## Building Blocks (Level 3)

Your friend builds a tower using at most $n$ blocks of each side length. For instance, if $n$ is $3$ then your friend selected blocks from a pile with edges of length $1, 1, 1, 2, 2, 2, 4, 4, 4, 8, 8, 8$, and so on. Also, the largest block in your friend’s tower has length $c$, which equals $2^k$ for some value of $k$.
11. What is the largest possible number of cubes that could be in your friend’s tower?
12. What is the smallest possible number of cubes in your friend’s tower?
13. What is the largest possible height for your friend’s tower?
14. What is the smallest possible height?
Source: Julia Robinson Mathematics Festival

SOLUTION
11. What is the largest possible number of cubes that could be in your friend’s tower?
Your friend could use all of the blocks available to her from size $1$ up to size $c=2^k$
$n\left (2^0\right ),n\left (2^1\right ),n\left (2^2\right ),\cdots,n\left (2^k\right )$
Largest possible number of cubes that could be in your friend’s tower
$n\left (k+1\right )$

12. What is the smallest possible number of cubes in your friend’s tower?
Your friend uses only one block of size $c$.
Smallest possible number of cubes in your friend’s tower
$1$

13. What is the largest possible height for your friend’s tower?
$n\left (2^0\right )+n\left (2^1\right )+n\left (2^2\right )+\cdots+n\left (2^k\right )$
$=n\left (2^0+2^1+2^2+\cdots+2^k\right )$
$=n\left (2\cdot 2^k-1\right )$
$=n\left (2c-1\right )$
The following pattern shows why $2^0+2^1+2^2+\cdots+2^k=2\cdot 2^k-1$
$2^0+2^1=3=2\cdot 2^1-1$
$2^0+2^1+2^2=7=2\cdot 2^2-1$
$2^0+2^1+2^2+2^3=15=2\cdot 2^3-1$
$\cdots$
$2^0+2^1+2^2+\cdots+2^k=2\cdot 2^k-1$

14. What is the smallest possible height?
Your friend uses only $1$ block of size $c$.
Smallest possible height
$c$

11. $n\left (k+1\right )$
12. $1$
13. $n\left (2c-1\right )$
14. $c$