Problem:
A solid rectangular block is formed by gluing together N congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly 231 of the 1-cm cubes cannot be seen. Find the smallest possible value of N.
Solution:
Let the dimensions of the block be p cm by q cm by r cm. The invisible cubes form a rectangular solid whose dimensions are p−1,q−1, and r−1. Thus (p−1)(q−1)(r−1)=231. There are only five ways to write 231 as a product of three positive integers:
231=3â‹…7â‹…11=1â‹…3â‹…77=1â‹…7â‹…33=1â‹…11â‹…21=1â‹…1â‹…231
The corresponding blocks are 4×8×12,2×4×78,2×8×34,2×12×22, and 2×2×232. Their volumes are 384,624,544,528, and 928$$ , respectively. Thus the smallest possible value of N is 384​.
The problems on this page are the property of the MAA's American Mathematics Competitions