I’m not quite sure what you mean by the " ‘search the vertex’ game" reference, so perhaps the following advice is not usefull to you, but anyways, here I go :
The middle “line” of vertices should all be vertically aligned. If this is not the case you can bring them into line by doing a constrained scaling.
[>] First select a vertex that you wish to align the “line” of vertices to,
[>] snap the 3Dcursor to that vertex by pressing Shift_s > Cursor to selection
[>] Press the dot/period key on your main keyboard set of keys (this will cause scaling/rotating operations to be performed with respect to the 3Dcursor)
[>] Start scaling by pressing [s], then while dragging your mouse horizontally, press your middle mouse button, and while holding down the Ctrl key, scale down to 0 .
(Now all your mesh’s middle vertices should be vertically aligned)
Now press Shift_d to duplicate the vertices. Scale [s], and [x] OR [y] (depending which view you are mirroring in, to flip the vertices across the axis.)
[!] Note that since we were in ‘scale with respect to the cursor’ mode, all vertices were flipped exactly across from where the cursor is.
Now all that remains is to do a ‘remove doubles’ operation ( [w], [4]) to weld the two halves together.
(By the way to put your scaling rotating mode back to the default, press the main keyboard [,] key - you should see the icons in the 3Dwindow header change when this is done - which is another way to change this option).
Hope that helps - it takes alot longer to explain then to actually do it 
To comment on the horse chess piece, it looks nice except for the ears (a little too pointy) and where the top part meets the bottom part.
Nice work, keep going.