Surprises strewn around and along the path we walk.

Musings of the mentor for the Math Explorations class, Dr Utpal Chattopadhyay, on his first day at the Gifted Summer Program 2024 #GSP2024 by Genwise. Utpal's (enviable!) bio at the end of the article...

Occasionally I conduct workshops with early middle school students who have an interest in mathematics. One of my favorite activities in these workshops, particularly in the first day, is to ask them the following question:

The Question

Your friend says she has been offered 1 Crore Rupees if she can give three prime numbers between 1001 and 10001 which will add up to 9000. She thinks you are good at Math and is asking for your help. What will be your advice to her? Can you help her win 1 Crore?

The class reaction and approach to solving the problem

These students in my workshop this year are very sweet and quite green. Confronted with this strange question, they initially sit there with stunned silence, almost frozen. So I tell them can they simplify the problem a bit to take a crack at it. For example, can they try to solve the following?

Find three prime numbers between 10 and 100 which will add up to 120

With this new version of the problem, there is movement among them. They are very busy trying out with numbers. One student raises her hand and said I found three primes: 47, 71 and 2. I gently point out 2 is a not a prime between 10 and 100. Another students murmurs to herself... does not seem possible. I lean over to hear her thoughts and ask her to write that down in complete sentences.

Another student suddenly shouts, Sir it is not possible. All prime numbers above 2 must have 1, 3, 7, or 9 as their unit's digit. You take any three of these for example 3, 7 and 9 and add them it will not have 0 in the unit place. His conclusion is correct. His line of argument is interesting and non-standard. ( I will give a simpler reason why this is not possible, in a moment) . So I prod him and ask him how may combinations of three numbers out of the four 1,3,7,9 he has tried. So he says there are three combinations and he tried them all. I tell him to slow down and check carefully if there are three combinations or four. Others are listening to all this. So I ask them can they see how many ways you can choose 3 objects out of four? They write the four combinations (3,7,9), (7,9,1), (9,1,3) and (1,3,7). So I tell them, let us digress a bit and understand this choosing 3 out of 4 a bit more carefully. Can they "see" that every time they are choosing three, they are actually choosing a single number to leave out. And of course there are only 4 numbers and they can leave out only one number, one by one, so the number of choosing 3 out of 4 has to be 4, without writing down all the combinations of 3 out of 4. At this stage I tell them one of the beauties of math is to streamline thinking to such an extent that they can see an immediate generalization of this result. If they have n distinct object, THEN the number of ways to choose (n-1) out of n IS THE SAME ) as number of ways of choosing ONE out of n which is EQUAL TO n.

I introduce them to the notation nCr, given n objects choose r, (they ask what can r be and they answer their own question), and I tell them we have proved collaboratively the result that nC1 = nC(n-1) = n.

When I gave them the problem I HAD NO IDEA that I would be discussing basic counting of combination and permutation, a subject usually taught in the 9th or 10th standard. But it came naturally and they understood all this at that moment. Obviously retention would be low, but that is not the point. The point is they saw the beauty in the discussion and when they learn combination again in later years, those topics will sing music to their ears. The vitality of students' novel and unanticipated ways of thinking.

And now the boring simpler solution. Three primes between 10 and 100 are all odd numbers. Three odd numbers must add up to an odd number hence there are no three primes between 10 and 100 which will add up to 120.

They can ask their friend not to waste her time.

-- end of Utpal's musings

Utpal Chattopadhyay

Utpal is senior mentor and course designer at GenWise.

His passion in life is "to firmly establish science as a wonderful culture in developing young minds."

Since 2010, Utpal has been teaching Advanced Physics to talented undergraduate students at the Indian Statistical Institute in Bangalore. He was one of the founders of Curiouscity Science Education, where he conducted numerous science sessions with middle school children. Utpal has been also been facilitating courses on Physics and Mathematics to gifted school students for the past several years. Utpal has a bachelor's degree with honours in Physics from IIT Kharagpur (1st in his class), and a PhD from the State University of New York at Stony Brook. His corporate/ professional work experience includes stints at Bell Labs and Motorola (where he was a Director). Post 2005, he cannot imagine life without a chalk and the blackboard!

Since 2016, Utpal has facilitated numerous programs for GenWise - across our year-round and summer programs - encompassing topics in Mathematics and the Sciences.