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?
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?
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Approach Solution - 1
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.
Approach Solution -2
The figure gives a top view showing the spheres arranged in a tightly packed square-based layer, and a side (elevation) view showing the same stack rising as a triangular, pyramid-like arrangement of 4 layers. To count contact points, it is easiest to separate them into contacts within one layer and contacts between two layers stacked on top of each other, then add both totals.
Contacts within a single layer: in a closely packed layer like the one shown in the top view, each interior sphere touches 4 neighbours (up, down, left, right) in a square arrangement, while spheres along the edges touch fewer. Summing the touching pairs across all the spheres in one layer, without double counting any pair, gives 56 contact points spread across the layers.
Contacts between layers: every sphere in an upper layer rests in the pocket formed by 4 spheres in the layer directly below it (since the pyramid narrows one row at a time going up), so each sphere in an upper layer adds 4 more contact points with the layer beneath it. Summing this across all 3 layer-to-layer boundaries in the 4-layer pyramid gives 40 more contact points.
Adding the two totals together, \[ 56 + 40 = 96, \] the total number of points where the spheres touch each other in this arrangement is 96.
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