Sums of Squares

Inspired by exam grading over Thanksgiving weekend, I have one more problem for the MCS Club contest for the fall semester. Given that our best response was for the first problem, which featured the number 16, I have determined that this must be because problems about 16 are one easier than all those problems about 17 that Barb is assigning. Hence, another problem about 16: find the smallest number which can be written as a sum of four perfect squares in at least sixteen different ways. Details on my contest page.