A math problem for the Christmas season
Britain's Telegraph has an amusing article on the song "On the first day of Christmas, my true love sent to me, a partridge in a pear tree". Professor Marcus du Sautoy of Oxford asks just how many gifts did the true love send in all of twelve days. How do we compute for the total number of gifts given in any day without tediously adding them all up?
Our second-year math class was given the same problem as a homework. When the day arrived for the homework to be checked, it turned out nobody--except me--in the class figured out the formula: nx(n+1)/2, with n being the number of days. The teacher got so incensed with the class that he called me up at asked me to go to the blackboard and write my formula. Perhaps not knowing that I got the right one, she made the promise that she will give the highest grade of 95 on the grading card--effectively disregarding the rest of the student's performance in the class-- for that school year quarter to whoever will get the answer correctly. Boy, was I happy then.
How did I get it? For anybody with rudimentary knowledge of quadratic equations or mathematically brilliant like Karl Friedrich Gauss as the article points out, the problem would be a cinch to solve. Now I was neither of the two then (as now), but what I did was add up all those numbers and, by trial and error, practice different equations that might fit the total for each number of day. By staying up very late and waking up very early, I was lucky to find that nx(n+1)/2=total number of gifts was the equation that fits.
The teacher gave me the highest grade possible on that grading period. I was glad though that she did not ask in detail how I came up with the formula. I would have had to reply that I added all the numbers up and looked for a fitting equation, a roundabout time-consuming way that is hardly the mark of a numerate man.