## Quarter the Cross

Yesterday, I noticed that some of the math teachers I follow on Twitter were challenging their students, and themselves, with the #quarterthecross problem.  The problem is simply to find interesting regions of a five-square cross that take up exactly one-quarter of its area.  This looks like a great way to get kids to explore fractions and geometry.

A solution that occurred to me would be to use an infinity of nested crosses.  If the area of the cross shrinks at each step by one-third, then the combined area of the colored regions converges to one-quarter of the large cross.  This works because the alternating geometric series

$\dfrac{1}{ 3} - \dfrac{1}{ 3^{2}}+\dfrac{1}{ 3^{3}}- \dfrac{1}{ 3^{4}}+\dfrac{1}{ 3^{5}}...$

converges to $\dfrac{1}{4}$.  Note that if the area shrinks by one-third, the width of the cross must scale down by the square root of 3.  So every two steps the width shrinks by one-third, placing a little cross precisely inside the middle square of a larger cross.