Triangular Treasure Mystery Cache

In basic number theory, we learn about the Triangular Numbers, a sequence of numbers of the form n(n+1)/2, where n is some positive integer.

The triangular numbers have a remarkable way of showing up in real world problems when you least expect it. For instance, if ten strangers attend a dinner gathering and each one shakes hands with every other person in attendance, then how many handshakes would occur? While you could count each person as shaking the hands of nine other attendees, you would be double-counting handshakes. One way to count all the handshakes would be to consider that person one shakes nine other hands, then person two shakes eight other hands (not counting person one who we've already counted), then person three shakes seven other hands, etc. The total number of handshakes is 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 handshakes. Other similar problems can be solved this way, such as finding the number of games to be played in a round-robin tournament of 10 teams, or the number of potential relationship problems in a social group of 10 people.

One fun seasonal problem related to the Triangular Numbers comes from the Christmas carol, The Twelve Days of Christmas. For instance, in the popular song how many gifts were given on the fifth day of Christmas? The common quick answer is five gold rings, but of course the song actually states that on the fifth day of Christmas my true love gave to me five golden rings, four calling birds, three french hens, two turtle doves, and a partridge in a pear tree - that's a total of 5 + 4 + 3 _ 2 + 1 = 15 total gifts on the fifth day alone! Fifteen is the fifth Triangular Number, the sum of 1+2+3+4+5. An interesting fact about the song is that by the time it's over, your true love has given you a total of 364 gifts (but of course you've got a yard full of pear trees and a lot of partridge poop to deal with). If your true love had spread them out over the year, they could have given you a different gift on every day of the year but Christmas!

So let's get to the cache, which is not placed at the posted coordinates. You've got to solve this simple puzzle to get two three-digit numbers, which you can use to replace the "000" in each of the North and West coordinates. Suppose that there were more than just twelve days of Christmas, and that the many days of Christmas began on January 1 and continued throughout the year. On January 4, your true love would be giving you ten total gifts, 1+2+3+4.

The rating is for the puzzle, although some may have a much easier time with this depending on your mathematical comfort level. Finding the final cache shouldn't be too difficult. You can check your answers for this puzzle on Geochecker.com.