Problem 1 · The tower of 84 spheres
Seven triangular layers: tangencies within each layer and between layers.
Integer answer, at most 4 digitsWith 84 identical spheres we build a tower shaped like a regular tetrahedron, similar to the one in the picture but with more spheres, leaving as little space as possible between them. How many points of tangency are there among all the spheres?
Copa Cangur · SCM
Hard
Closed answer
Reasoned solution
Key idea: tetrahedral numbers are $T\!e(n) = \tfrac{n(n+1)(n+2)}{6}$ and $T\!e(7) = 84$: the tower has $7$ layers, layer $k$ being a triangle of $T_k = \tfrac{k(k+1)}{2}$ spheres.
Within a triangular layer of side $k$ there are $3T_{k-1} = \tfrac{3k(k-1)}{2}$ tangencies (three per upward-pointing small triangle). Between layer $k$ and layer $k+1$, each sphere of the upper layer rests on $3$ of the lower one: $3T_k$ tangencies.
$$\sum_{k=1}^{7} \frac{3k(k-1)}{2} + \sum_{k=1}^{6} \frac{3k(k+1)}{2} = 3\,T\!e(6) + 3\,T\!e(6) = 6 \cdot 56 = 336.$$
Answer: 336