Nine cookies are cut from a circle of cookie dough: eight cookies of radius $1$ inch are arranged in a ring, each tangent to its two neighbors in the ring and to the edge of the dough, and a ninth cookie in the center is tangent to all eight ring cookies. The leftover scrap is reshaped to form another cookie of the same thickness. What is the radius of the scrap cookie?

1. Let $d$ be the distance from the center to each ring cookie's center. Adjacent ring cookies (radius $1$, spaced $\frac{360^\circ}8=45^\circ$ apart) are tangent, so the chord between neighboring centers is $2$: $2d\sin22.5^\circ=2$, so $d=\dfrac1{\sin22.5^\circ}=\sqrt{4+2\sqrt2}$ (using $\sin22.5^\circ=\frac{\sqrt{2-\sqrt2}}2$).
2. The center cookie is tangent to each ring cookie, so its radius is $r_c=d-1$.
3. Each ring cookie is tangent to the dough's edge, so the dough radius is $R=d+1$.
4. The scrap area is $\pi R^2-8\pi(1)^2-\pi r_c^2=\pi\left(R^2-r_c^2-8\right)$. Since $R^2-r_c^2=(R-r_c)(R+r_c)=(2)(2d)=4d$, this is $\pi(4d-8)$.
5. Setting this equal to $\pi r^2$ for the scrap cookie's radius $r$: $r^2=4d-8=4\sqrt{4+2\sqrt2}-8$.
6. $r=\boxed{2\sqrt{\sqrt{4+2\sqrt2}-2}}\approx1.566$.