# 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. Sep 30 '13 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. Sep 30 '13 at 1:44
• btw, I'm voting your answer as correct. Just waiting for the time Sep 30 '13 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. Sep 30 '13 at 1:53
• Thank you, for the answer. Always learn something new here :) Sep 30 '13 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
Sep 30 '13 at 1:58

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." Sep 30 '13 at 2:16
• @Guest123456 it is something called mathjax / latex Sep 30 '13 at 2:16
• @BenjaminDickman the puzzle is extremely ambiguous, it's like saying; let's define a sequence as $a_0 = c$, what is $a_n$? Sep 30 '13 at 2:20
• @user2798694 Agreed! Sep 30 '13 at 2:46

Think it simply. Count the total amount of squares.

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.

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