Compute the smallest positive integer $n$ such that， for any given integer $p\geq n$, we can partition a given square into $p$ number of squares (the small squares are not necessarily congruent)
I think the answer is 4, clearly square can be partition into the perfect square number pieces.
Given any dissection, we can find one with three more squares by cutting one of the squares into $4$. We can find dissections with $6, 7,$ and $8$ squares, so we can find one with all larger numbers. We can't find one with $5$, so $n=6$. The $6$ is a $2 \times 2$ plus five $1 \times 1$ squares, the $7$ is three $2 \times 2$ squares plus four $1 \times 1$, and the $8$ is a $3 \times 3$ plus seven $1 \times 1$ squares as shown below.
