More understandable explanation for Cat-Rabbit-Dog quiz This is a quiz from Brilliant.org Algebra Fundamentals courses.
Quiz:

At Step 40, how many cats will there be in the 7th row?
Answer: 14
Explanation given by Brilliant.org is not clear to me so if someone can give a better explanation it will be helpful.
Explanation by Brilliant.org

In rows where dividing by 3 has a remainder of 1, the pattern is
cat-rabbit-dog.
7 when divided by 3 has a remainder of 1, so cats occur at 1, 4, 7,
etc., i.e. at points where dividing by 3 has a remainder of 1.
In step 40, the grid is 40 columns wide. The full cat-rabbit-dog
pattern happens for $39\div 3=13$ times, but since the pattern
at row 7 starts with a cat, column 40 has one extra cat. So there are
$13 + 1 = 14$ cats.
Note: A more compact way of saying "dividing $x$ by 3 has a remainder
of 1" is $x \mod 3 = 1$.

 A: In Step 40, there will be $40$ rows each containing $40$ animals. In each row, each group of three consecutive animals consists of one of each type. You can separate the seventh row into the first animal followed by $13$ groups of three. The $13$ groups of three will contain $13$ cats, so the answer will be $14$ if the first animal in the seventh row is another cat, and $13$ otherwise.
Counting down the first column, we see that the first, fourth, seventh, etc. (going up by $3$ each time) are cats. So the answer is $14$, because there is an extra cat at the start of the $7$th row.
What the explanation is trying to add is that you would get the same answer if the row was changed to any other number in the sequence $1,4,7,...$ (but no bigger than $40$). These numbers are the numbers that leave a remainder $1$ when divided by $3$. Other rows - those whose position divides exactly by $3$ or leaves remainder $2$ - will not have a cat at the start, so would have $13$ cats.
A: It strikes me that this is a case where talking about division and remainders clouds the issue. I would explain things as follows:
Every third row is identical, so the seventh row is the same as the fourth is the same as the first.
At the fortieth step, the first row will have $13$ groups of three (cat, rabbit, dog) and one extra cat, for a total of $14$ cats. So the seventh row will also have $14$ cats.
A: By reading all the answers I able to understand more and here is my simple explanation.

*

*Its a sequence.

*1st row and 4th row were same in Step 4,
so the pattern is same in every 3rd row,
pattern in 1st and 4th row is Cat-Rabbit-Dog-Cat-Rabbit-Dog-...

*In Step n we have n2 animals and n animals in each row.

*Question is, in Step 40 how many cats are in row 7th?

*In Step 40 each row has 40 animals.
A: Apparently, in step $n$, there is an animal in position $(r,c)$ iff $1\le r,c\le n$, and that animal is a cat iff $r+c\equiv 2\pmod 3$. So for $n=40$ and $r=7$, we want to find the number of $c$ with $1\le c\le 40$ and $7+c\equiv 2\pmod 3$. Those $c$ have the form $c=1+3k$ where $k$ is allowed to range from $0$ (making $c=1$) up to $13$ (making $c=40$).  So there are $$14$$ valid choices for $k$.
