A one rupee coin is placed on a piece of paper. How many more coins of the…

Answer: 6 coins

Reasoning:

• Let the radius of each coin be r. The center of any outer coin must be at distance 2r from the central coin's center, so all outer centers lie on a circle of radius 2r around the center.

• If n identical outer coins are placed evenly around that circle, the distance between centers of two adjacent outer coins equals the chord length for angle 2π/n, which is 2(2r)·sin(π/n) = 4r·sin(π/n).

• For adjacent outer coins to touch, their center distance must be 2r. So we need 4r·sin(π/n) = 2r, which simplifies to sin(π/n) = 1/2.

• The equation sin(π/n) = 1/2 is satisfied when π/n = π/6, so n = 6.

• Thus exactly six coins can be placed so that each outer coin touches the central coin and its two adjacent outer coins. Fewer than six leaves gaps, and more than six would force overlap.

A simple geometric interpretation: the six outer coin centers form a regular hexagon around the center, giving the familiar close-packed ring of six coins.