By Rashmi Jejurikar
Dhruv and Rhea were walking back from school, humming the tunes of their favourite songs. Suddenly, Rhea stopped walking, as if struck by a question.
Dhruv (D): What’s up Rhea? All okay?
Rhea (R): Yeah, all okay. I was just thinking about my math class today. There is something that’s been bothering me.
D: Care to explain? Maybe I can help!
R: Well, umm.. Alright. So, we were discussing consecutive numbers in our lesson today. Anu ma’am gave us a whole bunch of pairs of consecutive numbers and asked us to add each pair and see if we can find any patterns.
D: Yeah, I remember this lesson from last year. Did you find the pattern?
R: Yes, we did at once! That the sum of two consecutive numbers is always odd.
D (confused): Cool! So what’s bothering you?
R: Umm.. Let me try to explain. You see, when many of us had found this pattern, Anu ma’am asked us if we can prove it. At first, I said something like, “we’ve seen this happen in so many pairs! And no matter which pair of consecutive numbers we try this out with, we never find a pair that results in an even sum.” And then...
D: Well, that’s a bit problema…
R: Dhruv, you’re interrupting me AGAIN! Can you stop doing that pleeease?
D: Oops, I’m so sorry! Continue..
R: So, as soon as I said this, Jyoti raised her hand. She said, “how can we be sure there isn’t a pair out there that yields an even sum?” This confused me. I guess, if we try to prove it my way, we can’t ever be sure. Umm.. Because we can never check every pair, since there is no limit to the number of numbers!
D: You’re right.. Did you bring this up in class?
R: Yeah. Anu ma’am said, “Exactly! Now, can you think of ways in which we can prove this such that we are totally certain that we are right?” There was a lot of discussion, and then Ronak brought up something interesting.
As Rhea was talking, they had reached Dhruv’s house. Upset that their conversation was about to be cut short, Dhruv invited Rhea home so they could continue with the discussion.
D: Do you want to just come home? I’m interested to know what happened next, and a whiteboard and some hot pakodas might be useful.
R: That would be great!
Belly full of pakodas, Dhruv and Rhea settled down in front of the whiteboard. Rhea began to write.
R: So, I was saying, Ronak found a way to prove that the pattern we found would hold for any number. He said something like:
Any number, no matter how large or small, can have only 10 possible last digits (0, 1, 2, 3, 4, 5, 6, 7, 8, or 9).
An even number can only have 0, 2, 4, 6 or 8 as its last digit.
If we show that adding all possible (10) pairs of consecutive digits from the list above (0-1, 1-2, ... 8-9, 9-0) results in a sum with an odd last digit (1, 3, 5, 7 or 9), we can say that the sum of any two consecutive numbers is odd.
D: That’s quite cool! Let’s try a few numbers. For example, the sum of 93 and 94 is 187. The last digits of 93 and 94 are 3 and 4. So when we add them, we get 7, which is an odd number, and is also the last digit of 187.
R: Yeah, so once we show that the sum of 3 and 4 is odd, we will always get an odd sum if we add any pair of numbers with 3 and 4 as their last digits: 83-84, 73-74, 123333-123334. All these sums would have 7 as the last digit, which would make them odd.
D: That’s a really neat trick, isn’t it?
R: I thought so too, and so did the rest of the class. But Anu ma’am said that this is trickery.
D: Trickery, you say?
R: Yeah.. I’ve been wondering about this ever since.
D: Why didn’t you bring this up in class?
R: The bell rang at just that moment, and we had to stop our discussion there. Now I can’t ask her until Monday!
D: Did she give you a hint at least?
R: She did. Although.. It doesn’t seem like much of a hint to me. She said, “you are thinking of a different notion of even numbers. Try to think of a concept of even numbers without using numerals.”
D: Cryptic, indeed.
The smell of pakodas and rain floating through the air, Dhruv and Rhea started to feel sleepy. Rhea walked back home, still thinking about one question, “How is it possible that numbers and numerals are different from each other?”
Pause and Reflect: How would you answer Anu Ma’am’s question? Why did she say that the proof discussed above is some kind of trickery?