msg455

Bianchetti's Study #46
Submitted by Patrick McCartney

In reference to Sinanovic's Puzzle #46, the key "entry points" are c5 and f4.

Using the theory of "corresponding squares," when the White’s King is on c5, Black’s must be on e7 (d7 is too far from the f4 entry point), White King on d4, Black must be on f7, White King on e3, Black must be on g6, White King on f4, Black must be on h5.

So c5 = 1 = e7, d4 = 2 = f7, e3 = 3 = g6, f4 = 4 = h5

Now d3 is adjacent to 2 and 3. The only legal square for the Black King that is adjacent to both 2 and 3 on his side is g7, So d3 = 5 = g7

Now c4 is adjacent to 1, 2, and 5. Only square for Black adjacent to 1, 2, and 5 is f8, So c4 = 6 = f8

Now c3 is adjacent to 2, 5, and 6. Only square for Black adjacent to 2, 5, and 6 is g8, So c3 = 7 = g8

Next, e2 is adjacent to 5 and 3. Only squares adjacent to 5 and 3 are f7, h7, and h6. From f7, White has Kd2, and Black has no square to go to adjacent to 3, 5, and 7. Same problem if the Black King goes to h7. So that leaves only h6. So e2 = 8 = h6

Next, d2 is adjacent to 3, 5, 7, and 8. Only h7 meets this quota. So d2 = 9 = h7

Last, c2 is adjacent to 5, 7, and 9. Only h8 meets this quota. So c2 = 10 = h8

Now b4 also gets a 2, b3 gets a 5, b2 gets a 9.
Now, the theory of corresponding squares is a form of reciprocal zugzwang, and so 3 things end the game. If the White King penetrates thru c5, White wins. If the White King penetrates thru f4, White wins. If the 2 Kings are ever on the same number square, it’s Reciprocal Zugzwang, and so if White is then to move, it’s a draw, while if it’s Black to move, White wins.

Therefore:

With Black to move, 1…Kg8 draws as 2.Ka2 Kh8 3.Ka3 Kg8 4.Ka4 Kf8 and White can’t penetrate while 2.Kb1 Kg7 3.Kc1 Kg8 4.Kd1 Kg7 5.Ke1 Kg6 and no penetration.

Alas, it’s White to move, so:

1.Ka2! Kg8 (Only move, 1…Kh7 2.Kb2 and 1…Kg7 2.Kb3 are both Zugzwang for Black)

2.Ka3 Kf8 (Again only move, can’t step on 2, 5, or 9, and 2…Kh8 3.Kb4 and White penetrates)

3. Kb2 Kf7 (Forced as 5 and 7, or g7 and g8 are off limits, and Ke8 or Ke7 gives White Kc3, where he can’t go to the f-file, so he stays on the e-file and White plays Kd3 where Kf7 allows Kd4 and anything else allows penetration with Ke3)

4.Kc2 Kf8 (Forced again. Kg7 and Kg8 are off limits, Ke8 or Ke7 again allows Kd3, same problem as before, and Kg6 allows Kc3)

5.Kd2 Kf7 (Forced again as 7, 5, and 3 are all taboo, i.e. g8, g7, and g6)

6.Ke2 +-

Either Black is forced to go to g6 or g7, giving White Ke3 and Kd3 respectively, or else he will be too far away, and White will penetrate with Ke3-f4.

Patrick McCartney