## No Words Allowed

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### No Words Allowed

Here's a puzzle I thought up the other day that I don't know how to answer:

What's the longest string of letters you can make, where no letter can be used more than once, where no part of the string can be rearranged to make a word?

So you couldn't have "abcdefghijklmnopqrstuvwxyz" because you could rearrange the "b", the "d", and the "e" to make "bed".

I can reasonably assume that the longest string with no vowels is 26, minus the 5 vowels, - 1 (the letter "y" for words like "cry"), which is 20 characters. But what about with one, two or three vowels? ("a" or "i" cannot be included as they make words by themselves).

0 vowels - bcdfghjklmnpqrstvwxz - 20 characters
1 vowel - ?
2 vowels - ?
3 vowels - ?

What do you guys think?
Last edited by RubbishyUsername on Thu Dec 22, 2011 1:58 pm UTC, edited 1 time in total.

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### Re: No Words Allowed

0 vowels - bcdfghjklmnpqrstwxz - 19 characters

Cwm.

xkcdfan

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### Re: No Words Allowed

I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson

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### Re: No Words Allowed

Arguably I should restrict it to simple English Words - I don't really think "tsktsk" is a word. (Sorry)

And according to that Wikipedia page, "cwm" is more often spelt "coombe" in English as opposed to Welsh.

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### Re: No Words Allowed

How about "hmm"? That one sees common usage. I would also include "nth" as a common word, but maybe that's just my background biasing me.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson

skeptical scientist
closed-minded spiritualist

Posts: 5920
Joined: Tue Nov 28, 2006 6:09 am UTC

### Re: No Words Allowed

I can reasonably assume that the longest string with no vowels is 26, minus the 5 vowels, - 1 (the letter "y" for words like "cry"), which is 19 characters.

0 vowels - 1b2c3d4f5g6h7j8k9l10m11n12p13q14r15s16t17w18x19q20z - 19 characters
1 vowel - ?
2 vowels - ?
3 vowels - ?

What do you guys think?

I think your math-fu needs work All joking aside, with 1 vowel you already need to row just about the entire alphabet out.
Let's take U as only vowel. We'll need as many 1 syllable words as possible, with as many different consonants as possible..
run, buck, gull, null, full, (l would be good to get rid of) fun, buff, cup, cut, put, bud, mud, gun, gut, hull, hut, hug, jug, just,
My english vocabulary isn't strong enough to make all mutually exclusive letters... And I think this will boil down to good scriptwork.

t1mm01994

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### Re: No Words Allowed

Somehow, you have 2 "q"s in there. So yes 19. And although you are right that we will lose most of the string in the first vowel, by just removing "l" and "t" from you list of examples we remove nine of the words. In fact, it might be sensible to start off with the letters that don't form two letter words and work up.

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### Re: No Words Allowed

Two "q", but no "v", and 26-5-1=20 is right.
Including vowels kills the string - you have to remove at least one consonant for each new vowel (realistic: much more than 1):
"us" => include u, remove s
"on" => include o, remove n
"be" => include e, remove b
and a and i cannot be included.

To make things more interesting:
What is the longest string of letters without repetition, where no substring can be reordered to form a complete word (and nothing else)?

That means that bcdfghjklmnpqrstwxzu (21 letters) is fine, as long as "zu", "uz", "zux", "xuz", "wxzu", "uwxz" and so on are not words.
"a" and "i" are excluded with that rule, too, but I think 24 are possible.

How many strings of maximal length exist?
mfb

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### Re: No Words Allowed

OK, you're right about the 20 character thing. But remember that sometimes removing one consonant for a vowel eliminates far more words than another consonant - consider "but", "tut", "nut", "hut", "put" and "cut": you don't have to remove the letters "b", "n", "h", "p" and "c" to eliminate all of those words, just "t".

Also, you're subcase entirely relies on the position of your vowel. By putting it among letters that don't appear often, you can have much longer strings as your word not only has to use all the letters, but some of those letters are things like q and z. Consider the two vowel set of strings with "i" and "u" - you are (I'm sure almost) completely unable to use "i" with "q" until you eventually find "u" - which you've placed at the other end of the string.

Assuming I haven't misunderstood anything.

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### Re: No Words Allowed

"qi" is actually a word (alternate spelling of chi).
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson

skeptical scientist
closed-minded spiritualist

Posts: 5920
Joined: Tue Nov 28, 2006 6:09 am UTC

### Re: No Words Allowed

Hrmm... Would it not be possible to make a program to definitively answer this, given a hypothetical virtual dictionary?
I'm somewhat more interested in the answer than the exercise <.<
aerion111

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