Neznayka draws a rectangle, divides it into 64 smaller rectangles by drawing $7$ straight lines parallel to each of the original rectangle's sides.
After that, Znayka points to $n$ rectangles of the division at the same time, and Neznayka names/gives/reveals the areas of each of these rectangles.
What is the smallest value of $n$ for Znayka to be able to name/give/reveal the areas of all the rectangles in the division?
On
The answer is $n=15$.
To convince yourself it is true try to prove using induction the more general property: For a partition of the rectangle into $m\times m$ smaller rectangles (using $m-1$ straight lines parallel to each of the original rectangle's sides) the minimal number is $n=2m-1$ (In your case $m=8$).
After this, try and go even further: what about partitions into $m\times k$ smaller rectangles?
Again, use induction to prove $n=m+k-1$
BASIC LIMIT :
There are $8$ unknowns along the length of the rectangle.
There are $8$ unknowns along the width of the rectangle.
We require $8+8=16$ Equations to get those unknowns.
When we know those $16$ unknowns , then all areas are known.
The areas we might choose are :
$8$ Diagonal Elements
$7$ Elements just below the Diagonal
$1$ Element at the top Corner
UPDATED LIMIT :
Shown in the Diagram , we have $16$ Elements which are selected to highlight that we have to take ratios of neighboring terms which are left-right & which are up-down.
Imagine that we double all widths while halving all lengths : Since $2 \times 1/2=1$ , we will still have same areas for all Parts.
This implies that we have $1$ less unknown , hence we have only $16-1=15$ unknowns & hence we require only $15$ Equations via $15$ Elements.
We can skip the top-right green Element in the Diagram.
Every row & every Column can be obtained via ratios of neighboring terms.
Hence $n=15$ here.