12. Five cards have the numbers $101,102,103,104$ and 105 on their fronts.
101
102
103
104
105
On the reverse, each card has one of five different positive integers: $a, b, c, d$ and $e$ respectively. We know that $a+2=b-2=2 c=\frac{d}{2}=e^{2}$.
Gina picks up the card which has the largest integer on its reverse. What number is on the front of Gina's card?
SolUTION
104

We know that $d=4 c, d=2 a+4$, and $d=2 e^{2}$.
So we can quickly see that $d>c, d>a$ and $d>e$.
Also, $d=2 b-4>b$ for $b>4$. We can eliminate the $b \leq 4$ case, for $b$ cannot be odd (for then $c$ would not be an integer); $b$ cannot be 4 (for then $a$ would be zero); and $b$ cannot be 2 (for then $c$ would be zero). Therefore, $d>b$.
Therefore $d$ is the largest of $a, b, c, d$ and $e$, so Gina has 104 on the front of her card.