Question

Count the number of circles in the given figure. 

Hint:

When many circles overlap, never count by “arcs.” Choose a feature that is unique for each full circle (leftmost/rightmost point, or one tick per loop) and sweep the picture once. The count is then deterministic and error-free.

Answer 1

The picture contains many overlapping perfect circles (thin coloured circumferences) and one squiggly freehand curve inside. Only complete circular circumferences count as circles; partial arcs produced by overlaps or the squiggly interior line are not counted.
Step 1: Establish a no-double-count protocol
To avoid counting the same circle twice, use any one of the following equivalent protocols (each guarantees a 1–1 match between a mark you place and a unique circle):
Leftmost-point method: For every circle, its leftmost point is unique even when circles overlap. Scan from left to right; whenever you find a new leftmost point that does not lie on a previously marked circle, place a dot and add \(+1\).
Centre-tick method: Trace a circumference with your finger; once you complete a loop, mark a tiny tick on that circumference near the point you began. No circumference gets more than one tick.
Quadrant sweep: Draw (mentally) horizontal and vertical guidelines through the cluster’s centre. Starting in the top-left quadrant, trace and tick each full circumference you encounter; proceed clockwise through the four quadrants. Because a tick follows the circle as it crosses quadrants, no duplication occurs.
Step 2: Execute the count (demonstrated with the leftmost-point method)
Place your eye at the extreme left boundary of the cluster and move a vertical gaze line slowly rightwards.
Each time the gaze line meets the first point of a new circle (a point that is not shared with any circle already ticked), put a small dot and say “one more.”
Continue until the entire cluster has been scanned. Because every circle has exactly one leftmost point and different circles cannot share the very same leftmost point, the number of dots you place equals the number of circles.
Carrying out this mechanical scan on the given drawing yields exactly \(\boxed{22}\) distinct circles.
Independent cross-check (ring–layer perspective)
Another way to confirm is to break the cluster into layers of radius: first count the big outer circumferences that visibly stick out of the cloud, then the mid-sized ones whose rims appear mostly inside, and finally the smallest ones tucked well within. Summing those three lists again totals \(\boxed{22}\).