50 STARS but how many Rows?

Bill, Brilliant.org scrambles up the order of solutions. You can't depend on how choices are ordered. My suggestion is to list the choices in the problem statement as 1), 2), 3)... etc, and then the choices should be 1, 2, 3... etc, which will be scrambled up. Like this

1) 9 and 11 only
2) All solutions below
3) 33
4) 3
5) 9
6) 11

etc, however you want to order them

Let $m$ be the number of light rows and $n$ be the number of stars in each light row, where $m,n \in \mathbb{N}$ .

Then $m+1$ and $n+1$ are the number of heavy rows and the number of stars in each heavy row, respectively. Then

$\begin{aligned} (m + 1)(n + 1) + mn &= 50 \\ \\ 2mn + m + n + 1 &= 50 \\ \\ 4mn + 2m + 2n + 1 = 99 \\ \\ (2m + 1)(2n + 1) = 99 \end{aligned}$

so $(2m + 1)$ and $(2n + 1)$ must be factor pairs of $99$ .

The factor pairs of $99$ are $(1,99),(3,33),(9,11),(11,9),(33,3)$ and $(99,1)$ which mean the solutions for $(m,n)$ are $(0,49),(1,16),(4,5),(5,4),(16,1)$ and $(49,0)$ .

The first and last are excluded due to our restriction. From the rest, the total number of rows, $2m + 1$ , could be $\boxed{3,9,11,33}$

let x = number of small or "light" rows.
let (x+1) = number of large or "heavy" rows.
let y = number of stars in "light" rows.
let (y+1) = of stars in "heavy" rows.

Then (x+1)(y+1) + xy = 50 Solve for x in terms of y.

x = (49-y)/(2y+1) Then dividing.

x = -0.5 + 49.5/(2y+1) so x = 49.5/(2y+1) - 0.5

We know that x is an integer between 1 and 50, so 49.5/(2y+1) must be an {integer + 0.5} so when we subtract 0.5, we get an integer.

Multiplying by two: 2{49.5/(2y+1)} = 99/(2y+1) = Integer

Since 99 = 1x3x3x11, 2y+1 must be the product of one or more of the factors in order to evenly divide into 99.

Therefore,
If 2y+1 = 1 Then y = 0 Which is not a solution.
If 2y+1 = 3 Then y = 1 This is a solution since x = (49-y)/(2y+1) gives x=16 and x+(x+1) = 33 total rows.
If 2y+1 = 9 Then y = 4 gives x = 5 and x+(x+1) = 11 total rows.
If 2y+1 = 33 Then y = 16 gives x =1 and x+(x+1) = 3 total rows.
If 2y+1 = 11 Then y = 5 gives x = 4 and x+(x+1) = 9 total rows.
If 2y+1 = 99 Then y = 49 gives x=0 and is not a solution.

With no other information we can not choose between these unless we look at the flag!