It is TM12. As shown clearly in the given figure, the number of half-cycle present in the x-axis is one, and the number of half-cycles present in the y axis is two. As tangential components of electric field present over the surface, it is the transverse magnetic mode.
There seems to be no variation of the electric field along the patch (y), except for end effects, so one mode number is 0. There is one half wave variation across the patch (x), so the other mode number is 1.
From the diagram it appear that That there is two half cycle variation in x-axis direction and there no variation in y-axis direction so the mode excited shell be T20
From the diagram attached as a PNG file it is clearly visible that number of half cycle variation in x-axis direction is two, having respective vector direction are opposite. and as Malcom White Sir has correctly point out there is some variation in y-axis direction at near the edges corner, but it can not be consider predominate variation. Thus no variation in y-axis direction. Hence excited mode appear to be TM20
You are wrong. The boundary condition at the edge of the patch is open circuit. The electric field will be a maximum here. Across the x direction it is a maximum at both edges and zero in the middle. This is ONE half cycle.