An Inspiring Problem

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Maryam Mirzakhani

In 2014, Maryam Mirzakhani became the first woman awarded the Fields Medal, the most prestigious mathematics award in the world.   Though phenomenally successful in math,  as a child, Maryam was interested in becoming a writer.  It took one inspiring problem to turn her attention from writing to mathematics.

Not Interested in Elementary or Middle School Math

Though she is known for her mathematical ability now, Maryam Mirzakhani had no interest in math in primary school.  In an interview with  Clay Mathematics Institute, she noted:

“As a kid, I dreamt of becoming a writer. My most exciting pastime was reading novels; in fact, I would read anything I could find. I never thought I would pursue mathematics before my last year in high school.”

As you can see from her quote, interest in mathematics can blossom in late high school rather than in elementary or middle school.  This  later blossoming does not restrict a student’s mathematical ability.   All it takes sometimes is an interesting problem.

An Intriguing Problem

“My first memory of mathematics is probably the time that he [her brother] told me about the problem of adding numbers from 1 to 100. I think he had read in a popular science journal how Gauss solved this problem. The solution was quite fascinating for me. That was the first time I enjoyed a beautiful solution, though I couldn’t find it myself.”

Her interest was peaked with an inspiring problem:
$$1 + 2 + 3 + … + 99+ 100 =\: ?$$

When I give this type of problem to my elementary or middle school students, they often start adding the numbers one at a time:
\begin{align} 1 + 2 &= 3\\ 3+ 3 &= 6\\ 4 + 6 &= 10\\ 5 + 10 &= 15\\ &\vdots \end{align}
If continued until the problem is solved, this method requires 99 sums. Doing the addition problem one sum at a time is beyond most student’s patience level. However, my students know me well enough to realize that there must be another solution — asking them to do 99 sums in a row does not fit my teaching style.

One Elegant Solution

One such method involves changing the order of addition (see PEMDAS for more info on when changing the order of operations is valid). Numbers that sum to 100 are paired. With this change,
\begin{align}1 + 2 + 3 + &… + 99+ 100\\ &=100 + (1+99) + (2 + 98) + (3+97) + … + (49+51) + 50\\ &= 100 + 49 \times 100 + 50\\ &= 50\times 100 + 50\\ &= 5050\end{align}

Do your students have an opportunity to be inspire by such problems? See Math Tidbits for more inspiring problems.