Apparently Simple Puzzle

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?

• Well if you are just asking why they got 14, I guess Jared has already answered that. But you could also argue that the answer is 11.25. It depends on how you view the puzzle. I could say that the answer is 8 and have my own reasoning why. Commented Sep 30, 2013 at 1:55

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.

• Would my answer be incorrect? The problem was very vague. I would say that 11.25 is another possible answer from an algebraic standpoint. Commented Sep 30, 2013 at 1:44
• btw, I'm voting your answer as correct. Just waiting for the time Commented Sep 30, 2013 at 1:50
• @Guest123456: You've given a fine answer. The problem is not at all well-posed. I was just explaining what I believe to be the reasoning behind the common answer of $14$. I'm sure there are ways to reason that the answer should be one of many different numbers. Commented Sep 30, 2013 at 1:53
• Thank you, for the answer. Always learn something new here :) Commented Sep 30, 2013 at 1:53
• @Guest123456: The question is vague, and your answer is valid. (Another "wrong" ---but perfectly valid--- approach would be to set up a proportion such that $5$ is the total length of the line segments.) The puzzle would be better-posed if it included a $1\times 1$ square with answer "$1$", to help rule out what the puzzler considers "wrong" answers. (And, really, the puzzler shouldn't call alternative answers wrong; instead, he should say, simply, "That's not the solution I had in mind.")
– Blue
Commented Sep 30, 2013 at 1:58

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.

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

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

• Instead: Try turning each into an $n \times n$ multiplication table. Then the problem becomes: Sum the diagonal (to get the total number of "sub-squares"). More generally, you could sum all entries in the multiplication table to get the total number of "sub-rectangles." Commented Sep 30, 2013 at 2:16
• @Guest123456 it is something called mathjax / latex Commented Sep 30, 2013 at 2:16
• @BenjaminDickman the puzzle is extremely ambiguous, it's like saying; let's define a sequence as $a_0 = c$, what is $a_n$? Commented Sep 30, 2013 at 2:20
• @user2798694 Agreed! Commented Sep 30, 2013 at 2:46

Think it simply. Count the total amount of squares.

My answer is 20. If the first image... or pattern is valued at five then the second image has four patterns inside of it that equal 5. So 4×5=20. I believe this is correct but i also believe that there is more than 1 correct answer to the riddle