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?

Answer 1

Correct Answer: 96

Step 1: In a closely stacked arrangement, each sphere touches several others. Start by counting how many points of contact each row has.
Step 2: Given the triangular arrangement, each sphere touches 6 others in the next layer. By examining the structure, we calculate the total number of touching points across all layers.
Step 3: The total number of points where the spheres touch each other in this configuration is 96.

Answer 2

This method treats the stack as four square layers of spheres. It counts contact points twice over, once between spheres in the same layer and once between spheres in touching layers, then adds the two totals.

The stack has four layers arranged as squares of side \( 4, 3, 2, \) and \( 1 \). These layers hold \( 16, 9, 4, \) and \( 1 \) spheres, for \( 30 \) spheres in total. Within a single square layer of side \( n \), each sphere touches its neighbors along rows and columns only, giving \( 2n(n-1) \) contacts inside that layer. This works out to \( 24 \) contacts in the bottom layer, \( 12 \) in the next, \( 4 \) in the next, and \( 0 \) in the single sphere on top. Together these give \( 40 \) contacts within layers.

Between two neighboring layers, every sphere in the smaller upper layer rests inside a pocket formed by four spheres directly beneath it, so each upper sphere touches four spheres below. The upper layers here hold \( 9, 4, \) and \( 1 \) spheres, giving \( 4 \times 9 = 36, 4 \times 4 = 16, \) and \( 4 \times 1 = 4 \) contacts across the three gaps between layers, adding to \( 56 \) contacts between layers.

Adding the within layer and between layer totals gives \( 40 + 56 = 96 \) points of contact. the answer is 96