a) A cube $3 \times 3 \times 3$ was constructed with 27 smaller cubes $1 \times 1 \times 1$. For each pair of touching faces of two cubes, one drop of glue is used. How many drops of glue were used in total?

b) A cube $10 \times 10 \times 10$ was constructed with 1000 smaller cubes $1 \times 1 \times 1$. For each pair of touching faces of two cubes, one drop of glue is used. How many drops of glue were used in total?
Solution

a) From the figure below, to form a face $1 \times 3 \times 3$, 12 drops of glue are needed, corresponding to the 12 contact faces between the 9 cubes.

![](https://cdn.mathpix.com/cropped/2024_05_01_ffb48c2069f209c1311cg-06.jpg?height=348&width=347&top_left_y=1925&top_left_x=736)

To place one face on another, 9 drops of glue are needed, corresponding to one drop for each square. Thus, to glue the three $1 \times 3 \times 3$ blocks together, 18 drops are needed. Therefore, the total number of drops is $3 \cdot 12 + 2 \cdot 9 = 54$.
b) Again, we start the count by considering a block $1 \times 10 \times 10$. There are 9 rows and 9 columns between the squares of a $10 \times 10$ board, and these segments determine $9 \times 10 + 9 \times 10 = 180$ segments that are sides of $1 \times 1$ squares. Thus, to glue the squares of one block, 180 drops are needed. To glue two blocks together, 100 drops are needed. Therefore, to place 10 blocks $1 \times 10 \times 10$ together, $100 \cdot 9 = 900$ drops are needed. Therefore, the total number of drops used is $10 \cdot 180 + 900 = 2700$.

![](https://cdn.mathpix.com/cropped/2024_05_01_ffb48c2069f209c1311cg-07.jpg?height=352&width=352&top_left_y=649&top_left_x=924)