The December, 2007 Problem of the Month

 

 

On the twelfth day of Christmas, 
my true love sent to me 
Twelve drummers drumming, 
Eleven pipers piping, 
Ten lords a-leaping, 
Nine ladies dancing, 
Eight maids a-milking, 
Seven swans a-swimming, 
Six geese a-laying, 
Five golden rings, 
Four calling birds, 
Three French hens, 
Two turtle doves, 
And a partridge in a pear tree! 

 

How many gifts did my true love send to me on the twelfth day of Christmas?

 

How many gifts did my true love send to me in total (total sum of gifts over days 1-12)?

 

Find a general formula for the total number of gifts given in terms of n for the n days of Christmas.  The pattern is assumed to be 1 gift the first day, 1 + 2 gifts on day 2, etc.

 

SOLUTION

On the 12th day of Christmas, my true love gave 12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 12*13/2 = 78 gifts.

Notice that by starting from the ends and working in, we are adding 13 a total of 6 times.  Or if you average, we are adding 13/2 a total of 12 times.  In general,

1+ 2 +3 +… (n-1) + n = n(n+1)/2

 

The total number of gifts over the 12 days is:

1 + (2 + 1) + (3 + 2 + 1) + … + (12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1)

= 1*2/2 + 2*3/2 + 3*4/2 + … + 12*13/2 = 364

 

And most generally, the number of gifts over n days is

1 + (2 + 1) + (3 + 2 + 1) + … + (n + (n-1) +( n-2) + … + 3 + 2 + 1)

= 1*2/2 + 2*3/2 + 3*4/2 + … + n*(n+1)/2

= 1*(1+1)/2 + 2*(2+1)/2 + 3*(3+1)/2 + … + n*(n+1)/2

= [12 + 22  + 32  …+ n2]/2 + [1 + 2 + …+ n]/2

= [n(n+1)(2n+1)/6]/2 + [n(n+1)/2]/2

= n(n+1)(2n+1)/12 + 3n(n+1)/12

= [(2n3 + 3n2 + n) + (3n2 + 3n)]/12

= (2n3 + 6n2 + 4n)/12 =  (n3 + 3n2 + 2n)/6

= n(n+1)(n+2)/6

(We used the equation 12 + 22  + 32  …+ n2  = n(n+1)(2n+1)/6, which can be proved by various methods, one such way is by induction.)

Note that when n = 12, we get the above answer 12*13*14/6 = 364.

 

 

 

 

 

 

 

 

 

 

 

Submit solutions in PSA 216 to Problem of the Month care of Richard Reynolds, or via email at rich@math.asu.edu

 

See http://math.asu.edu/~rich/puzzles/main.html for a web page view of this problem and past problems with solutions.