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: The stack has four square layers of side \( n = 1, 2, 3, 4 \), holding \( n^2 \) spheres each, for \( 30 \) spheres total.

Step 2: Contacts within a layer of side \( n \) number \( 2n(n-1) \). Summed over the four layers this gives \( 2(0+2+6+12) = 40 \) contacts.

Step 3: Contacts between two layers equal \( 4 \) times the sphere count of the smaller upper layer. Summed over the three gaps this gives \( 4(1+4+9) = 56 \) contacts.
\[ \boxed{40+56=96} \]

Answer 2

As a check, this approach climbs the stack from the single sphere at the top down to the base. It adds new contacts at each stage, instead of splitting the count into within layer and between layer totals from the start.

The lone sphere at the top has no neighbor beside it, so it starts with \( 0 \) contacts of its own. It rests in the pocket of the \( 4 \) spheres below it, adding \( 4 \) contacts. Those \( 4 \) spheres touch each other along their own square layer for \( 4 \) more contacts. They also rest on the \( 9 \) spheres beneath them, adding \( 4 \times 4 = 16 \) further contacts. The \( 9 \) spheres touch each other for \( 12 \) contacts, and rest on the base layer of \( 16 \) spheres for \( 4 \times 9 = 36 \) contacts. The base layer itself has nothing beneath it, so it only adds its own \( 24 \) internal contacts.

Running the total as the layers are added gives \( 4 + 4 + 16 + 12 + 36 + 24 = 96 \). This matches the count found by working from the bottom up. the answer is 96