Tuesday, February 14, 2006

How Many Letters?

I found this riddle on the net: How many letters does the answer to this question have?

It is kind of a strange question, isn't it? If you want a chance to solve it for yourself you should stop reading now!








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I would say the cool thing about this question is that it supposes a test to see if it is right. The question starts with 'How many' so the answer must be a number (perhaps 18), or at least describe a value (like 'too many' or 'none'). So then the test is wether this description (assume a number also describes a value) matches up with the number of letters in that description. It is not so in English but there are likely languages where such a description does not exist. Also not in English but there are likely languages where there is more than one such description. In English there is exactly one.

We can test the answers proposed so far, 'too many', 'none' and 18.

'too many' -> 7 letters
'none' -> 4 letters
18 -> eighteen -> 8 letters

As we can see none of these descriptions match the number of letters it takes to write them and so none of them fit. But something strange happens if we recursively take the value of letters as a new description to be tested. We can even start with an answer that is completely absurd and it will take us to the correct one. Suppose we try to answer 'How many letters does the answer to this question have?' with something absurd like:

'The quick brown fox jumps over the lazy dog.' -> 35 letters
thirty five -> 10 letters
ten -> 3 letters
three -> 5 letters
five -> 4 letters
four -> 4 letters! We got it.

'Four' is a description of a number that takes exactly that number of letters to write (in English). I have also proven to my satisfaction that all descriptions will settle on four when applied to this algorithm.

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UPDATE: 18 Feb 2006

On closer examination it is fairly simple to prove that all numbers must settle on four. The theoretical possibilities are limited to the numbers settling on a number other than 4(1), finding a loop(2), growing infinitely small(3), growing infinitely large(4), perhaps continuing in some other irrational way(5) or settling on 4(6).

Observation 1:
All numbers greater than 4 point to a number smaller than itself.

Observation 2:
All numbers less than 4 point to a number greater than itself.

Observation 3:
3 is the only number less than 4 that points to a number greater than 4, namely 5, which points to 4.

Observation 4:
Letter counts are positive integers.


Possibility (1) cannot happen because of observations 1 and 2.
Possibility (2) cannot happen because of observations 1, 2 and 3.
Possibility (3) cannot happen because of observation 4.
Possibility (4) cannot happen because of observation 1.
Possibility (5) cannot happen because of observations 1, 2, 3 and 4.
Possibility (6) is all that remains.

4 comments:

  1. Oh Moe! You know other people can read this. Now it is on the record. What do you think this is going to do to my egomania?

    Thank you.

    ReplyDelete
  2. Heh, you lack some imagination. :) Here are some more perfectly valid answers:

    * Exactly ten
    * Five plus eight
    * One less than twenty-one

    There are also inexact ones, they are valid answers but don't specify a clear number:

    * Less than two hundred
    * More than one

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  3. This comment has been removed by the author.

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