# Ponder This

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## March 2020 - Solution

March 2020 Challenge

In order to force a draw, the players cannot leave more than one unplayed square in any given three consecutive squares.

It is easy to check that to force draw, one cannot have more than five unplayed squares in a 5x5 board.
Therefore there can not be a solution with more than 20 unplayed squares in a 10x10 board.
You can find 20 unplayed squares and then solve for the remaining 80 squares.

A possible solution is:

```X.+O+.XO.X
.XO.+XOO+.
O+OXO++XOO
+XXO.+O+.X
.O+XXO.+X+
OXO.+XXO+.
X.O+O.+OXO
++XXO+X+XX
XO.+XO.+O.
.O+X.O+X.+
```

The second '*' goes for finding a solution for a 15x15 board with 41 unplayed squares:

```X.+X+.X+X+.X+.O
.O+.XOO+.XOO+X.
+XXOO+.XOO+XXO+
O+O+XX+X+.XOX.O
.+XX+.XOO+X.O++
+O+.XOO+.XO++X.
O.XOO+.X+X.OXOO
X+X+.X++OO+OX.+
.XOO++OOX.OX+XO
OO+.XOX.O++X.O+
+XX+X.O++X.OX+.
XO.XO+OX.OXXO+O
.X++OOX+XX++OX+
++O.X+X.O+O.X+.
O.XOX.+XO.+OX.O
```

Bertram Felgenhauer found an 18x18 board with 58 unplayed squares:

```+OX.O+O.+X+X.O++X.
X.+XO.+XXO.+O+.XOX
+XXO+X+O.+O+XXOX.O
.O+O.+OXO+XXO.XO++
XO.+XXO.+XO.X++OX.
++X+O.+O+.X++O.X+X
X.+OXO+XXOXO.XOX.+
OOXO.+XO.X++OX++OO
.X++O+.X++O.X+O.X+
+OX.+XOXO.XOX.+OX.
X.+OOXX++OX++OOX+O
OXOX.+XO.X+O.X+X.+
.+O+XXO+OXO+OX.+OO
+OX.O+O.X+O.X+OOX.
X.+OX.+O+.XOX.+O+X
OXX++OXX+OO++OOX+O
.+OOX+.OX+.XO+.XO.
X.OX.+O+.OX+.OX+.X
```

and a 20x20 board with an optimal 71 unplayed squares:

```O.+X+X.+XXO.+XO.+X.O
+XXO.XOO+.XOX.+OXO+.
.O++OO+.XOO++OO+X+OX
XXO.X+X+X+.XOX.+OX.+
O.+XO+.XOO+X.O+X.O+O
XO+O.XOO+.XO+OXO+OX.
.+XX+O+.X+X.O+X.OX++
OO+.XOX+XOO++XO++X.O
+.XOO+.XO+.XOX.OXOX+
X+X+.X+XOO+X.O++X.+O
.XOO++OO+.XO++X.OX+.
OO+.XOX.O+X.OXOXX+OX
+.X+X.O++XO++X.+O+.O
O+XOO++XOX.OXOX+.OXX
.XO+.XOX.O++X.+OXX+.
+X.O+X.O++X.OX+.O+O+
XO++XO++X.OXX+OXO+.X
+XXOX.OXOXX+O+.O+XXO
.O+.O++X.+O+.OXX+O+.
O.+XOX.OX+.X+O+.OX.+
```