Algorithms on partitioning

Hey everyone do you sense anything different? Well the Project Euler counter breezed from 166 to 175 in these one or two days, and it's like I can reach 200 or 225 without much difficulty. The main reason is that a new difficulty rating is out (it may not be new for you if you visit the site frequently though), and we can access to relatively easier later numbered question easier.

In fact many of them relies on the same searching technique or a number theory trick...here let's look at one of the classic algorithm and its variation:

We start from a classical example: In how many ways we can partition a number $n$ into sum of integers? (Don't cheat by looking at the OEIS!)

A trip to South Island, New Zealand (4): Tekapo

Without much resources except the land itself, the exploring trail along the Southern Alps 150 years ago remained to be the major populated area in the South Island. Tekapo is another early settlement beside the lake of the same name --- just as the triple point of matters, we have the triple point of New Zealand here --- the intersection of calm waters, great mountains and clear sky.