Singapore Math Guru

How many squares are there in this figure?

6by6squares

How many squares can you find here?

 

How many of you have seen this question in the popular advertisement in POPCLUB magazine? POPCLUB magazine has a circulation of approximately 250,000 copies. If you are a member of Popular Bookstore, chances are that you would receive a copy of the POPCLUB magazine.

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For those who are interested enough to find out the answer, here it is. And by the way, this question in aligned with MOE syllabus standards.

To find the answer to the question above, we must consider the figures below.

1square

1 by 1 square

 

Now let us examine the scenario where there is only 1 square. Just 1 square. If there is only 1 square, no matter how you look at it, there is only 1 square.

And so we say that there is only 1 one by one square:

1 x 1 = 1

 

2x2square

Let’s consider the scenario where there are only 4 squares as in the figure above. How many squares are there here? Is your answer 4?

If you observe carefully, there are actually 5 squares in total. 1 square is made up of 4 identical smaller squares and there are 4 smaller identical smaller squares.

1 + 4 = 5

There are actually 4 one by one squares and 1 four by four square.

And if you are trying to sieve out a pattern from this, you will find that you can represent 5 as:

1 x 1 + 2 x 2 = 5

Can you see a pattern emerging here?

3by3squares

Now, let us consider a scenario where there are 9 squares shown above. These 9 squares make up 1 big square. The total number of squares can be calculated as:

1 x 1 + 2 x 2 + 3 x 3 = 14

There are 1 three by three square, 4 two by two squares and 9 one by one squares.

Do bear with us for a moment here. But I believe that those of you reading this get the picture.

4by4squares

Now, let us consider 16 identical squares within 1 square. as shown in the figure above.

The number of squares is :

1 x 1 + 2 x 2 + 3 x 3 + 4 x 4 = 30

There is 1 four by four square, 4 three by three squares, 9 two by two squares and 16 one by one squares.

5by5squares

Now consider 25 identical squares making up 1 big square as shown above.

The number of squares can be represented as:

1 x 1 + 2 x 2 + 3 x 3 + 4 x 4 + 5 x 5 = 55

There is 1 five by five square, 4 four by four squares, 9 three by three squares, 16 two by two squares and 25 one by one squares.

6by6squares

Now, last but not least, let us consider 36 squares within a big square as shown in the figure above. Note that there are 6 identical squares along each side of the large square. This is the question whose answer we are seeking to find anyway. if you have been paying attention to this point, you will understand how to solve this right now.

The total number of squares can be represented by:

1 x 1 + 2 x 2 + 3 x 3 + 4 x 4 + 5 x 5 + 6 x 6 = 91

To make this clearer, there are:

- 36 one by one squares

- 25 two by two squares

- 16 three by three squares

- 9 four by four squares

- 4 five by five squares

- 1 six by six square

And there you have it. The answer to this question is 91 squares in total.

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