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 six_4off 12-13-2005 04:24 AM

Crossword Puzzle

This came up at a boring shift at work. If there is a crossword puzzle w/ a total of 170 blank spaces for letters, what are the odds of a person filling it in correctly w.out the use of any clues. i figure that the odds of guessing any single letter correctly is 1/25. I guessed (however i am not good at this sort of thing) that the answer would be 26 to the 170th power. (Laugh hard at me if i am wrong which i am sure that i am.) Just wondering if any1 knew the math bbehind it and could explain.

6/4

 Siegmund 12-13-2005 05:23 AM

Re: Crossword Puzzle

It depends how much work you're willing to do to improve your chances. Your answer is right if you randomly guess at every letter.

For a start, you can guess letters in proportion to their normal frequency of use (12% E, then T A O N R I S H downward), and roughly double your chance of getting each letter right. [If you wanted to maximize the number of correct letters, you'd fill in the whole puzzle with EEEEEE, but that would give up any chance of getting the whole puzzle right.]

Or you can go to a frequency table of digrams and trigrams, and fill in all the across clues with vaguely pronounceable-looking strings of letters. This would give you maybe a 7% chance at guessing the first letter of each word right, and then a 15 or 20% chance of getting each additional letter in the word right if you started with the right letter. That puts you at something like (.07)^40*(.15)^130 of getting the whole puzzle right.

You can do much better still if you check the digrams and trigrams in BOTH directions, and have chances approaching 25% of getting any given letter right.

Further improvements are possible if you include information about the general structure of crosswords (lower right corners of subsections will be "S" much more often than in a random letter grid for instance) or if you generated your digram frequencies by analyzing thousands of crosswords instead of using general-usage frequencies.

All of these improvements together ought to push your chances to somewhere in the vicinity of (.40)^170.

 LetYouDown 12-13-2005 11:19 AM

Re: Crossword Puzzle

Siegmund is obviously correct, but to answer your question...if you're truly filling in random letters, your figure is correct. At least in spirit. It does make the assumption that the letters in the puzzle are also randomly distributed and not biased with linguistic trends (such as E being much more common than X).

 12-13-2005 12:22 PM

Re: Crossword Puzzle

Actually, no assumption about the distribution of the letters is needed (besides that they are A-Z and not some specialties). Filling truly random letters guarantees that the events of correctness of a particular letter are independent and each has a probability of (1/26) of being correct.

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