There is an apple with radius 31 mm. It is in the shape of a perfect sphere. A worm gets into the apple and eats a tunnel of total length 61 mm, and then leaves the apple. (The tunnel need not be a straight line.)
Prove that you can cut the apple with a straight slice through the center so that one of the two halves is not eaten.
First consider the problem for a circle instead of a ball.
Let \(A\) and \(C\) be the entry and exit points of the worm. Let \(B\) be the point opposite the worm’s entry point.
Note that the worm can never travel from \(A\) to be \(B\) as the distance is 62 mm.
Draw a line that bisects \(C\) and \(B\) and goes through the circle’s center (hence cuts the circle in two). By construction, all points on the line are equally far from \(C\) and \(B\). Thus if the tunnel touches the line, then it could also have reached \(B\). Since a tunnel between \(A\) and \(B\) is impossible, the tunnel cannot touch the line.
Cutting the apple along the line leaves the tunnel completely on one side, leaving the other side untouched by the worm.
For a sphere, replace the bisecting line with a bisecting plane where all points on the plane are eqi-distant from the points \(B\) and \(C\). The rest of the proof carries through.