Problem:
A solid rectangular block is formed by gluing together congruent -cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly of the -cm cubes cannot be seen. Find the smallest possible value of .
Solution:
Let the dimensions of the block be by by . The invisible cubes form a rectangular solid whose dimensions are , and . Thus . There are only five ways to write as a product of three positive integers:
The corresponding blocks are , and . Their volumes are , and 928$$ , respectively. Thus the smallest possible value of is .
The problems on this page are the property of the MAA's American Mathematics Competitions