1990 AHSME Problems/Problem 25

Problem

Nine congruent spheres are packed inside a unit cube in such a way that one of them has its center at the center of the cube and each of the others is tangent to the center sphere and to three faces of the cube. What is the radius of each sphere?

$\text{(A) } 1-\frac{\sqrt{3}}{2}\quad \text{(B) } \frac{2\sqrt{3}-3}{2}\quad \text{(C) } \frac{\sqrt{2}}{6}\quad \text{(D) } \frac{1}{4}\quad \text{(E) } \frac{\sqrt{3}(2-\sqrt2)}{4}$

Solution

Let $r$ be the radius, let $C$ be the center of the cube, and let $P$ be the center of one of the eight outer spheres, noting that $PC=2r$ .

Pack, in the corner of the unit cube, a smaller cube whose inner corner coincides with $P$ . The cube is of dimensions $r\times r\times r$ and its space diagonal is of length $r\sqrt3$ . Thus the space diagonal of the unit cube is $2(r\sqrt3+PC)=2r(\sqrt3+2)$ . But this must equal $\sqrt3$ . Solving for $r$ , we get $\fbox{B}$ $[asy] draw((0,0)--(0,1)--(1,1)--(1,0)--cycle); for(int i=0;i<4;++i)draw(rotate(i*90,(.5,.5))*circle((0.25,0.25),0.225)); //draw(circle((0.75,0.25),0.225)); draw(circle((0.5,0.5),0.19),dashed); draw((0,0)--(.5,.5),dotted); dot((.25,.25));dot((.5,.5)); label("P",(.25,.25),S);label("C",(.5,.5),N); [/asy]$

1990 AHSME (Problems • Answer Key • Resources)
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1990 AHSME Problems/Problem 25

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