Thanks for all the replies folks! 
I thought about using the analog line of the arduino to sense the signal.. From what I have read, the signal strength from a flipped core can't really be expected to be above 40mV. The arduino's analog pins are 10 bit, 1024 states, represented by 0-5V right?
5V / 1024 states = 0,0048828125
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Ooooh! The analoge pins has a resolution of 4,88mV! I, for some reason or other, was thinking the resolution was 40mV until this very moment. -_-' In that case, I might actually be able to get a readout when the pin flips! =D If I get that working, I can move on to the next step. Wooho!
For those who might wonder, the next step is writing 0,0,1,0,0,1 and so on to a single core while manually decreasing the current through the core until I no longer get a proper "flip". Then I'll know what exact current is needed to flip the core and I can continue to my first multi-core test where half that current is used in an x-line and half in y. After some more adjusting with this 3-4 core setup, I will know what I need to know to build a full-fledged 32B memory!
Who knows? I might even throw in some nice, clicky relays. One for each line, since a transistor isn't very compatible with this setup where the current has to be able to flow both ways. ^^
Oh, btw, how would I go about hooking the sense-wire to the arduinos analog input? Could probably figure it out, but I'm not entirely used to this whole pull-up/pull-down resistor stuff yet so you guys could likely save me a lot of trouble by just telling me. Thanks!