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 Puzzle in Real Life: 4 Letter Cubes Goto page 1, 2  Next
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impossibleroot
Hi-Keeba!

 Posted: Sat Sep 26, 2009 1:04 am    Post subject: 1 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
Daedalian Member

 Posted: Sat Sep 26, 2009 1:24 am    Post subject: 2 256 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
Pitbull of Truth

 Posted: Sat Sep 26, 2009 1:33 am    Post subject: 3 ~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
Pitbull of Truth

 Posted: Sat Sep 26, 2009 1:39 am    Post subject: 4 Here's a 8 letter/16 word solution (though a couple are obscure): C/T, A/O, P/R, E/S
Zag
Unintentionally offensive old coot

Posted: Sat Sep 26, 2009 1:41 am    Post subject: 5

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.

 Code: P  A  T  S B  E  N  T W  I  S  H F  O  R  E

pats, path, pate, pans, pant, pane, pass, past, pars, part, pare
pets, pete, pens, pent, pest, pers, pert, pere,
pits, pith, pins, pint, pine, piss, pish,
pots, pone, post, posh, pose, port, pore
bats, bath, bate, bans, bane, bass, bash, base, bars, bart, bare,
bets, beth, bens, bent, bene, bess, best, bert, bere
bits, bite, bins, bine, bire,
bons, boss, bosh, bose, bore
watt, wets, wens, went, west, were,
wits, with, wite, wins, wine, wist, wish, wise, wont, wort, wore
fats, fate, fete, fens, fits, fite, fins, fine, fist, fish, firs, fire, font, fors, fort, fore
Zag
Unintentionally offensive old coot

 Posted: Sat Sep 26, 2009 1:44 am    Post subject: 6 That was simulposted with the others. Looking at mith's, I think that C instead of F produces a few more words.
Zag
Unintentionally offensive old coot

Posted: Sat Sep 26, 2009 1:51 am    Post subject: 7

 Code: P  A  T  S B  E  N  T W  I  S  H C  O  R  E

pats, path, pate, pans, pant, pane, pass, past, pars, part, pare
pets, pete, pens, pent, pest, pers, pert, pere,
pits, pith, pins, pint, pine, piss, pish,
pots, pone, post, posh, pose, port, pore
bats, bath, bate, bans, bane, bass, bash, base, bars, bart, bare,
bets, beth, bens, bent, bene, bess, best, bert, bere
bits, bite, bins, bine, bire, bons, boss, bosh, bose, bore
watt, wets, wens, went, west, were,
wits, with, wite, wins, wine, wist, wish, wise, wont, wort, wore
cats, cans, cant, cane, cast, cash, case, cent, cert, cits, cite, cist,
cots, cote, cons, cone, cost, cort, core

Or S, and you get these, but that does make three S's altogether.
sate, sans, sane, sass, sash, sars, sare, sets, seth, sent,
sits, sith, site, sins, sinh, sine, sirs, sire, sots, sons, sort, sore
impossibleroot
Hi-Keeba!

 Posted: Sat Sep 26, 2009 3:09 am    Post subject: 8 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
Hi-Keeba!

 Posted: Sat Sep 26, 2009 3:15 am    Post subject: 9 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
Unintentionally offensive old coot

Posted: Sat Sep 26, 2009 4:15 am    Post subject: 10

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.

 Code: P  A  T  S P  E  T  S N  I  T  S N  O  T  S
impossibleroot
Hi-Keeba!

 Posted: Sat Sep 26, 2009 12:59 pm    Post subject: 11 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
Hi-Keeba!

 Posted: Sun Sep 27, 2009 1:20 am    Post subject: 12 Here's a shot of the cubes as they are now, for reference. They're big.
PuzzleScot
Daedalian Member

Posted: Mon Sep 28, 2009 1:14 pm    Post subject: 13

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!)

 Code: B A L E M E N K P I R S W O S T

A mere 137 can be had from changing the 'S' on wheel 3 to a 'T'.

Edit: Checking for more up-to-date wordlists on-line, I found http://www.isc.ro/lists/sowpods4.html (Scrabble tournament list), where the original and less-rude versions above produce 167 and 162 words respectively.
Pigboy
Daedalian Member

 Posted: Mon Sep 28, 2009 2:10 pm    Post subject: 14 you seem to have gone for CVC starts, are these more common than VCV?
PuzzleScot
Daedalian Member

 Posted: Mon Sep 28, 2009 3:24 pm    Post subject: 15 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
Daedalian Member

 Posted: Mon Sep 28, 2009 4:25 pm    Post subject: 16 A semi-intelligent search produced [PMSD][AEIO][RNLC][TESK] for 167 also. Seems to be the limit, or very near it anyway...
Pigboy
Daedalian Member

Posted: Mon Sep 28, 2009 5:55 pm    Post subject: 17

 PuzzleScot wrote: 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.

where / how did you find that?
impossibleroot
Hi-Keeba!

 Posted: Mon Sep 28, 2009 6:36 pm    Post subject: 18 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
Daedalian Member

Posted: Mon Sep 28, 2009 7:26 pm    Post subject: 19

My program / word list didn't find any:

 Code: #!/usr/bin/perl use strict; # Consider a list of four pairs of distinct letters. # Is it possible to choose them so that all ways to choose one of each pair #    forms a word? @ARGV = ('/usr/share/dict/words'); my %lst = (); {     my @lst = <>;     foreach (@lst) {    chomp \$_;    if (\$_ =~ /^[a-z][a-z][a-z][a-z]\$/) {        \$lst{\$_} = 1;    }     } } sub check_lst {     my (\$prefix, @lst) = @_;     if (scalar @lst == 0) {    return defined \$lst{\$prefix} ? 1 : 0;     }     my \$first = shift @lst;     foreach my \$l (@\$first) {    if (check_lst (\$prefix . \$l, @lst) == 0) {        return 0;    }     }     return 1; } my @letters = split //, "abcdefghijklmnopqrstuvwxyz"; sub recurse {     my (\$i, @lst) = @_;     if (\$i == 4) {    print "[";    foreach (@lst) { print "(", @\$_, ")"; }    print "]\n";     } else {    #if (\$i == 3) { recurse (\$i + 1, @lst); return; }    my @let = @{\$lst[\$i]};    for (my \$j = 0; \$j < @letters; ++ \$j) {        if (\$letters[\$j] gt \$let[0]) {       \$lst[\$i] = [ @let, \$letters[\$j] ];       if (check_lst ("", @lst) == 1) {           recurse (\$i + 1, @lst);       }        }    }     } } foreach my \$w (sort keys %lst) {     recurse (0, [ substr (\$w, 0, 1) ], [ substr (\$w, 1, 1) ],         [ substr (\$w, 2, 1) ], [ substr (\$w, 3, 1) ]); }

There were a fair number that could do all eight when the fourth letter was singleton, and some could be extended; [st][io][np][gs] produces 12 words.
impossibleroot
Hi-Keeba!

 Posted: Mon Sep 28, 2009 10:00 pm    Post subject: 20 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
Daedalian Member

 Posted: Mon Sep 28, 2009 11:53 pm    Post subject: 21 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
Pitbull of Truth

 Posted: Tue Sep 29, 2009 1:45 am    Post subject: 22 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
Daedalian Member

 Posted: Tue Sep 29, 2009 3:57 am    Post subject: 23 Pire might not be a word, but it is the name of a Nobel Peace Prize recipient.
impossibleroot
Hi-Keeba!

 Posted: Tue Sep 29, 2009 11:16 am    Post subject: 24 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
Daedalian Member

 Posted: Tue Sep 29, 2009 2:46 pm    Post subject: 25 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
Daedalian Member

 Posted: Tue Sep 29, 2009 3:10 pm    Post subject: 26 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
Hi-Keeba!

 Posted: Wed Sep 30, 2009 11:11 am    Post subject: 27 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
Daedalian Member

 Posted: Wed Sep 30, 2009 11:24 am    Post subject: 28 I think that 'letters per face' and 'letters per cube' have been mixed up along the way somewhere...
ralphmerridew
Daedalian Member

 Posted: Wed Sep 30, 2009 2:16 pm    Post subject: 29 My name is "ralphmerridew", all lowercase. If you must abbreviate, use "rm".
Zag
Unintentionally offensive old coot

 Posted: Wed Sep 30, 2009 2:32 pm    Post subject: 30 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
Hi-Keeba!

 Posted: Wed Sep 30, 2009 3:08 pm    Post subject: 31 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
Daedalian Member

Posted: Wed Sep 30, 2009 3:42 pm    Post subject: 32

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/

 Code: \$ grep "^[PMSD][AEIO][RNLC][TESK]\$" sowpods.txt (167 words found) DACE DACK DALE DALS DALT DANK DANS DANT DARE DARK DART DECK DELE DELS DELT DENE DENS DENT DERE DICE DICK DICT DINE DINK DINS DINT DIRE DIRK DIRT DOCK DOCS DOLE DOLS DOLT DONE DONS DORE DORK DORS DORT MACE MACK MACS MALE MALS MALT MANE MANS MARE MARK MARS MART MECK MELS MELT MENE MENT MERE MERK MICE MICK MICS MILE MILK MILS MILT MINE MINK MINT MIRE MIRK MIRS MOCK MOCS MOLE MOLS MOLT MONK MONS MORE MORS MORT PACE PACK PACS PACT PALE PALS PANE PANS PANT PARE PARK PARS PART PECK PECS PELE PELT PENE PENK PENS PENT PERE PERK PERT PICE PICK PICS PILE PINE PINK PINS PINT PIRS POCK POLE POLK POLS POLT PONE PONK PONS PONT PORE PORK PORT SACK SACS SALE SALS SALT SANE SANK SANS SANT SARK SARS SECS SECT SELE SELS SENE SENS SENT SERE SERK SERS SICE SICK SICS SILE SILK SILT SINE SINK SINS SIRE SIRS SOCK SOCS SOLE SOLS SONE SONS SORE SORT

This list is nice because..
1) Rudest word possible is DICK, although not really that rude...
2) ALL combinations of cubes 1,2,4 can produce at least one word!
3) Any 3 first letters can make at least one word using the 4th cube! (well, apart from DIL, but that could be the challenge to find it!)
4) Only 'S' is used twice, otherwise unique.
5) No higher known number of valid words produced (167)
ralphmerridew
Daedalian Member

 Posted: Wed Sep 30, 2009 3:46 pm    Post subject: 33 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
Daedalian Member

 Posted: Wed Sep 30, 2009 5:15 pm    Post subject: 34 nice one... one ruder word exists there though...
Zag
Unintentionally offensive old coot

 Posted: Wed Sep 30, 2009 5:16 pm    Post subject: 35 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
Daedalian Member

 Posted: Wed Sep 30, 2009 5:23 pm    Post subject: 36 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
Unintentionally offensive old coot

Posted: Wed Sep 30, 2009 5:45 pm    Post subject: 37

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.
 Code: ABCDEFGHIJKLMNOPQRSTUVWXYZ VB_____HI__TWNO___SL__MX_Z

Mix of cases
D <-> P (instead of D -> _ and P -> _ )
N <-> U (instead of N <-> N and U -> _ )
F <-> F (assuming a scripty F. instead of F -> _ )

 Code: ABCDEFGHIJKLMNOPQRSTUVWXYZ VB_P_F_HI__TWUOD__SLN_MX_Z

Adding also, which can only hurt the word count, but does make it interesting if there are actually any words using the Q.
B <-> Q (instead of B <-> B and Q -> _ )
 Code: ABCDEFGHIJKLMNOPQRSTUVWXYZ VQ_P_F_HI__TWUODB_SLN_MX_Z
ralphmerridew
Daedalian Member

 Posted: Wed Sep 30, 2009 6:33 pm    Post subject: 38 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
Daedalian Member

 Posted: Wed Sep 30, 2009 7:08 pm    Post subject: 39 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
Pitbull of Truth

 Posted: Wed Sep 30, 2009 7:45 pm    Post subject: 40 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.)
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