing it.

-- Tim Chow <tycchow@math.mit.edu>

----------

Obligatory puzzles:

What is the next number in the following sequence:

3, 2, 7, 11, 2, 29, 47, 19, 41, 199, 2, 521, 281, 31, 2207,  ....

What is the next number in the following sequence:

5, 3, 11, 43, 683, 2731, 43691, 174763, 2796203, 3033169, .....

-- Bob Silverman <bs@gauss.mitre.org>

[ Editor: no solutions to Rice's counter-puzzle, to Silverman's two
sequences, or to the intended rule for 99, 120, 143, 201 were found in
the collected postings. ]