troydaniels: It will be tricky to programmatically determine which squares have to be 0; it's easy to come up with examples, but it'll be hard to find all possibilities.

Here's a table with expected scores, assuming you guess, and subsequent guesses are only done if they give a positive score.

    +-------+-------+-------+-------+-------+-------+-------+-------+
    |   1   |   2   |   3   |   4   |   5   |   6   |   7   |   8   |
+---+-------+-------+-------+-------+-------+-------+-------+-------+
| 1 | +5.00 | -1.50 | -0.33 | -1.00 | -1.40 | -1.67 | -1.86 | -2.00 |
+---+-------+-------+-------+-------+-------+-------+-------+-------+
| 2 |       |  0.00 | +2.33 | -0.17 | +0.20 | -0.47 | -0.71 | -1.00 |
+---+-------+-------+-------+-------+-------+-------+-------+-------+
| 3 |       |       | +5.00 | -0.00 | +1.80 |  0.00 | +0.43 | -0.27 |
+---+-------+-------+-------+-------+-------+-------+-------+-------+
| 4 |       |       |       |  0.00 | +3.40 |  0.00 | +1.57 |  0.00 |
+---+-------+-------+-------+-------+-------+-------+-------+-------+
| 5 |       |       |       |       | +5.00 | -0.00 | +2.71 | +0.00 |
+---+-------+-------+-------+-------+-------+-------+-------+-------+
| 6 |       |       |       |       |       |  0.00 | +3.86 | +0.00 |
+---+-------+-------+-------+-------+-------+-------+-------+-------+
| 7 |       |       |       |       |       |       | +5.00 | -0.00 |
+---+-------+-------+-------+-------+-------+-------+-------+-------+
| 8 |       |       |       |       |       |       |       |  0.00 |
+---+-------+-------+-------+-------+-------+-------+-------+-------+

'+0.00' means there's a small positive expected score, '-0.00' means there's a small negative expected score.