Solution for "A Little Algebra" 

Let's denote the value of a cow by the letter \( c\), and let's denote the value of a ship by the letter \(s \).  The cheese is a little different, since sometimes there are two wedges of cheese, and sometimes there are three wedges.  So it's reasonable to assume the value of cheese will depend on the number of wedges present.  With this in mind, we'll denote the value of a single wedge of cheese by the letter \(g\).  Thus, two wedges will have value \(2g\), and three wedges will have value \( 3g\).  (Why \(g\) for cheese?  Well, \(c\) was already used for the cow, so we're using \(g\) because cheese is so good!)  With these variables, the equations in the puzzle can be rewritten as :

Equation 1:  \(c \times c \times 2g =64 \)
Equation 2:  \(3g+3g \times 3g=42 \)
Equation 3:  \(s \times 3g=48 \)
Equation 4:  \(2g \times s+c=? \)

​Step 1:  Use Equation (2) to find \(g\)

\[ \begin{align} &3g+3g \times 3g=3g+9g^2=42\\
​\\
& \implies g+3g^2=14\\
& \implies  3g^2+g-14=0\\
& \implies (3g+7)(g-2)=0\\
& \implies  g=2 \mbox{  or  } g=-\frac{7}{3}
\end{align}\]

Now, usually these types of puzzles are looking for nice, integer solutions.  So we're going to assume (yes, I'm making an assumption here) that the author of the puzzle was intending to use \(g=2\).

Step 2:  Use Equation (3) to find \(s\)
Substituting \(g=2\) into Equation (3), we obtain 
\[\begin{align} &s \times 3(2)=48\\
​\\
& \implies  6s=48\\
& ​\implies  s=8 \end{align}\]

Step 3:  Use Equation (1) to find \( c\)
Substituting \(g=2\) into Equation (1), we obtain 
\[\begin{align} & c \times c \times 2(2)=64\\
​\\
& \implies 4c^2=64\\
& \implies c^2=16\\
& \implies c=4 \mbox{  or  }c=-4
\end{align}\]
Again, we're going to assume that the puzzle author wanted the positive solution to be used.  So we'll say \(c=4\).

Step 4:  Plug \(g \), \(s\), and \(c \) into Equation (4)
Plugging \(g=2 \), \(s=8\), and \(c=4\) into Equation (4), we obtain
\[\begin{align} &2 (2) \times 8+4=?\\
​\\
&\implies 4 \times 8+4=?\\
&\implies 36=?
\end{align}\]

Therefore, the solution is 36.

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