Puzzle in Real Life: 4 Letter Cubes

[impossibleroot]

It's been years since I was a regular poster and puzzler here (Diagram Puzzles, anyone?), but I was presented a challenge recently and knew where my first stop would be for help.

I teach, and one of my friends on campus is the art teacher. He's built a large outdoor installation of cubes, arranged horizontally along a metal bar, which allows four faces of each cube to be seen when spun. He paints images on each of the four faces, allowing people on campus to rearrange the artwork at will.

Last night, he talked to me about his next plan: painting fanciful letters on each cube, allowing people to make word combinations. So the question is this:

Is there a set of 16 letters that would allow a valid four letter word (not that kind!) to be visible given any random spin? If so, the answer could be presented as a list of 4 words, where any letters within a column are interchangeable.

If it's not possible, can it be proven that it is not? If it is possible, what word combos work? Bonus points for the most letter variation between the cubes (as in the fewer repeated letters, the better).

I do not know the answer to this, either way, but I'm working on it. I have a list of Scrabble-fair 4-letter words and a frequency count in Excel (second letter should probably be a vowel).

If you help with a strategy or get a working answer, I'd be happy to take pictures of the resulting piece and send them your way or post them! Help me, Grey Labyrinth, you're my only hope.

[L'lanmal]

256 words is pretty ambitious... though you might find something from people who create word-ladder puzzles.

It seems you might be able to create a set of 8 letters (2 in each location) that would work, and have your pieces be two-sided. Or perhaps design letters which have multiple interpretations. (Like WOOP might be a combination which is debatedly not a word, but it spells dOOM when turned upside-down.) Or one of your choices could be a blank, creating three-letter words if selected, rather than four.

First attempt: B/H A/? N/R D/K.

[mith]

~wave~

It's certainly possible with repeated letters on a single cube, obviously. It's probably possible with two different letters per cube; I'm pretty sure it's not possible with each cube having four different letters (though I haven't done anything involving a word list to check either).

Probably wouldn't be too difficult to write a program to look for a combination of letters that gives the most valid words out of the 256 spins; or to look for a combination with repeats that always gives valid words and gives the most. (Which isn't to say I could write one quickly, if I even had the time. But which of those solutions would you be more interested in?)

[mith]

Here's a 8 letter/16 word solution (though a couple are obscure):

C/T, A/O, P/R, E/S

[Zag]

It is clearly possible, if you are allowing duplicates, though it's not clear that there is an interesting solution. (You could always make all four of every block the same.)

Here's my best try, in 10 minutes.

[Zag]

That was simulposted with the others. Looking at mith's, I think that C instead of F produces a few more words.

[Zag]

[impossibleroot]

Wow, you guys rock!

I will pass these along to see what he thinks. Now that I look at it, non-obscure words would probably be preferred over non-repeating letters.

In the meantime, I wrote a simple program with a 4 step nested 4 loop and arrays with realistic letters. I still have to compare the output to a word list, but I'll hopefully be able to compile a decisive list of options.

Keep 'em coming if anyone has 'better' solutions, and thanks to all.

[impossibleroot]

So, Zag, you're pretty convinced that a full 256 total word solution is possible? Is this your gut instinct or do you have a logical reason for it? I only ask because I *don't* have a sense for it -- I feel like it'd be easy to get pretty close (as you demonstrated), but I have no idea how hard it might be to get a real, full solution.

[Zag]

No, I was saying that, with duplicates, it is possible to make it so that any way the blocks were spun, it would be a word. But it might not be very interesting. This set, for instance, makes a word in every configuration. But I think you'll agree that there are too many duplicates, and the arrangement I suggested above is more interesting.

[impossibleroot]

Ah, gotcha. Well, I'll pass it along to the artist, to see if he really want it all to work, regardless of duplicates, or if he's okay with many interesting words with some non-working combos.

I'm going to keep looking for the perfect solution, though, if it exists.

[impossibleroot]

Here's a shot of the cubes as they are now, for reference. They're big.

[PuzzleScot]

Grepping my mwords list, I can increase the word count from 125 to 140. this now gives a probability > 0.5 of a valid word showing after any random spin of the 4 cubes. (Unfortunately it has two rude words, and would consequently be the ones left displayed 99% of the time!)

[Pigboy]

you seem to have gone for CVC starts, are these more common than VCV?

[PuzzleScot]

by nearly 10:1, yes.

CVC* 3099
VCV* 331

CCCC 47
VCCC 141
CVCC 2044
CCVC 628
CCCV 73
VVCC 76
VCVC 288
VCCV 126
CVVC 797
CVCV 1055
CCVV 85
VVVC 5
VVCV 24
VCVV 43
CVVV 21
VVVV 1 (EOUI, by the way)

Note that I'm not claiming my previously quoted solution is the optimal configuration. That would need a fairly exhaustive check of most possible configurations.

[PuzzleScot]

A semi-intelligent search produced [PMSD][AEIO][RNLC][TESK] for 167 also. Seems to be the limit, or very near it anyway...

[Pigboy]

[impossibleroot]

Superb, thank you all!

Now, how about a second part? What if we put two letters on each face instead of one? Would it then be possible to cover all 256 spin possibilities with at least one legitimate 4 letter word?

I'm thinking that each face could be divided with a horizontal or diagonal line, with a letter on either half. That way, people interacting with the piece could play a kind of 'find the word' game.

In other news, I'm also planning on passing along another design using numbers and mathematical operations that will generate numbers. I'll try to come up with a way to make these numbers into a code that could be deciphered by the students for a prize or award of some sort. When I come up with something, I'll post it here for testing by the puzzle experts...

[ralphmerridew]

My program / word list didn't find any:

[impossibleroot]

But zero complete solutions, eh? Well let me put this out there, then:

If *you* had say over this 4 cube setup and wanted to put something interesting with letters or words on the 16 faces, what would you do? Bonus points if it's some kind of puzzle situation (fairly straightforward), but any elegant, letter-based ideas would be welcomed and quite possibly used. Word game creators, I know you're out there!

[Scurra]

I have to confess that I would probably try to do the reverse of this puzzle - which is to put a set of letters on the cubes that couldn't make any valid four-letter words but which would look as if they could... (so including vowels.)

I wonder if this is harder or easier than the original challenge?!
_________________
_{still Quiz Olympiad champion. Must get a life.
New definitions: COFFEE - someone who is coughed upon}

[mith]

The first extension that came to mind for me was "How many valid 4-letter combinations are there if we can reorder the dice?".

(For example, in PuzzleScot's example PIRE is not a legal word but would be a legal combination since RIPE is a word.)

[Courk]

Pire might not be a word, but it is the name of a Nobel Peace Prize recipient.

[impossibleroot]

Ah, good one, Mith. So we could come up with a complete working set given anagrams -- I'm sure that's possible. Back to the programming! I like that idea.

[ralphmerridew]

My program failed to find any 3 on a cube answers, even allowing anagrams.

I found plenty of full 2-2-2-2 (16) and 3-3-3-1 (27) answers, but no 3-3-3-2.

I did find a handful of 3-3-2-2 (36) answers
[(ao)(ct)(ers)(dnp)]
[(dst)(ao)(mnp)(es)]
[(bmp)(ai)(dlr)(es)]
[(dst)(aio)(mn)(es)]
[(mnp)(aio)(ds)(es)]
[(fmp)(aei)(rs)(et)]
[(fs)(lr)(eio)(aet)]

[ralphmerridew]

Okay, I relaxed the restrictions a bit:

At most four letters on each die, and more than 36 words, and this jumped out:
48: [(aio)(cdls)(mnpr)(e)]

Any number of letters per die, at least 50 words
50: [(ai)(cmptw)(e)(dlnrs)].
50: [(cdety)(rs)(a)(bmntw)].
54: [(s)(aeo)(fglmnprtw)(et)].
54: [(s)(ept)(aio)(lmprtw)].
56: [(s)(aeio)(flmnprt)(et)].
56: [(s)(aeio)(flmnprw)(et)].
56: [(s)(aeio)(flmnptw)(et)].
56: [(s)(aeio)(flmnrtw)(et)].
56: [(s)(aeio)(flmprtw)(et)].
56: [(s)(aeio)(flnprtw)(et)].
56: [(s)(aeio)(fmnprtw)(et)].
56: [(s)(et)(aeio)(lmnprtw)].
64: [(s)(aeio)(flmnprtw)(et)].

At least two letters per die, more than 36 words:
40: [(io)(cmnrs)(es)(dt)].

[impossibleroot]

Ralph, you said you found plenty of full 2-2-2-2 solutions? Do you mean a complete set of 256 solutions using anagrams with 2 letters per face? If so, I am very interested in those... but perhaps I'm not understanding you.

[PuzzleScot]

I think that 'letters per face' and 'letters per cube' have been mixed up along the way somewhere...

[ralphmerridew]

My name is "ralphmerridew", all lowercase. If you must abbreviate, use "rm".

[Zag]

As a "customer" of this installation, I can tell you that I would not really consider anagrams. Also, having more than one letter per face would seriously detract from the effect. I would much rather that you go for the maximum number of combinations that work, on a setup where every face has a single letter and no one die has any duplicates.

I think that there was a solution above where half of the combinations were real words. This is easily enough -- I think that 1/3 would be plenty -- for a fun thing that will get plenty of attention.

[impossibleroot]

Sorry about that, ralphmerridew.

Zag, I'm now leaning your way, as is the artist. Single letter per face, lots of working solutions, plenty of non-words that *could* be words. Maybe I'll give a prize to the person who figures out the real total number of words...

[PuzzleScot]

http://en.wikipedia.org/wiki/Sowpods

"SOWPODS is the word list used in tournament Scrabble in most countries except the USA, Thailand and Canada."

I used sowpods.zip from http://www.isc.ro/lists/

[ralphmerridew]

sowpods is a bigger list than /usr/share/dict/words. I wrote a genetic algorithm to maximize number of words with four dice / four letters, and it usually topped out at 115 words; I switched to sowpods and it quickly found two 171 word answers:

([d, m, b, p], [o, i, a, u], [n, r, s, l], [s, k, t, e])
([d, m, b, p], [o, i, a, u], [n, r, s, t], [s, k, t, e])

[PuzzleScot]

nice one... one ruder word exists there though...

[Zag]

Hmmm. I was just considering that this has two sides. If one is looking at it from the other side, all the letters will be upside down. If the artist is clever with upper/lower case and tweaking the letters a bit, he could have even more words, when you count looking at it from both directions. So, from rm's best list below:

B, O, I, S, K, E can remain the same.

d and p become each other
M becomes W
A becomes V (hard to use, though, where it is)
u and n become each other. Or N becomes itself and we don't use U
T and L can become each other by making the T a little lopsided

R is a problem.

rm, what's the best you can do if you count both sides? and avoid the letters that don't work?

[ralphmerridew]

Could you give an exact list of which mapping you want me to use, in the form

abcdefghijklmnopqrstuvwxyz -> vb...
For letters which do not invert, give an underscore.

[Zag]

Hmmm. I originally wrote this thinking of a mirror flip, rather than a rotational flip. (Or, more accurately, I was mixing both, taking whichever was most useful.) But, sadly C, E, K, and uppercase D really don't flip to anything useful. I'm assuming that B can be drawn such that it still looks like a B when upside down. (upper left and lower right are sharp corners, upper right and lower left are rounded corners. It'll work.)

Strictly upper case.

[ralphmerridew]

Out of curiosity, I decided to see how much the anagrams allowed answers improved when I used SOWPODS. I found 79 different 4-3-3-3 answers.
108: [(aeo)(lrs)(fgkm)(aeo)].
108: [(bfgk)(aeo)(aeo)(lrs)].
108: [(bfgm)(aeo)(aeo)(lrs)].
108: [(bfkm)(aeo)(aeo)(lrs)].
108: [(bgkm)(aeo)(aeo)(lrs)].
108: [(bmn)(aego)(aio)(lrs)].

Will work on "inversions allowed" next.

[ralphmerridew]

First pass with strict uppercase, I got
([K, E, l, s], [l, n, R, s], [U, o, i, v], [w, t, P, b]) -> 180
([m, b, l, p], [i, u, o, a], [t, n, r, s], [k, s, t, e]) -> 180
("K" is an inverted "k", and is nothing normally.)

[mith]

Aren't those essentially the same solution?

Would be pretty cool if there was an optimal solution that needed an inverted letter, though.

(For the cubes the puzzle is based on, allowing anagrams doesn't make sense - they can't easily be rearranged. But if we're allowing inversions *and* anagrams, orientation don't really matter for any practical application. Perhaps try it where an "A" can either be an A or a V, whichever is needed. As is, if you had two cubes with A and T on them, you can only spell something with A+T or L+V, but not V+T or A+L. Would be a bit surprised if it gained you more than a word or two, but would be interested to see if it even does that.)