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