Three Rings
Read the puzzle
Three equal rings overlap like a classic three-circle diagram: two side by side at the top and one centred below them. Every pair of rings overlaps, and there is one small middle area inside all three rings. 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.
Count the areas inside one ring, two rings and all three.
7
3 parts inside one ring, 3 where two rings overlap, and 1 middle where all 3 meet: 3 + 3 + 1 = 7.
How to solve it
Sort the areas by how many rings they are inside. That way none is missed and none is counted twice.
Inside exactly one ring: each ring has an outer part that no other ring covers. There are 3 rings, so 3 areas.
Inside exactly two rings: each pair of rings shares a lens-shaped overlap, minus the very middle. There are 3 pairs (top-left with top-right, top-left with bottom, top-right with bottom), so 3 more areas.
Inside all three rings: the small curved triangle in the very centre, 1 area.
Add them up: 3 + 3 + 1 = 7 closed areas.
You can check with a count of crossing points. Each pair of rings crosses twice, so there are 3 x 2 = 6 crossing points, which cut the three rings into 12 arcs. For a drawing like this, areas = arcs - points + 1 (not counting the outside), so 12 - 6 + 1 = 7.
The usual wrong answers are 3, from counting only the rings, or 6, from forgetting the shared centre. Some people say 8 because they also count the space outside, which the question rules out.
Takeaway: group the areas by how many shapes cover them.
