10}

Points $A, B, C, D$ are marked on a sheet of paper. A recognition device can perform two types of operations with absolute accuracy: a) measure the distance in centimeters between two given points; b) compare two given numbers. What is the minimum number of operations this device needs to perform to definitely determine whether the quadrilateral $A B C D$ is a rectangle?
To determine whether $A B C D$ is a rectangle, it is sufficient to check the equalities $A B=C D$, $B C=A D$, and $A C=B D$ - a total of 9 operations (3 operations for each equality: two measurements and one comparison).

A rectangle $ABCD$ will be a square if $A B=B C$ - for this, one more, the 10th, operation of comparing the lengths of segments $A B$ and $B C$ is needed. We will prove that a smaller number of operations will not suffice in both cases, i.e., that all the specified operations must be performed. Indeed, if we do not know that two opposite sides of a quadrilateral are equal, we cannot even assert that $A B C D$ is a parallelogram, since an isosceles trapezoid (its diagonals are equal) is also possible. Therefore, checking the equalities $A B=C D$ and $B C=A D$ is necessary. If we do not know that $A C=B D$, then $A B C D$ can be any parallelogram. Thus, checking the equality of the diagonals is also necessary. Finally, in the case of a square, it is also necessary to check the equality of two adjacent sides.