The Empty Block

Spring 2020

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In the days and weeks following the first surprise of that early November morning, we observed a dramatic change in his interest in mathematics. We believed it was partly due to an animation series on YouTube called “Number Blocks”. In a short amount of time he was able to do two digit summations as well as single digit multiplications. He did all these in his mind without writing them down. He was particularly interested in squares of numbers. Soon he started to calculate the squares of the numbers greater than 10, again in his mind and without writing them down. 

I was very curious about how he did the calculations in his mind. One day after he told me the square of 12 was 144, I asked him how he knew that. When I heard the answer, I couldn’t believe my ears. 

“It is the sum of 121 and 11 and 12” He replied in a confident tone. 

At first, I considered the possibility that he had just thrown random numbers at me, which was not like Mortaza. To verify his response, I added these numbers and it checked out. The sum of 11 and 12 and 121 was 144. So, I asked “How do you know these numbers add up to 12 times 12?” 

If I was surprised and somehow confused, it was nothing compared to what I felt when I heard his answer. “If I added 11 and 11, then one block would remain empty” he said in a matter-of-fact tone, as if nothing in the world was more obvious than this. It was clear that he was under the impression that my confusion was because I did not understand why we had to add the sum of 11 and 12 to 121, instead of adding the sum of 11 and 11 to 121. The rest seemed too obvious to him to even explain. 

Later on, when I thought about his answer, I realized that Mortaza had a graphical approach to calculate the square of numbers. He must have picked it up while watching the “Number Blocks” animations, in which numbers are in the forms of blocks. For example, number nine can be in the form of one column of nine blocks or a square of three by three blocks. Mortaza’s approach to calculate the square of a number was based on the square of the previous number.The image below, I believe, explains what he had in mind:

 

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In mathematical form:      n^2 = (n-1)^2 + (n-1) + n

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A few days later, I took Mortaza for a walk in the nearby woods. During our walk, I asked him if he could tell me what the square of 14 was. A few moments later he responded 196. Then I asked him how he had calculated it. I thought I knew what his answer would be:  "196 is the sum of 169 (square of 13) plus 13 plus 14." Nevertheless, I asked him because I enjoyed hearing he said it. His answer, however, was completely different. He said 196 was the sum of 100 and 40 and 40 and 16. It took me a while before I understood his logic. The interesting fact about his new method was that, you didn't need to know the square of the previous number. Again, he had used a graphical approach. He had divided a 14 by 14 square into 2 squares and 2 equal rectangles, the area of each of them was very easy to calculate. The figure blow explains this approach. 

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For numbers greater than 10, this approach can be stated in mathematical form as:

n^2=100+2*(n-10)+(n-10)^2