Jayachandran/Mint
Jayachandran/Mint

Opinion: Maths that I bet will light up an 11-year-old’s eyes

An 11-year-old might enjoy the way a seemingly intractable mathematical problem can be solved easily by approaching it from a different perspective

My generation—various election results over the last few years have got me to believe—is a lost cause. I’m not going to qualify that or debate it. Nor am I going to discuss any particular election results—besides anything else, I’m sure you’ve had your fill of such discussion.

What I am going to do instead here is focus on younger generations, in the hope and belief that they are not lost causes. In particular, I want to attempt a response to a request I ran across in the middle of May. An English professor of mathematics, Peter Rowlett, posted it on Twitter: “An 11-year-old has shown interest in what I do (teach maths at uni). She asked me to set her a problem so she could see what maths at uni is like (not necessarily to solve). What would you show her?" Rowlett followed up with this: “This comes from her surprise that I don’t do big sums in my head, and I don’t use a protractor at work."

Since I don’t teach maths at a university—though neither do I use a protractor, nor do big sums in my head—I’m going to take the liberty of modifying that request somewhat. How might I persuade an 11-year-old that maths is not (just) about big sums and protractors? What might I show her about maths that would make her eyes light up? So that just maybe, she will choose to pursue maths in years to come?

Jumping right in: I would start with that institution beloved to all 11-year-olds and to some of us older folk: Birthdays and their parties. Specifically, the famous birthday paradox that, if you know your arithmetic, isn’t a paradox. It goes like this. I’m in a room with a gaggle of friends at a birthday party. You look in on the lot of us and say: “I bet 500 that at least two of you have the same birthday." Should I scoff at your naïvete and jump to take the bet, sure I’m going to make an easy 500 bucks?

That depends, of course, on how many there are in my gaggle. If it’s just two of us, for example, I can be nearly certain we have different birthdays: Mine might be 31 May, and pal Azad’s might be on any of the other 364 days of the year. If it’s three of us, I can still be pretty certain of winning the 500: Mine on 31 May, Azad’s on one of 364 other days, and Madhu’s on one of the remaining 363 days of the year. I would probably be similarly sanguine about winning if we were four, or five, or 10, or perhaps even 30, in our gaggle. But what if there were 365 of us crammed in that room? Now I might not take your bet so quickly. With that many people, I think, it must be pretty certain that two of them share a birthday. Which it is. I’d be a fool to take your bet.

The chances of you winning, in other words, go from being very small when there are just two of us, to near certainty when there are 365 of us. So what about in between? There must be a count of guests at which it becomes better than 50-50 that you will win. What is that number?

Answer: 23. Your fairly average-sized 11-year-old birthday party. Make it 30 kids and the chance goes up to 70%. Many’s the time that I’ve been with that many kids—party, classroom, school trip—and played out this little puzzle and watched a few dozen pairs of eyes go round with amazement. Some then ask for an explanation—not hard, so I’ll leave it for you to work out—and that’s when I like to think maths has got them hooked.

Another beauty: The Möbius strip. Lay three long strips of paper, perhaps 2-feet long and an inch wide, flat on a table. Lift up the two ends of the first strip and tape them together to make a simple loop. Do the same with the second and third, except in each case twist one end over before taping it to the other end—so both are now loops with that twist. They may not look very different from the first strip, but hold that thought.

Give your 11-year-old a pair of scissors and tell her to start cutting. First, the first loop—right down the middle, all the way around. Sure enough, she will have two separate loops, and will probably be looking at you a little strangely: What’s with this silly taping and cutting?

Thus, it’s exactly the right time to give her the second loop and tell her to cut it right down the middle, all the way around. Get ready for some round-eyed amazement, because what results is definitely not two separate loops. Don’t let her rest on her laurels, either. Get her started on the third loop, but this time she must cut not down the middle, but a third of the way from one edge, all the way around. More round eyes, I assure you. Again, I’ll leave you to work out exactly what results from the second and third cuts. (Try it yourself). Not what the 11-year-old expects, that’s for certain.

There’s plenty more magic possible with loops of paper, twisted or otherwise, and a pair of scissors. Experimenting with it all is a good way for an 11-year-old to get familiar with ideas of surfaces and edges—and who knows, that might develop into an interest in topology, the mathematical field that studies them.

Next: For something just a bit more challenging and yet still fun, I might choose adding up numbers. Especially because the great German mathematician, Carl Friedrich Gauss, figured out something about that as a child himself. Here’s the question to ask the 11-year-old: Can you add up the numbers from 1 to 100 in no more than 15 seconds?

Or, to put it mathematically:

S = 1 + 2 + 3 + 4 + 5... + 99 + 100.

Can the 11-year-old find S in less than 15 seconds?

Well, that’s what Gauss did. The story goes that he had been a pest in class, so his annoyed teacher sent him to a corner and gave him exactly this problem as a punishment. A hundred numbers to add! Surely it would take young Carl forever, the teacher reasoned. But young Carl had an answer in a trice. How?

First, he wrote his sum this way:

S = 100 + 99 + 98... + 2 + 1.

Now he added both those expressions for S, each of the 100 pairs at a time:

S + S = (1 + 100) + (2 + 99) + ... + (99 + 2) + (100 + 1)

That is:

2 x S = 101 + 101 + 101 + ... + 101 + 101 = 100 x 101.

Which then gives us:

S = 50 x 101 = 5,050.

Young Carl had his answer in double quick time, but he had something more valuable: A method, a formula, to find the sum of any such sequence, meaning with any number of terms. Want to add up the numbers from 1 to 4,109? Answer: 8,443,995 and no, I didn’t actually add them, I just applied the formula. And, there’s a lesson that’s still more valuable: If you find a different way of looking at a problem—flip it over if you like, as Gauss did—you might just find a solution.

That’s typical of the way mathematics goes. Faced with a seemingly intractable problem, mathematicians will search for different angles of attack, apply techniques from other fields of maths. Often enough, that gives them new perspective, new knowledge and a possible path to a solution. Whole new areas of mathematics have opened up this way.

An 11-year-old might enjoy the way Gauss played with numbers. But I think the broader idea, of considering problems in different ways, will appeal to her even more, besides being a good prescription for even an 11-year-old’s life.

Finally: A quick crack at measurement. Give the kid a piece of string a centimetre long, ask her to measure the dimensions of the room. She’ll likely giggle. “What’s the matter," you ask her, “that’s too easy for you?" Tell her to use the same piece of string to measure the height of the Qutb Minar. She’ll crack up then, amazed at these ridiculous tasks you’re setting her. But why is she laughing? Because without even thinking about it, she knows something about a crucial life skill that’s founded on mathematics: Estimation. Here’s your chance to tell her more.

So there you are: Four different peeks into the wide world of mathematics—numbers, topology, probability, estimation—that I’m sure an 11-year-old will appreciate. So sure, I’ll bet 500 on it.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His latest book is Jukebox Mathemagic: Always One More Dance. His Twitter handle is @DeathEndsFun

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