Missing the Marks

7C2 -2 ...REASON:there are 7 marks that is, 0,4,9,16,25,36 taking combinations of any two we get the readings by their difference so 7 c2 but by inclusion exclusion we need to remove the common differences which are two in number .

Consider all numbers of the form $x^2-y^2$ , where $x,y$ are integers such that $6 \geq x > y \geq 0,$ . The number of different ordered pairs $(x,y)$ that fulfil the inequality are ${7 \choose 2}=21$ . However, there exists two integers $n$ such that there exists two different pairs of integers $(x_1,y_1),(x_2,y_2)$ satisfying the above criteria where $x_1^2-y_1^2=x_2^2-y_2^2=n$ (these two numbers being $16=5^2-3^2=4^2-0^2, 9=5^2-4^2=3^2-0^2$ ).

The total number of distinct lengths that can thus be taken are thus those that fulfil the above conditions, excluding repetitions, which gives us $21-2=\fbox{19}$

This is what I thought initially as well, but then I realised that the question was asking which lengths can be measured without moving the ruler. Every number has a factor of one, so if we could move the ruler we could measure up to infinity inches.