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Hm. Sorry to have wasted your time. I'd picked up on this bit of the OP
... since A and E are restricted to only two values between them, they must consume those two values;
... and hung my hat on it, but that obviously doesn't apply in the same way to the two examples above.
Question: Would this example also decompose into the (same?) two groups as the above?
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In reply to Re^3: decomposing binary matrices