## Self-Referential Puzzles

This set is a bit longer than usual. I have compiled several puzzles having a unique quality - the answers affect the answers. For example, if asked:
• "How many letters are in the answer to this question?"
only the spelled number "Four" could be an answer, since it's the only number with the same number of letters as the number ("Five" has 4 letters, "Three" has 5 letters, etc.) Of course "0" would also be a valid answer, since it doesn't include letters.

Spelling puzzles
• How many letters are in the answer to this question? (Answer other than "Four" or "0".)
• There are exactly _____ E's in this sentence.
• There are _____ E's in this sentence.
• This sentence uses _____ E's.
• And, this sentence has _____ E's.
• Can you find a representative "E" sentence for "two" through "twelve"?
• This sentence has _____ letters.
• I hesitate to ask this, but I am giving you three weeks. If only one of a letter exists, the letter should not be plural, as in "one Z", not "one Zs" (I have suggested some letters are singular below.) If some supporting words need to be changed/added/deleted to make it work, please do.

This sentence has _____ A's, _____ B's, _____ C's, _____ D's, _____ E's, _____ F's, _____ G's, _____ H's, _____ I's, _____ J, _____ K, _____ L's, _____ M's, _____ N's, _____ O's, _____ P's, _____ Q, _____ R's, _____ S's, _____ T's, _____ U's, _____ V's, _____ W's, _____ X's, _____ Y's, and _____ Z.

From Philippe Fondanaiche:
• Replace the ? and ?? by the appropiate words.
THERE ARE ? H'S IN THE ?? SENTENCE
THERE ARE ? F'S IN THE ?? SENTENCE
THERE ARE ? T'S IN THE ?? SENTENCE
THERE ARE ? E'S IN ALL THE SENTENCES

? represents any integer number, spelled.
?? is "first" or "second" or "third" or "last"

Digit Puzzles (fill the blanks with digits 0-9, or possibly multi-digit numbers):
• This sentence has __ 0's, __ 1's, __ 2's, __ 3's, __ 4's, __ 5's, __ 6's, __ 7's, __ 8's, and __ 9's.
• This sentence has __ zeros, __ ones, __ twos, __ threes, __ fours, __ fives, __ sixes, __ sevens, __ eights, and __ nines.
From Philippe Fondanaiche:
• Consider the following table:
```      -----------------------------------------
| 8 | 2 | 5 | 9 | 3 | 2 | 1 | 8 | 7 | 2 |
-----------------------------------------
|   |   |   |   |   |   |   |   |   |   |
-----------------------------------------```
In the first row there is the 10-digit number 8259321872. In the second row, the first square contains the digit (< 10) which shows the number of 0's in the 2 rows, the second square contains the digit which indicates the number of 1's in the 2 rows....., the last one shows the number of 9's. Field this table.
• Field the third row of the table whose the 2 first rows are 4942149894 and 1581847443.
• Build the last row of a table having the maximum number of 10-digit rows. [Remember each entry in the last row is a single digit, referring to the entire table.]

Self-Referential Numbers
• 213 can be described as having "One 1, One 2, One 3", or 111213. In a successive list, with each line describing the previous, we get:
213
11 12 13
41 12 13
31 12 13 14
41 12 23 14
31 22 13 24
21 32 23 14
... and the last number (21322314) describes itself.
• What is the largest self-referential number? If you can show they can be arbitrarily large, what is the largest with less than 20 digits?
• Does every list, as above, terminate in a single self-referential number? Or can you find a non-terminating list, or an especially long list?
• Can you find a cycle of numbers (A describes B describes C ... describes A)?

If you find other puzzles of this sort, feel free to submit them. I'll add them when I can.

Source: Unless noted, these are original.

Solution
Mail to Ken