  We classify the connection between $t$-cores and self-conjugate $t$-cores to
sums of squares. To do so, we provide explicit maps between $t$-core partitions
and self-conjugate $t$-core partitions of a positive integer $n$ to
representations of certain numbers as sums of squares. For example, the
self-conjugate $4$-core partition $\lambda=(4,1,1,1)$ corresponds uniquely to
the solution $61=6^2+5^2$. As a corollary, we completely classify the
relationship between $t$-cores and Hurwitz class numbers.
  Using these tools, we see how certain sets of representations as sums of
squares naturally decompose into families of $t$-cores. Finally, we construct
an explicit map on partitions to explain the equality
$2\operatorname{sc}_7(8n+1) = \operatorname{c}_4(7n+2)$ previously studied by
Bringmann, Kane, and the first author.

Combinatorics;Number Theory