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John has started an Einstein's-like puzzle ("$n$ people of $n$ ages prefer $n$ foods, etc.; determine which is which from the following facts"), using a matrix for his work. He shows me his completed work, asking whether he's made any errors. There are no obvious errors (two accepted boxes in the same set, or all boxes in a set eliminated). I know of two ways to determine whether he's erred:

  1. Continue solving the puzzle (without error) where John has left off, and see whether I arrive at an impossible state. If so, John erred. If not, I'll complete the puzzle, and John hasn't erred.
  2. Start the puzzle over again and solve it (without error) until either (a) I have eliminated every square John has, and accepted every square he has, in which case he hasn't erred; or (b) I have accepted some square John has eliminated or vice versa, in which case John has erred.

Is there another way to determine whether John has erred? Specifically, is there a way that doesn't involve redoing or continuing the solution, but, rather, involves merely examining what John has done?

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Yes.

But only if the matrix contains contradictory information.

If the errors have yet to cause contradictory information to appear in an incomplete grid, then the answer really is no, you have the 2 approaches (apart from cheating and checking the answers page of your puzzle book) that can determine if the grid is bad, and which to use, as you have probably guessed, depends on how much progress has been made on the puzzle.

Just as the matrix can be used in several ways to spot connections it can also be used to spot information that is obviously bad by similar analysis of the pairs of crosses that share rows and columns.

The partial grid below shows crosses that assert truth. (eg the Dane lives in the white house)

Einstein grid errors example

Each colour being a group that together are bad. Each group containing at least one cross that is wrong. In these examples you can remove a single cross and the other crosses of that colour are no longer contradictory.

Remember testing for obvious errors can only prove an answer to be bad. It can not prove it to be good.

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Check whether each of the two traits that meet at an accepted square meet others with the same results (eliminated/accepted). Say, if John has concluded that Alice is 7, which he marked with an accepted square, and accepted or eliminated some of the other traits/variables that meet with one of "Alice" or "7" (eye color etc.), he must not accept for 7 what he eliminated for Alice (and vice-versa). But just leaving a square empty without accepting or eliminating it shouldn't count as an error.

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