Four Rings
Read the puzzle
Four equal rings overlap in a 2 by 2 square layout: top-left, top-right, bottom-left and bottom-right. Each ring overlaps its side and top or bottom neighbours, and the diagonal rings also overlap in the middle, where all four rings cover one small centre area. How many closed areas are there inside the rings? Do not count the space around them.
How many closed areas are there inside the rings? Do not count the space around them.
Check the middle: the diagonal rings overlap too.
13
4 parts in one ring, 4 in two rings, 4 in three rings and 1 centre in all four: 13.
How to solve it
Group the areas by how many rings cover them, working from the outside in.
In exactly one ring: each of the 4 rings has an outer part that no other ring touches, so 4 areas.
In exactly two rings: where side-by-side or top-and-bottom neighbours overlap, there is a lens on the outer edge of the picture. There are 4 such neighbour pairs, so 4 areas.
In exactly three rings: closer to the middle, the diagonal rings also overlap, which creates 4 small pieces covered by three rings each.
In all four rings: the tiny centre, 1 area.
Total: 4 + 4 + 4 + 1 = 13 closed areas.
Check with crossing points. The 4 neighbour pairs cross twice each (8 points) and the 2 diagonal pairs cross twice each (4 points), giving 12 points. Each ring is crossed 6 times, so it splits into 6 arcs: 4 x 6 = 24 arcs. Areas inside = arcs - points + 1 = 24 - 12 + 1 = 13.
The usual wrong answer is 9, from counting it like a 3 by 3 grid and missing that the diagonal rings overlap too. Another is 14, from including the outside space.
