HMMT 2011 U19

Find the least positive integer $N$ with the following property: If all lattice points in $[1,3] \times[1,7] \times$ $[1, N]$ are colored either black or white, then there exists a rectangular prism, whose faces are parallel to the $x y, x z$, and $y z$ planes, and whose eight vertices are all colored in the same color.