Let the center point be $H$. $ADH$ is an isosceles right triangle.  Since $ABCD$ has area $16$, $AD$ has length $4$.  So $DH$ has length $\frac{4}{\sqrt{2}}=2\sqrt{2}$.  Let the marked points on $DH$ and $DC$ be $F$ and $G$ respectively.  Since angle $ADH$ is $45^{\circ}$, so is angle $FDG$.  Thus, since $e$ is a square, triangle $DFG$ is an isosceles right triangle.  Thus $HF=DF=FG$.

Since $DH=2\sqrt{2}$, these are equal to $\sqrt{2}$.  So $DG$ has length $2$, and as $CD$ has length $4$ this means that $CG$ has length $2$.  Since angle $FGD$ is $45^{\circ}$ and $e$ is a square, if we label the marked point on $BC$ as $J$ then angle $CGJ$ is $45^{\circ}$.

Thus triangle $CGJ$, the grey piece, is an isosceles right triangle, and one of its legs is $2$, so its area is $\frac{2^2}{2}=\boxed{2}$.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Quadrilateral $ABCD$ is a square with area 16 square inches. The figure represents the pieces of a Chinese tangram in which all the triangles are isosceles and piece "e'' is a square. What is the area of the gray piece, in square inches?

[asy]
fill((2,0)--(4,2)--(4,0)--cycle,gray(.7));
draw((0,0)--(0,4)--(4,4)--(4,0)--cycle,linewidth(1));
draw((0,0)--(4,4),linewidth(1));
draw((1,1)--(2,0)--(4,2),linewidth(1));
draw((0,4)--(3,1)--(3,3),linewidth(1));
label("$A$",(0,4),W);
label("$B$",(4,4),E);
label("$C$",(4,0),E);
label("$D$",(0,0),W);
label("e",(2,1));

[/asy]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
2
Quadrilateral $ABCD$ is a square with area 16 square inches. The figure represents the pieces of a Chinese tangram in which all the triangles are isosceles and piece "e'' is a square. What is the area of the gray piece, in square inches?

[asy]
fill((2,0)--(4,2)--(4,0)--cycle,gray(.7));
draw((0,0)--(0,4)--(4,4)--(4,0)--cycle,linewidth(1));
draw((0,0)--(4,4),linewidth(1));
draw((1,1)--(2,0)--(4,2),linewidth(1));
draw((0,4)--(3,1)--(3,3),linewidth(1));
label("$A$",(0,4),W);
label("$B$",(4,4),E);
label("$C$",(4,0),E);
label("$D$",(0,0),W);
label("e",(2,1));

[/asy]
test