[math-fun] Question about difficulty of online Set game
We've discussed the game of Set here in the past, but for anyone unfamiliar with it I'll summarize: There is a pack of cards, each marked with a graphic having 4 characteristics (color, number, shape, shading). Each characteristic occurs in one of 3 possible ways (color = red, green, or purple; number = 1,2,or 3; shape = oval, diamond, or squiggle; shading = outline, hatched, or solid). Every combination is represented on just one card, for a total of 3^4 = 81 cards. The game is played by someone's turning over 12 random cards in a 3x4 rectangular array. The object of the game is to identify a "set" of 3 cards, defined as follows: 3 cards form a set if for each of the 4 characteristics, the 3 cards display only 1 variant, or else all 3 variants, among them. E.g, the three cards "3 green outline ovals", "2 red outline diamonds", and "1 red outline squiggle" form a set. A set may be seen to be isomorphic to the concept of an affine line in the vector space (F_3)^4. --------------------------------------------------------------------------- The New York Times website now has a daily solitaire Set game, with 4 levels of play from easy to hard. The harder levels 3 and 4 work this way: You see an image of 12 Set cards in a 3x4 array. When you identify a "set" you click on all 3 cards in turn, and the software confirms that you're right. The array of 12 cards is pre-chosen to have 6 "sets" among them, and the object is to identify all 6 of them. The online game is here: < http://www.nytimes.com/ref/crosswords/setpuzzle.html >, (but it may require a subscription to access).
QUESTION: What makes one 3x4 array with 6 sets among them "harder" than another? (I often find this online puzzle's "level 4" to be easier than its "level 3", so this has me wondering.)
--Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
For anyone not familiar with the math of Set, let me shamelessly plug "The card game SET", by Ben Davis and Diane Maclagan, in Math. Intelligencer 25 #3 (Summer 2003), for which I was (a very hands-on) editor. You can find a pdf of it on my publications page, http://people.brandeis.edu/~kleber/pubs.html . Dan asks:
The online game is here:
< http://www.nytimes.com/ref/crosswords/setpuzzle.html >,
QUESTION: What makes one 3x4 array with 6 sets among them "harder" than another? (I often find this online puzzle's "level 4" to be easier than its "level 3", so this has me wondering.)
All sets (ie lines in (F_3)^4) are affinely isomorphic, certainly, but from the card game point of view, they can be easier or harder to spot, depending on which attributes vary and which are fixed. In my experience playing the game, the hardest sets to spot seem to be those in which three of the four attributes vary and one stays fixed, though I don't know why this should be true (and maybe it isn't really; I only have a subjective feeling, not any actual data). That said, the number of fixed attributes in the six sets from today's level 3 and 4 puzzles were (3,2,2,1,1,0) vs (3,3,2,2,1,0) respectively, so I'd have guessed level 3 to be the harder one on these grounds. So much for that hypothesis. Which attributes they are probably matters too -- I know I feel more stupid about being slow to see a set when it turns out to be monochromatic, for example :-). --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.
On Feb 9, 2008 2:24 AM, Dan Asimov <dasimov@earthlink.net> wrote:
QUESTION: What makes one 3x4 array with 6 sets among them "harder" than another? (I often find this online puzzle's "level 4" to be easier than its "level 3", so this has me wondering.)
I think that harder sets are ones where all four attributes are different, and the easiest are the ones where only one attribute varies. So the level 4 ought to have a higher proportion where three or four attributes are different. --Joshua Zucker
participants (3)
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Dan Asimov -
Joshua Zucker -
Michael Kleber