Apparently Simple Puzzle

Question

So a picture appeared on FB asking for the answer to the picture below, to which many people responded "14".

Everywhere I looked, people answered 14. However, I stood out because I was the only one who said that the answer is 11.25.

The problem can be represented algebraically like this:

4x = 5

x = 1.25

9x = ?

9(1.25) = 11.25

So my question is..how the heck did people get 14? Am I using the wrong approach?

Answer 1

The first picture shows $5$ squares, the $4$ smaller ones, along with the $2\times 2$ square. Counting $1\times 1$, $2\times 2$, and $3\times 3$ squares, the second picture shows $14$ squares.

With this reasoning, a $4\times 4$ grid would equal $30$.

In general, an $n\times n$ grid would equal $1^2+2^2+\ldots+n^2=\frac{n(n+1)(2n+1)}{6}$.

These are the square pyramidal numbers, because they are the number of cubes needed to build a pyramid with a square base $n$ levels high.

Answer 2

I could say it is numbering the smaller boxes from $1$ like below and then adding the diagonal from top left to top right and still be correct.

Note that this is an $AP$ and can be generalized to

$$\cfrac{n(n^2 + 1)}2$$

Answer 3

I see the answer being the total number of squares, regardless of their size, but squares. Of those I see 14. 9 small squares 4 squares (overlapping) of 4 squares each ( 1,2,4,5 - 2,3,5,6 - 4,5,7,8 - 5.6.8.9) and 1 large square.