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
converges to . 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.