65194 – We Solve Problems

#### Dissections with certain properties , Pigeonhole principle (area and volume)

Every day, James bakes a square cake size $3\times3$. Jack immediately cuts out for himself four square pieces of size $1\times1$ with sides parallel to the sides of the cake $($not necessarily along the $3\times3$ grid lines$)$. After that, Sarah cuts out from the rest of the cake a square piece with sides, also parallel to the sides of the cake. What is the largest piece of cake that Sarah can count on, regardless of Jack’s actions?

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