I was not thinking about etching the capacitors; that would give you just a few pF, and the resonances would all be in the VHF range. I was thinking more about frequencies around 1MHz, mounting a big fat capacitor on the component side, soldered to copper patches there (no holes, but the patches are so far apart that SMD would not work, and you would need capacitors with wires). They would form resonance circuits with the etched spiral coil. (These are indicated only schematically; I suppose it would be advantageous to have many loops there to drive up the inductance, so the copper traces and their spacing would probably be narrower than in the picture.)
The idea of having a spiral that continues pretty far inwards is to distribute the magnetic flux more homogeneously over the area inside the coil. For a simple wire loop the flux concentrates near the wire, but the inner loops create more flux in the center. In principle it should be possible to make the radial density of the loops vary in such a way that the detuning gets independent in lowest order from the exact positioning of the metal object relative to the center of the coil. Ideally only the size of the metal object would matter. Completely covering the spiral coil with a conducting sheet would make its inductance almost completely vanish. I was hoping the largest patch of metal used on a piece could reduce the inductance by a factor 4, reasonably independently of the the exact positioning as long as it is inside the loop, which would then double the resonance frequency.
The purpose of the outer guard loop is to confine the 'return flux' outside the spiral coil, which induces a counter-current in it (as it is short-circuited) that would keep the total flux through that outer loop zero, preventing the field to leak into the neighboring squares. This would basically cancel the magnetic dipole moment, so that the field would decay much faster with distance, making it much less sensitive to distant metal objects. That the guard loops are connected to the trace that feeds the outer side of the spiral coil is just incidental; it did not seem to gain anything by making them cross through a wire bridge.
If it is easy for you to etch such a coil and measure the resonance decay, it would be great. I haven't etched any PCB myself since I was a student in the eighties. Everything depends on how narrow the resonances are. Readout would be done by forcing a frequency-swept rf current of given magnitude through one (board-file-wise) series of resonatotors; those in resonance would hugely amplify the current through the coil, and as all magnetic flux of the latter passes through the pickup loop on the top side, the induced voltage there will increase proportionally.
What chips would you recommend to demodulate the rf signal, to detect which of the eight resonators are in resonance? Would it be better to have a dedicated modulator on each (board-rank) detection line, or would it be better to use an analog multiplexer to selectively connect one of the detection lines to the demodulator, and probe them one after the other? The resonances will need time to develop, limiting the rate of the frequency sweep. But I guess demodulation can in principle be instantaneous. (We could mix the driving signal with that from the detection line in a four-quadrant multiplier.) Anyway this is of later concern; first we should test if it works on a single square, and how many different piece tyoes we could reliably detect. If that looks good we could try a tic-tac-toe board
, to test how bad the cross-talk problem is.
BTW, it would probably be better to rotate the spiral coils 45-degrees compared to the situation I have drawn, so that the wire from its center and the copper trace from the outside go towards the square's corner, where there is more room to connect the capacitor to a patch outside the guard loop.
As to the LEDs: did you plan to use one LED per square, or LEDs in the square corners (so that you would have a 9x9 array of LEDs)?