Here is a problem of my own invention; I hope you enjoy it:

*At a festival, every guest presented a gift to every older guest. The ten youngest guests combined gave thirteen times as many gifts as they received. What was the total number of gifts presented at the festival? *

Suppose there are n guests in total. The youngest guest gives a gift to everyone else, that is n-1 people, but they receive none. The second youngest gives to n-2 people, and receives only 1 gift (from the youngest guest). Continuing in this fashion, we find that the total number of gifts presented by the ten youngest guests is

(n – 1) + (n – 2) + … + (n – 10) = 10n – 55,

and the number of gifts they receive is

1 + 2 + … + 9 = 45.

We are told that the first of these totals is thirteen times the second. So

10n – 55 = 13 x 45 = 585,

and we have that n = 64. Finally, the total number of gifts presented across the entire festival is

1 + 2 + … + 63.

Here we use the fact that the sum of the first k positive integers is k(k+1)/2, to obtain a final answer of

63 x 64 / 2 = **2016**.

### Like this:

Like Loading...

*Related*

DR N KORPELAINENMarch 7, 2017 / 10:41 pmShouldn’t this be 55, not 45?

LikeLike

Puzzle CriticMarch 7, 2017 / 11:45 pmYes, it should! Thanks for pointing it out. It’s now correct; thankfully the answer is still the same.

LikeLike