Question

Shown below is an arrangement of closely stacked spheres. Assume each one to be in contact with its immediate neighbour. What is the total number of points where the spheres touch each other?

Hint:

In problems involving closely packed arrangements, visualize the pattern of connections, and use geometric relationships to count the points of contact.

Answer 1

Step 1: Split the count into two types.
In a stacked pyramid of spheres, contacts happen either between neighbours sitting in the same layer, or between a sphere and the spheres directly below it in the next layer down.

Step 2: Count layer-by-layer.
Working from the top view and the side (elevation) view together, each layer of the pyramid is a square-packed sheet of spheres, and each sphere in that sheet touches up to 4 immediate neighbours within the same layer. Adding up same-layer contacts across every layer of the stack gives 56 contacts.

Step 3: Count between-layer contacts.
Each sphere resting in an upper layer sits nested in the hollow formed by 4 spheres directly beneath it, so every sphere above the bottom layer contributes 4 contacts downward. Adding this across all the layer boundaries in the pyramid gives 40 more contacts.

Final Answer:
\[ 56 + 40 = 96 \]
The spheres touch each other at 96 points in total.

Answer 2

A different way to check the same total is to count contacts sphere by sphere instead of layer by layer, then divide by 2 so each touching pair is not counted twice.

Take any one sphere in the stack. If it sits inside a layer, away from the pyramid's outer slope, it can touch up to 4 spheres in its own layer and up to 4 more spheres in the layer directly below it, so an interior sphere can be involved in as many as 8 contacts.
Spheres along the outer slope of the pyramid touch fewer neighbours, since there is no layer or neighbour on the open side.
Adding up every sphere's individual contact count across the whole pyramid this way gives a running total of 192 sphere-to-contact instances.
Since each physical contact point is shared between exactly 2 spheres, this running total has counted every contact twice, so the true number of distinct contact points is \[ \frac{192}{2} = 96. \]

So the spheres touch each other at 96 points, matching the layer-by-layer count.