Puzzle #16 – On the nth Day of Christmas

yuleguideAccording to the song, on the nth day of Christmas your true love gives you to a total of

\displaystyle N(n)=\sum_{i=1}^n i

presents.  That is, on the fifth day, for example, you receive 1+2+3+4+5 presents. Here’s a guided way of working out this sum.

Consider the n\times n array of presents shown on the right.

Questions:

i) How many presents are there in total in the array?

ii) How many presents are there along the diagonal of the array?

iii) How many presents are there not on the diagonal of the array?

iv) Therefore how many brown presents are there?

v) Therefore how many gold presents are there?

vi) Using your answer, how many presents do you receive on the nth day of Christmas?

\

Extension: how many presents have you received by the nth day of Christmas?

Hint: you may need this formula.

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