asimovd@aol.com wrote:

<< A partridge in a pear tree?

I have to cry fowl. The puzzle was posted before Xgiving, so would not have a Xmas reference.

<< Let v_n be the vector in R^(n-1) defined as (1/n, 2/n,...,(n-1)/n). Let w_n be the vector in R^(n-1) with the same coordinates in reverse order. Find the limit as n -> oo of the angle between v_n and w_n. (Try this without using numerical approximation or summation-of-powers formulae.)

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I was referring to the number of gifts in the song, which matched the dot product of the two vectors for n=12 (well, to within a power of 2): 1*12 partridges 2*11 turtle doves 3*10 french hens 4*9 calling birds 5*8 gold rings 6*7 geese 7*6 swans 8*5 maids 9*4 ladies 10*3 lords 11*2 pipers 12*1 drummers -- Mike Stay staym@datawest.net