Endgames in Fog of War Chess

Assume White wins in his next move. Therefore, White must achieve either a stalemate or checkmate (as Black does not move voluntarily into a square covered by White):

As Black is not at the edge of the board, White has to cover all eight squares adjacent to the Black king. However, the rook can only cover four squares, the White king only three (when not directly adjacent to the Black king, here Black would win immediately):

$◀$

It remains to show that Black cannot be pushed to an edge. To do so, we will start with three technical lemmas that describing situations which can be forced by Black.

In three situations, the Black king will be forced out of the safe space, and can not directly return. Here, however, the Black king can always move to a cluster of squares, from which he can either return to the safe space, capture material of White (leading to a draw or victory for Black), or the cluster of squares reoccurs (thus Black survives in an endless loop).

We start with an important lemma, which we will use multiple times in the following.

White is to move.

• $\mathrm{■}$

  If White moves the king, then the square combinations (1), (2), or (3) sustain themselves, or Black can take the White king. If the White king moves to the $B$ or $G$ file (which is safe when the rook covers $C$ resp. $F$) and blocks squares of some clusters; note that the rook cannot be protected by the king when moving to the left or right on line $3$ and will be caught, and that the Black king can advance to a safe space when the rook moves on other lines. Thus, the White king stays behind and covers the rook.

• $\mathrm{■}$

  If White moves the rook

  1. 1.

     left or right, it might uncover the Black kings location if placed between the $C$ to $F$ file. In any case, Black can in its next turn move the king to any square of another, undiscovered cluster. If the rook moves to the $C$ or $F$ file, we can assume that the Black king was in the invisible file. Then, or if the rook moves to the $D$ or $E$-file, Black can move the king. If, e.g, the rook is moved to $D$, then

     1. (a)

        if the Black king is in (2) or in (3), the Black king is on $D2$. The Black king then moves into cluster (1), and this lemma reapplies.

     2. (b)

        if the Black king is in (1), the Black king can stay there, and the assumptions remain unchanged.

  2. 2.

     If the rook moves down to $(A\text{–}H,1\text{–}2)$, then move the Black king to the safe space, here: $(C\text{–}F,3)$, which is adjacent to (1), (2), and (3). This is possible because the White king can cover at most $(C\text{–}E,3)$ or $(D\text{–}F,3)$. If the rook blocks the last one ($C3$ or $F3$), the Black king can capture the rook or move to the safe space.

  3. 3.

     If the rook moves up or down to $(A\text{–}H,3\text{–}7)$, the Black king can remain in his cluster.

Thus, either this lemma reoccurs, Black captures a piece, or moves to the safe space. $◀$

1. 1.

   Moving the king leads to the capture of it, or Black moves to the safe space ($C3$).

2. 2.

   Moving the rook to the right or up allows us to reapply this Lemma.

3. 3.

   Move the rook left to the $A-C$-file. Black then might not have been on the now visible squares, but might be on $(D,1\text{–}2)$. It is now Black’s turn, and the uncertainty extends to $E1$, $E2$. Thus, Lemma 14 applies.

4. 4.

   Moving the rook down can be treated analogously to (3) or by symmetry.

$◀$

1. 1.

   Let the White king be on any square in ranks 5–8 $(A\text{–}H,5\text{–}8)$.

   1. (a)

      If the rook is on $(E\text{–}H,3)$:

      • $\mathrm{■}$

        If the king is moved, the Black king can remain everywhere on $(A\text{–}D,1\text{–}2)$ and this lemma reapplies.

      • $\mathrm{■}$

        Moving the rook

        • –

          up or down allows Black to return to the safe space.

        • –

          left or right; then either the rook can be captured, or the Black king can stay in $(A\text{–}D,1\text{–}2)$, and this lemma reapplies.

   2. (b)

      If the rook is on $(E\text{–}H,4\text{–}8)$:

      • $\mathrm{■}$

        If the king moves, this lemma reapplies as above.

      • $\mathrm{■}$

        Moving the rook

        • –

          up lets Black move to the safe space $C3$.

        • –

          down to $(E\text{–}H,3\text{–}7)$ leads to the Black king can stay on $(A\text{–}D,1\text{–}2)$ and this Lemma reapplies.

        • –

          down to $(E\text{–}H,1\text{–}2)$ let Black move to the safe space.

        • –

          to the left or right allows the Black king to move to the safe space.

2. 2.

   Let the White king be on $(A\text{–}C,4)$.

   1. (a)

      If the rook is on $(E\text{–}H,4\text{–}8)$:

      • $\mathrm{■}$

        Moving the king leads to either the king being captured, or the Black king can stay on $(A\text{–}D,1\text{–}2)$ and the lemma reapplies.

      • $\mathrm{■}$

        Moving the rook

        • –

          down either results in capture of the rook or Black can move to the safe space $E3$.

        • –

          up or right let Black move to $E3$ (safe space) or reapply the lemma.

        • –

          left either let Black move to the safe space ($E3$) or the Black king can stay in $(A\text{–}D,1\text{–}2)$ and the lemma reapplies.

   2. (b)

      If the rook is on $(E\text{–}H,3)$:

      • $\mathrm{■}$

        Moving the king either allows Black to capture it, or the Black king can stay on $(A\text{–}D,1\text{–}2)$ and this lemma can reapply.

      • $\mathrm{■}$

        Moving the rook

        • –

          down either results in capture or allows Black to move to the safe space $E3$.

        • –

          up or right allows the Black king to stay on $(A\text{–}D,1\text{–}2)$ and reapply this lemma.

        • –

          to the left:

          • *

            Moving it to $(E\text{–}G,3)$ allows Black to stay on $(A\text{–}D,1\text{–}2)$ and reapply this lemma.

          • *

            Moving it to $(A\text{–}C,3)$: Black chooses not to be there, extending the uncertainty to $E1$, $E2$, while maintaining its position on $D1$, $D2$, leading to Lemma (***).

          • *

            Moving it to $D3$: Black chooses to be on $D2$, then moves to $C1$, $C2$, $E1$, $E2$, leading again to Lemma (***).

3. 3.

   Let the White king be on $(E\text{–}H,4)$. The cases are similar to 2, except that Black moves to the safe space square $C3$ instead of e.g. $E3$.

4. 4.

   Let the White king be on $D4$ (dangerous (!) because the White king blocks the reachable squares of the safe space).

   • $\mathrm{■}$

     Moving the king either allows Black to capture it, or the Black king can stay on $(A\text{–}D,1\text{–}2)$, and this lemma can reapply.

   • $\mathrm{■}$

     Moving the rook left, up, or right results in Lemma (***), the rook being captured, or the Black king can stay on $(A\text{–}D,1\text{–}2)$ and this lemma reapplies.

   • $\mathrm{■}$

     Moving the rook down to $(E\text{–}H,3\text{–}7)$, then the Black king can stay on $(A\text{–}D,1\text{–}2)$ and this lemma reapplies.

   • $\mathrm{■}$

     Moving the rook down to $(E,1\text{–}2)$ allows Black to take the rook.

   • $\mathrm{■}$

     Moving the rook down to $(F\text{–}H,1\text{–}2)$:

     • –

       to $F2$: Black chooses to be on $B2$, and can then move to $(A\text{–}C,1)$, or $(A\text{–}B,3)$.

       • *

         Move the White king

         • ·

           Left/ Left-down, then capture.

         • ·

           down: this situation is described as $(X1)$ at the end of this proof.

         • ·

           anywhere else: Move to safe space ($C3$).

       • *

         Move the rook

         • ·

           up: Then Black decides to be on $(A\text{–}C,1)$, then expanding to $(A\text{–}D,2)$, so this lemma reapplies.

         • ·

           left/ right: either Black can take the rook, or Black decides to be on $A3$, $B3$ and expands to $A4$, $B4$ (Situation $(X2)$, see below).

         • ·

           down, Black decides to be on $(A\text{–}B,3)$ and expands to $(A\text{–}B,2\text{–}4)$ (Situation $(X3)$, see below).

     • –

       The squares $G2$, $H2$ behave analogously.

     • –

       Move the rook to $F1$. Then Black chooses not to be on $(A\text{–}D,1)$. So, black can now be on $(A\text{–}D,2)$, $A3$, $B3$, and E2 (!).

       • *

         If White now moves the rook up (more than one square), then either Black takes or we can apply Lemma (***) as the king can now expand to $D1$ and $E1$.

       • *

         If White moves the rook up one square, then the previous case with the rook on $(F\text{–}H,2)$ can be applied as Black might be on $B2$. So, the Black king can stay on $(A\text{–}D,1\text{–}2)$ and this lemma reapplies.

       • *

         If White moves anything anywhere else, then Black can move to the safe space, as he can be on $E2$ or capture.

     • –

       Analogously for $G1$, $H1$.

5. 5.

   If the king is on $(F\text{–}H,1\text{–}3)$

   • $\mathrm{■}$

     Move the king: capture or Black can stay on $(A\text{–}D,1\text{–}2)$ and reapply this lemma.

   • $\mathrm{■}$

     Move the rook:

     • –

       down to $(E\text{–}H,1\text{–}2)$: move to safe space.

     • –

       left to $(A\text{–}D,3)$: take the rook.

     • –

       left to $(A\text{–}D,4\text{–}8)$: move to safe space.

     • –

       otherwise the Black king can stay on $(A\text{–}D,1\text{–}2)$ and reapply this lemma.

Then Black can move to any of $(A\text{–}D,1)$ and $(A\text{–}B,3\text{–}4)$. Now, White is to move.

• $\mathrm{■}$

  If the White king moves, then either capture it or Lemma (***) ($A4$, $A5$, $B4$, $B5$ which by rotating the board results in Lemma (***)).

• $\mathrm{■}$

  If the rook moves:

  • –

    down; then Lemma (***) by moving to $(A\text{–}B,5)$.

  • –

    left; then take or Lemma (***).

  • –

    right; then ends up in Lemma (***).

  • –

    up to $(F,3\text{–}4)$ (one or two squares): then Black decides to be on $B4$, and expands to $(A\text{–}B,3)$ and $(A\text{–}B,5)$, resulting in Lemma (***). Otherwise, results in this Lemma (**) by moving to $(A\text{–}B,1\text{–}4)$, rotate board.

It is White’s turn.

• $\mathrm{■}$

  Moving the king, either take, or Lemma (***).

• $\mathrm{■}$

  Move the rook:

  • –

    down, left, right, then take or Lemma (***).

  • –

    up three squares to $(E\text{–}H,5)$ (otherwise Lemma (***)): expand to $(A\text{–}D,1\text{–}2)$ which leads to Lemma (***).

It is White’s turn.

• $\mathrm{■}$

  Move king: either Black captures or Lemma (***) (could also move to safe-square).

• $\mathrm{■}$

  Move rook: Up four squares, then this lemma reoccurs by expanding to $A1$, $B1$, otherwise: capture or Lemma (***)

$◀$

Finally we can use those observations to provide a strategy for Black:

We will show that if the White player starts (and cannot immediately take the Black king), then the Black king can always remain either inside the safe space or on squares from which his moves cannot be reduced to one edge (undesired square). Black always tries to prevent this by going to a square inside the safe-zone whenever there exist uncovered ones.

We distinguish between a corner case and an edge-but-not-corner case. The corner squares of the safe space are $C3$, $C6$, $F3$, and $F6$. The edge-but-not-corner squares are all other squares on the edge of the safe space.

If he is somewhere on $(D\text{–}E,4\text{–}5)$, then White cannot force him out of the safe space in the next move, and he cannot be captured if he is not on the edge.

Thus, the Black king is at the edge of the safe space, but not in a corner. To be forced down from here, the board must look like Figure 13.

Black can only be forced out of the safe space, if the rook attacks the rank or file of the Black king, and the White king covers the remaining squares inside the safe space.

Let the White king be on $D5$, the rook on $H3$, and the Black king on $D3$ (as on Figure 13). If the Black king is forced down, he can always move to $D2$, or $E2$, regardless. White cannot know where the Black king moved; therefore, he might be on any of those squares.

• $\mathrm{■}$

  Then White can move the king, afterwards Black can move from $D2$ to $E2$, $D1$, or $E1$ and from $E2$ to $D2$, which allows the application of Lemma (***)

• $\mathrm{■}$

  If White moves the rook up or down: Black returns to the safe space. If the rook is moved left or right, either the rook is captured or the Black king can move as in the case the king moved, therefore we can apply Lemma (***).

Without loss of generality, let this corner be $C3$. To force the Black king outside the safe space, White has to cover all squares adjacent to the Black king in the safe space:

We note that the White king must always cover at least one square in the safe space adjacent to the Black king. Otherwise, the rook must be on $D4$ to cover all. Then either the king must be adjacent (but then he would cover $D4$), or Black can take the rook.

We now consider all combinations of the White king’s and rook’s positions that cover all safe spaces adjacent to the Black king. We then analyze the situation after White’s next move.

1. 1.

   Place the White king on $B5$. Then $C4$ is covered. The rook then has to cover the $D$-file.

   1. (a)

      If the rook is placed on $D2$, $D3$, or $D4$, the Black king would capture it.

   2. (b)

      If the rook is placed on $D1$, then Black moves to $C2$; the rook then must retreat. After that, Black can return to $C3$ (if the rook moved to $D3$, Black would capture it).

   3. (c)

      If the rook is placed on $(D,5\text{–}8)$, the Black king could move to any square among $B3$, $B2$, or $C2$, which introduces uncertainty for White. White is to move.

      1. i.

         Let White move the king. If the king moves downward, Black can capture, as he might be on $B3$. If the White king moves elsewhere, the Black king returns to $C3$.

      2. ii.

         Let White move the rook. If the rook moves up, to the right, or down one square, Black can return to $C3$. If the rook is moved further down, Black, being on $C2$, captures. If the rook is moved left, then Black, being on $C2$, moves to $D3$.

2. 2.

   White king on $C5$. Then $C4$ and $D4$ are covered. The rook only needs to cover $D3$.

   1. (a)

      Rook on $D4$: White moving anything anywhere leads to Black moving back to the safe space or capturing material of White.

   2. (b)

      Rook on $D2$ or $D3$: then the Black king captures it.

   3. (c)

      Rook on $D1$: then Black moves down to $C2$. The White rook must move; he can only block the safe space $C3$ by going to $D3$, where the Black king captures it.

   4. (d)

      Rook on $(D,5\text{–}8)$. Black moves to one of $B2$, $B3$, $C2$.

      1. i.

         If White moves the rook, Black can either capture it or move back to $C3$.

      2. ii.

         If the king moves down-right, the Black king could move to any of $(A\text{–}B,1\text{–}4)$, $(C\text{–}D,1\text{–}2)$. This case is discussed in Lemma 15.

      3. iii.

         If White moves the king anywhere else, then Black can return to $C3$.

   5. (e)

      Rook on $A3$: Then moving the Black king would threaten the rook, forcing it to move; after this, Black can either capture the rook or return to $C3$.

   6. (f)

      Rook on $B3$, or $D3$: in each case, the Black king captures it.

   7. (g)

      Rook on $(E\text{–}H,3)$: Black moves to either $B2$, $C2$, or $D2$. White is to move.

      • $\mathrm{■}$

        Moving the rook up or down allows the Black king to return to $C3$.

      • $\mathrm{■}$

        Moving the king lets the Black king expand to $A1$, $A2$, $B1$, $C1$, or $D1$, while also possibly being on $C2$, $D2$. This case is discussed in Lemma 16.

      • $\mathrm{■}$

        Moving the rook to the left or right either results in the rook being captured or restricting the Black king, as in the last case. This case is discussed in Lemma 16.
3. 3.

   Place the White king on $D5$. This case is analogous to (Case 2.).

4. 4.

   Place the White king on $E5$. Then the rook must be on $D4$. In this case, the Black king moves to either $B3$, $B2$, or $C2$.

   • $\mathrm{■}$

     If the rook moves down or left, it is captured.

   • $\mathrm{■}$

     Otherwise, the Black can return to $C3$ (safe space).

Hence, either the Black king cannot be forced to one edge or White cannot win. $◀$

Note that when Black starts on $(X,Y)$, there is at most a $5\times 5$-square $([X\text{–}2]\text{–}[X+2],[Y\text{–}2]\text{–}[Y+2])$ in which Whites figures are at risk in Blacks second move.

Therefore, there are at least two ranks/files available which are safe from the opponent’s king after its first move and without the own White king. Note that the Black king cannot block any move of the rooks, as if he could, the rooks would be able to capture him.

In the first two moves, White can move the two rooks to one of the ranks/files. $◀$ Now we will see that the two rooks can assist their own king to move to a corner.

Note that rooks can move freely on the rank/file where the White king is not. This can be done, as the rooks cover each other, and their own king cannot block them.

White moves their own king towards an edge by repeating the following strategy:

• $\mathrm{■}$

  Cover two files/ranks ahead of the own king, such that he can move freely there. If there is only one file/rank ahead, cover that one.

• $\mathrm{■}$

  Move the own king one square towards one edge (not diagonal).

The second step can be done, as the rooks cover all squares which are adjacent to the new position of the own king (which are not covered by the king itself). One might note that diagonal moves are not always safe, and that it is possible for the Black king to hide behind the White king such that he does not get seen by the rooks.

Thus, when the White king arrives at the edge, he cannot move away from the rooks without the risk of running into Black’s king. Instead, he has to move towards his own rooks.

When the White king arrives at the field adjacent to the corner (which is blocked by a rook), it is easy to see that White can safely rotate its own pieces to free the corner from the rook and move the king here and the rooks adjacent to him. $◀$ Finally, we see that the king and two rooks can finish by forcing a checkmate.

Without loss of generality, let the White king be on $A1$, and the two rooks on $A2$ (Rook 1) and $B1$ (Rook 2). The following strategy resembles the staircase mate in classical chess, but we have to be more careful, as we do not know the exact position of the Black king. Therefore, we have to cover the rooks with the White king such that the Black king cannot take one of them. The strategy proceeds as follows and can be repeated until checkmate:

• $\mathrm{■}$

  Move Rook 2 up one square.

• $\mathrm{■}$

  Move Rook 1 up one square.

• $\mathrm{■}$

  Move the White king one square up.

As in classical chess, this forces the Black king to move up (and right), as otherwise White would capture him. Eventually, the Black king has no moves left to escape. $◀$