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        riddles >> medium >> Find the Missing Number
        
(Message started by: shawnG on Aug 20th, 2005, 6:35pm)
Title: Find the Missing Number
Post by shawnG on Aug 20th, 2005, 6:35pm
Can you find the missing number in the following sequence?

10, 11, 12, 13, 14, 15, 16, 17, 20, 22, 24, 31, 100, ???, 10000
Title: Re: Find the Missing Number
Post by JocK on Aug 20th, 2005, 7:05pm
I go for [hide]121[/hide] ...

and the next number (after 10000) will contain 16 identical digits...  8)
Title: Re: Find the Missing Number
Post by Icarus on Aug 22nd, 2005, 3:39pm
OUt of curiosity, Jock: did you recognize it yourself, or did you solve it by  The On-line Encyclopedia of Integer Sequences (http://www.research.att.com/~njas/sequences/)? (A very useful tool.)
Title: Re: Find the Missing Number
Post by JocK on Aug 22nd, 2005, 4:07pm

on 08/22/05 at 15:39:43, Icarus wrote:
OUt of curiosity, Jock: did you recognize it yourself, or did you solve it by  The On-line Encyclopedia of Integer Sequences (http://www.research.att.com/~njas/sequences/)? (A very useful tool.)


I'm sure we did the same...  ;D
Title: Re: Find the Missing Number
Post by JocK on Aug 22nd, 2005, 4:10pm
... which - I guess - is what shawnG did...  ;)
Title: Re: Find the Missing Number
Post by Icarus on Aug 22nd, 2005, 4:50pm
Obviously I did. I don't enjoy what amounts to "guess what I'm thinking" puzzles that much (a generalization - a few I have found interesting). "Find the missing term" problems are generally this sort, so having a tool to locate the answer quickly is nice.
Title: Re: Find the Missing Number
Post by JocK on Aug 23rd, 2005, 4:25am

on 08/22/05 at 16:50:50, Icarus wrote:
Obviously I did. I don't enjoy what amounts to "guess what I'm thinking" puzzles that much (a generalization - a few I have found interesting). "Find the missing term" problems are generally this sort, so having a tool to locate the answer quickly is nice.


I agree. It is very difficult to construct such puzzles that are both non-ambiguous, and at the same time non-trivial.
Title: Re: Find the Missing Number
Post by rmsgrey on Aug 23rd, 2005, 7:01pm
Myself, I recognised the sequence after looking at JocK's initial response. I agree with Icarus that "missing number" type puzzles (of which Mensa seems inordinately fond) generally reduce to "guess which of several possible answers I had in mind" which is rarely satisfying.

For instance, one puzzle magazine (I can't remember which) ran a number of sequences where the desired function in each case was of the form:

f(n) = anf(0) + bn

with only 5 terms given. If you knew the form they were looking for, then it was just a matter of finding the constants. If you weren't familiar with the way the puzzle setter thought, then it was effectively impossible
Title: Re: Find the Missing Number
Post by SWF on Aug 24th, 2005, 5:30pm
What I am thinking is that the missing number is 883. The n'th number is given by rounding the following to the nearest integer:

8.6 + 1.1*[ n + f( n-8 ) ] + f( 11.68n / 252.75 - 1 )

f(x) is the function that truncates negative numbers: f(x) = x if x>0, and is 0 otherwise.
Title: Re: Find the Missing Number
Post by JocK on Aug 25th, 2005, 3:31pm
rite SWF...! ;D

Someway I think this riddle represents a missed opportunity. Would a riddle along the same line have been posted, such as:

??, 1001, 100, 21, 14, 13, 12, 11, 10,  9, 9, 9, 9, 9, 9 ...

I would have found it a much more elegant problem. (Perhaps a personal preference?)

Related to this: didn't you guys notice the obvious error in the above and all the related sequences in the OLEIS?  8)
Title: Re: Find the Missing Number
Post by shawnG on Aug 27th, 2005, 1:26pm
Jock - I can certainly understand your disdain for these types of puzzles.  I have sort of a soft spot for number sequences, so I enjoy it.  I in fact, never looked it up on OLEIS - although I would have if it had taken me longer.

One thing I like about these types of puzzles is the ambiguity.  For instance, how many people would have come up with SWF's answer? Nice job.

Anyway, I'm new here and this was the shortest puzzle I could think of to post as a first.

BTW, this is a great place. :)
Title: Re: Find the Missing Number
Post by JocK on Aug 27th, 2005, 2:01pm
Welcome..! This forum is indeed a great place.

(As long as you don't let grumpy riddlers like me put you off..  8) )

By the way: I certainly don't have a disdain for guess the next number puzzles it is just that I don't like spending time on puzzles if I don't have the feeling that some unique solution is guarenteed to exist.

Title: Re: Find the Missing Number
Post by Icarus on Aug 28th, 2005, 3:35pm
Indeed, ShawnG, don't let any of our comments here discourage you! Just because I don't particularly care for a type of puzzle doesn't mean that there aren't others around who enjoy them.

My problem with ambiguous puzzles is that usually the poster is demanding a particular solution for them, and you often find yourself tossing out at random various solutions while the poster keeps shooting them down with out giving any further hints as to what they are thinking of, while acting smug at how they've "stumped" you.

But this obviously does not apply to you and this puzzle. In particular, you've shown appreciation for SWF's alternative solution, whereas the sort of poster I described gets upset when people post alternate solutions after the "correct" one is revealed (read around awhile and you will come across several threads wherein posters - sometimes the original, and sometimes others - get downright nasty because someone posted a different solution). This is particularly tiresome, as one of the favorite pastimes of many regulars is finding alternate solutions (SWF is king at this).

Since you are not one of this posters of limited imagination, please feel free to post any puzzle that you find intriguing, without worrying what anyone else thinks. If you found it intriguing, I promise you that others will as well.


on 08/25/05 at 15:31:07, JocK wrote:
Related to this: didn't you guys notice the obvious error in the above and all the related sequences in the OLEIS?  8)


I'm afraid that I don't see it. The closest thing I can see is that the sequence starts at n=0 instead of the more customary n=1, but I don't consider that an error.
Title: Re: Find the Missing Number
Post by JocK on Aug 28th, 2005, 4:54pm

on 08/28/05 at 15:35:41, Icarus wrote:
I'm afraid that I don't see it. The closest thing I can see is that the sequence starts at n=0 instead of the more customary n=1, but I don't consider that an error.


Well, a number in base-b should be written using the symbols 0, 1, 2, .., b-1 isn't it..?  ;)
Title: Re: Find the Missing Number
Post by JocK on Aug 28th, 2005, 4:56pm
So it was very prudent of ShawnG not to write down the 16-th term in the sequence...
Title: Re: Find the Missing Number
Post by towr on Aug 29th, 2005, 12:10am

on 08/28/05 at 16:54:26, JocK wrote:
Well, a number in base-b should be written using the symbols 0, 1, 2, .., b-1 isn't it..?  ;)
Except for base 1 , yes.. Otherwise every number (except 0) would have to be represented by an infinite numbers of zeros (or in other words can't be represented).
And the usual calculation works fine if you use 1's.

dn*bn+ .. +d2*b2+d1*b1+d0*b0
Title: Re: Find the Missing Number
Post by JocK on Sep 1st, 2005, 12:19pm

on 08/29/05 at 00:10:42, towr wrote:
Except for base 1 , yes.. Otherwise every number (except 0) would have to be represented by an infinite numbers of zeros (or in other words can't be represented).
And the usual calculation works fine if you use 1's.

dn*bn+ .. +d2*b2+d1*b1+d0*b0


True, but one could as well argue that base-1 is simply a tally system. To keep consistent with the generally accepted base-b notation, a number (tally) in base-1 should consist of a sequence of zero's.

In any case, I would argue that base-1 is ill-defined. Attempts that try to incorporate base-1 notation using the symbol "1" into a generic base-b approach do exist: http://my.tbaytel.net/forslund/rrf01.html  but are unlikely to be taken seriously... ::)
Title: Re: Find the Missing Number
Post by Grimbal on Sep 1st, 2005, 1:55pm
There are more exotic bases than those using digits 0 to b-1.

You could use base -2 with symbols 0 and 1 to get rid of the annoying + or - sign.
For example 1101 is 1*(-2)3 + 1*(-2)2 + 0*(-2) + 1 or -3.
1 + 1 = 110 = 4-2 = 2 which means you have to carry over to the next 2 digits.
10 = -2
110 + 10 = 120 (2 becomes 110 and 11 adds to the 1 in front) 120 = 1200 = 12000 etc. which is zero!

There is also the famous base (i-1), where i = sqrt(-1).
Symbols are 0 and 1.
If you take 11001, for instance, it means
1*(i-1)4 + 1*(i-1)3 + 0*(i-1)2 + 1*(i-1) + 1 which amouts to 2i-1.

In this way, you can express not only negatives, but even complex numbers (gaussian integers, or even any compex numbers assuming you use the - in this respect ill-named - decimal point) as a single string of digits.  You don't need to add differently depending if the numbers are positive or negative, and you don't have to combine multiplications and additions as in the formula (a+bi) * (c+di) = ac-bd + i(ad+bc).  A complex multiplication is just multiplying one number by the digits of the other (0 or 1) and add the columns.

Let's compute 2x2 = 1100 * 1100

     1100
     1100
     ----
     0000
    0000
   1100
  1100
  -------
  1210000 (2 becomes 1100)
111010000 which is 4.


As you can see, it is very simple...  :-[


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