Every day, Fröken Bok bakes a square cake measuring 3×3. Karlson immediately cuts out four 1×1 square pieces for himself from the cake, with the sides of the pieces parallel to the sides of the cake (not necessarily along the grid lines of the 3×3). After that, Little Man cuts out a square piece for himself from the remaining part of the cake, with the sides also parallel to the sides of the cake. What is the largest piece of cake that Little Man can expect, regardless of Karlson's actions?
On the left, the placement of Carlson's squares is marked, with which Little will obviously not be able to get a square with a side of more than $1 / 3$. Let's prove that Little will always be able to cut out a $1 / 3 \times 1 / 3$ square.
![](https://cdn.mathpix.com/cropped/2024_05_06_bca77ef9e87e9e47ba11g-43.jpg?height=450&width=922&top_left_y=478&top_left_x=566)

First method. Divide the cake into 81 $1 / 3 \times 1 / 3$ squares. Each of Carlson's squares will touch no more than 16 of these squares. Therefore, at least $81-4 \cdot 16=17$ squares will remain intact. Little will be able to cut out one of them.

Second method. On the right, five pieces of cake measuring $1 / 3 \times 1 / 3$ are arranged such that each of the four Carlson's squares will touch no more than one of them. Therefore, Little will always be able to take one of these pieces.

## Answer

$1 / 3 \times 1 / 3$.