A rectangular game board is composed of identical squares

A rectangular game board is composed of identical squares arranged in a rectangular array of $r$ rows and $r+1$ columns. The $r$ rows are numbered from $1$ through $r$ , and the $r+1$ columns are numbered from $1$ through $r+1$. If $r>10$, which of the following represents the number of squares on the board that are neither in the $4\t\h$ row nor in the $7\t\h$ column?

  1. $r^2-r$
  2. $r^2-1$
  3. $r^2$
  4. $r^2+1$
  5. $r^2+r$

So, you were trying to be a good test taker and practice for the GRE with PowerPrep online. Buuuut then you had some questions about the quant section—specifically question 10 of the second Quantitative section of Practice Test 1. Those questions testing our knowledge of Counting Methods can be kind of tricky, but never fear, PrepScholar has got your back!

Survey the Question

Let’s search the problem for clues as to what it will be testing, as this will help shift our minds to think about what type of math knowledge we’ll use to solve this question. Pay attention to any words that sound math-specific and anything special about what the numbers look like, and mark them on our paper.

We have a rectangular board divided into row and columns, and we’ll want to count the number of squares meeting certain criteria. So we will likely utilize what we’ve learned about Counting Methods and Quadrilaterals here. Let’s keep what we’ve learned about this skill at the tip of our minds as we approach this question.

What Do We Know?

Let’s carefully read through the question and make a list of the things that we know.


  1. We have a rectangular board containing $r$ rows and $r+1$ columns
  2. We want to express the number of squares on the board that are neither in the $4\t\h$ row nor $7\t\h$ column


Develop a Plan

A game board! This should be fun! Alright, so we want to count the number of squares that are NOT in the $4\t\h$ row NOR the $7\t\h$ column. And our game board has $r$ rows and $r+1$ columns. Seems like we can find our answer by: 1) finding the total number of squares and 2)subtracting the squares from the $4\t\h$ row and $7\t\h$ column. We need to make sure we don’t double count a square to remove in this second step though, as the $4\t\h$ row and the $7\t\h$ column have one square where they intersect.

Solve the Question

Well, to find the number of rows in the entire board game, it’s sort of like finding the area of a rectangle where we multiply the length and the width. So if we have $r$ rows and $r+1$ columns, then we’ll have $r·(r+1)$ total squares.

We need to subtract out the squares in the $4\t\h$ row and $7\t\h$ column now. Since each row has $r$ squares and each column has $r+1$ squares, it seems as if we’ll subtract out $r+(r+1)$ squares. However, each row and column pair has one square where they intersect, and we don’t want to subtract it out twice. So we’ll subtract away one less than what we previously thought, meaning we’ll subtract out $r+(r+1)-1$ squares, or more simply $2r$ squares.

So our final answer should be:

$$r(r+1) – 2r$$

The correct answer is A, $r^2-r$.

What Did We Learn

Well, not quite as fun as we might have expected a game board problem to be, but not too bad. The key to this question was to realize that we just needed to count the total number of squares and then subtract out the squares we didn’t want, making sure that we didn’t double count the square where the $4\t\h$ row and $7\t\h$ column intersected.


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