LII OM - I - Task 4

Determine whether 65 balls with a diameter of 1 can fit into a cubic box with an edge of 4.
Answer: It is possible.
The way to place the balls is as follows.
At the bottom of the box, we place a layer consisting of 16 balls. Then we place a layer consisting of 9 balls, each of which is tangent to four balls of the first layer (Fig. 1 and 2). The third layer consists of 16 balls that are tangent to the balls of the second layer (Fig. 4 and 5). Similarly, we place two more layers (Fig. 6).
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In total, we have placed $ 16 + 9 + 16 + 9 + 16 = 66 $ balls. It remains to calculate how high the fifth layer reaches.
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Let's choose any ball from the second layer; this ball is tangent to four balls of the first layer. The centers of these five balls are the vertices of a regular square pyramid, each edge of which has a length of 1 (Fig. 3). By the Pythagorean theorem, the height of this pyramid is $ \frac{\sqrt{2}}{2} $. Therefore, the highest point that the fifth layer reaches is at a distance of $ \frac{1}{2} + 4 \cdot \frac{\sqrt{2}}{2} + \frac{1}{2} = 1 + 2\sqrt{2} < 4 $ from the base plane. The 66 balls placed in this way fit into a cubic box with an edge of 4.