**Neeraj Naidu**

Numbers are beautiful. But some numbers are more beautiful than others. The question, however, is what makes a number beautiful? Is it the way it looks? Or the properties it exhibits? Or both?

Let’s take 135. A perfectly normal looking number. It looks just fine. Not too beautiful, not too bad. But look closely. What do you see?

• 135 is written using three numbers: 1, 3 and 5. These are the first three odd positive numbers.

• 135 = 1^{1} + 3^{2} + 5^{3}

The two properties listed above makes 135, for me, a pretty number. It is also a possibility that this number might have many more properties which I have totally overlooked.

135 might just not be as splendid as the Ramanujan-Hardy number 1729, but it clearly makes the point that numbers are beautiful.

Now I am going to write about my lesson (taken for 8^{th} graders) when I exposed my students to the beauty of 1089 and how naturally it opened the door to enter the realm of palindromes and palindromic numbers.

One of the many great things about mathematics is that one concept or a question leads to another and opens a window to a brave new world. Its sequential property makes it easy for a mathematics teacher to traverse from one thing to another just like a man wanders from A to B and thus nothing needs to be thrown to children unexpectedly out of thin air.

I did the following exercise with middle school children. But it can also be done at the primary level.

So, I was talking to the children about addition and subtraction and how we keep adding and subtracting numbers in our minds while playing a game of marbles or cricket or while shopping or when we are arriving late somewhere and keep glancing at the time or when we travel and keep looking at the milestone or counting the spoons of sugar while preparing tea. Our minds keep calculating things all the time. I told the children that we would do an interesting exercise and all they needed to do was to add and subtract and keep their eyes open to be receptive of any beauty that might arise.

This number game is popularly known as 1089. Most mathematics teachers know and do this exercise with their students. I am writing this piece not to explain the method (which can be found in textbooks or on the internet) but the magic children do with it and how a math teacher can conduct a mathematics class full on curiosity, inquiry, conjecture framing and proving a conjecture, finding faults and most of all, enjoyment and appreciation of the beauty of the very art of mathematics.

Ask the children to take any 3-digit number. In books, it is usually written that you should not take the number where the units and hundreds digits are same. Like you can’t take 232 because both units and hundreds digit numbers are the same, i.e., 2. If you take such a number, then you won’t be able to go further in this exercise. But the important thing is that you should not tell this to the children. Let them take whatever three-digit number they wish to and get stuck. Let them figure it out.

Let’s suppose the number is 461.

You can choose to do it on the board simultaneously while the children do it in their notebooks or not to do it, but just speak. That is the teacher’s choice. But sometimes, if you are doing it in a primary school then it is helpful for children as they see it being done on the board.

Now you need to reverse or flip the number. So, 461 becomes 164.

Next, subtract the two numbers. (Larger minus smaller)

That gives us: 461 – 164 = 297

Next, once again reverse the digits of the new number. So, you’ll get 792.

Now add both numbers.

792 + 297 = 1089

If children struggle with reversing then the teacher can help or show how it’s done. While subtracting, I have seen that many (younger) children try to subtract the larger from the smaller number which gives a wrong result.

Ask the children what answer/number they got. Many will be surprised that even though they chose entirely different numbers, the end result is the same – 1089. A teacher can also play magician and inform the children that s/he can tell the result without knowing what number the child has initially taken. Children will be boggled when they hear 1089 from the teacher.

Believe me, when I did this exercise for the first time, I was flabbergasted. You will see that the children will be so moved by this exercise that they will instantly choose another three-digit number and start repeating the procedure to confirm that the answer is 1089 for any other three-digit number. However, they will be surprised again to get 1089. This will create a sense of wonder in the child’s mind.

Now there will be two problems. Some children may not get 1089. This could be because they chose a number that has the same units and hundreds digit. For example, 575.

Let’s see what the child might have done.

Now if you reverse 575, you’ll get the same number 575.

Now when you subtract both numbers, you’ll get zero.

The child will be stuck here.

Here the teacher can ask the students if all three-digit numbers yield the result 1089 when processed as mentioned above. Some will say ‘yes’ and some ‘no’. To the children who say ‘yes’, you can ask if they have done this exercise with all three-digit numbers? If ‘no’ then how can we say that it is true for all numbers? Now ask the children who said ‘no’. They will show the class why they didn’t get 1089. That’s because when they subtract, the result comes to zero. You can push the children to look for a pattern here. What are the numbers which will yield zero after subtraction and what is the property of such numbers?

Children can easily see that the numbers which remain the same after reversing the digits yield zero after subtraction. Give them time to find this pattern.

Here the teacher might introduce ‘palindromes’ and ‘palindromic numbers’ because all palindromic numbers like 131, 252, 444…. will not be able to form 1089 through the above-mentioned process. However, I personally didn’t introduce palindromes at this stage. I let the children play more with 1089.

If children are able to find numbers (by finding the pattern of numbers) that do not yield 1089 then you can ask them what numbers will give them 1089. You need to push the children in such a way that they will be able to formulate the idea that the number should have different units and hundreds place digits. This is important. That is how children will frame the conjecture.

There maybe some children, who despite using different numbers in the units and hundreds place, didn’t get 1089. There is a good reason for this. To understand this, let’s take the number 231.

Now reverse the digits, you’ll get 132.

Now subtract the numbers: 231-132 = 99

Now if you reverse 99, you’ll get 99 and after adding both, you’ll get 198 and not 1089. Now there is no mistake in the process. But to get 1089, you have to tell the child to do it in a certain fashion.

2 3 1

1 3 2 (reverse of 2 3 1)

– ————

0 9 9

9 9 0 (reverse of 0 9 9)

+ ———–

1 0 8 9

So, by now most children in the class will be able to frame the conjecture of 1089. You can only prove the conjecture true if it applies for all three-digit numbers that have different units and hundreds digits. The children may point out that there are so many such numbers that it is a herculean task to check all such numbers. It would also be very boring. Meanwhile some interesting questions can be thrown to the children: How many such three-digit numbers exist that will yield 1089? And how many three-digit numbers exist that will remain the same when reversed and hence cannot yield 1089?

Now the question is how do you prove this conjecture to be true. There is no need to do it at the primary level. But in middle school, when children have a decent understanding of algebra, the teacher can prove it through it. I did this with my 8^{th} graders.

Any three-digit number could be written as: 100x+10y+z

Reversing the digits: 100z+10y+x

Subtracting both: 100x+10y+z – (100z+10y+x) = 99x-99z = 99(x-z)

Since, x and z are distinct single digit numbers, the possible values of (x-z) are: 1,2,3,4,5,6,7,8 and 9. That’s because x can be at most 9 (remember that x, y and z are digits from 0 to 9) and z can be at least 0, and thus, the highest value of (x-z) can be 9 and the lowest 1 (note that x and z cannot be equal otherwise the answer will be zero).

So, the only possible results of their products with 99 are: 099, 198, 297, 396, 495, 594, 693, 792 or 891.

When you add any of the numbers with the reverse of itself, you will always get 1089.

Doing the algebra depends on how much children want to know this. If the teacher feels that it’s not the right time, then this can be postponed and children can be asked to find out the reason why all such three-digit numbers give 1089. Is there any magic in it? Believe me, I have seen children perspiring to find out such reasons for hours without food and sleep. Once I showed a math magic trick to the children. One child got so obsessed with it that he didn’t sleep till 3 in the night because he was trying to dissect the problem, and eventually he did and showed me the trick the next morning. Interestingly he found a new way to work the trick. It was a revelation to the child. He fell in love with mathematics. It was no longer a fear for him but a source of wonder.

Now come the magical things that children do. Surprisingly, students in my class shouted out loud that we can do the same process with two-digit numbers and all the numbers (except when the units and tens digit are equal) would give us 99. This is again a nice conjecture. The teacher can push the children to prove it using algebra. It’s quite easy and similar to how we proved for three-digit numbers.

Another group of children did this with a four-digit number and got the result as 10890. Though we need to check that result and understand which numbers won’t yield that result, isn’t it inspiring to see children wildly moved by their natural curiosity to find and see beauty in a number and go on a roller coaster ride with its lovely properties?

How much of this happens in a mathematics classroom where children play with mathematics and come to a belief that everyone can think mathematically and do mathematics? When someone looks at a painting and finds it beautiful, it is not just because the painting is beautiful but also because we want to see beauty, we find beauty, perhaps our brain somehow understands the pattern of what is beautiful and desires to see it. Same thing happens with mathematics. Children get amazed when numbers exhibit beautiful properties and do magic.

There are many math-magic tricks that people do with 1089. You can find them on the internet. It just makes it fun to learn math. The next thing you can do with 1089 is to ask children to find the first 9 multiples of 1089.

1089 x 1 = 1089

1089 x 2 = 2178

1089 x 3 = 3267

1089 x 4 = 4356

1089 x 5 = 5445

1089 x 6 = 6534

1089 x 7 = 7623

1089 x 8 = 8712

1089 x 9 = 9801

First, the multiples should be written normally. The teacher can then ask the children if they can see any special pattern in this. Since 1089 itself is a special number, don’t you think it might show some more beautiful properties? Children easily see the beautiful pattern in the multiples. They will point the numbers in increasing and decreasing order. You can also write this in an elaborate way.

1089 x 1 = 1089 = 1000+ 0+80+9

1089 x 2 = 2178 = 2000+100+70+8

1089 x 3 = 3267 = 3000+200+60+7

1089 x 4 = 4356 = 4000+300+50+6

1089 x 5 = 5445 = 5000+400+40+5

1089 x 6 = 6534 = 6000+500+30+4

1089 x 7 = 7623 = 7000+600+20+3

1089 x 8 = 8712 = 8000+700+10+2

1089 x 9 = 9801 = 9000+800+ 0+1

You can also go further.

1089 x 10 = 10890 = 10000+900+(-10)+(0)

1089 x 11 = 11979 = 11000+1000+(-20)+(-1)

Children will point out the pattern of a constant increase of thousands and hundreds and constant decrease of tens and units.

Push children to look for more. One of the students in my class found this: if you add the ones and thousands digits, you’ll get 10 in the first nine multiples. And when you add the tens and hundreds digits, you’ll get 8.

Another student said that all first 9 multiples are divisible by 3 because the sum of all digits comes to 18 which is divisible by 3.

One pattern that I would like the children to find is how the numbers appear when reversed. The first multiple and the ninth multiple are reverse of each other. Similarly, the second multiple and the eighth multiple are reverse of each other. This goes on till the fifth multiple. If you reverse this number, you’ll get the same number. For me this pattern is very beautiful. And when children see this pattern they smile and laugh and even jump at times. That’s the power of mathematics.

Then one child said that one of the multiples appears different from the others and that’s 5445. I asked him why that is so? He said that the reverse of it is the same number. It’s like the numbers that we have struggled with in the previous exercise.

Here I take the opportunity to introduce the term ‘palindromic numbers’. The teacher can also introduce it before doing the multiples of 1089. Children, by that time, will know the two-digit numbers that won’t yield 99, three-digit numbers that won’t yield 1089 and four-digit numbers that won’t yield 10890. And all those numbers are palindromic numbers.

I reminded children to write some of the numbers that we have excluded from the previous exercise. Numbers that remain the same when reversed. Eg: 11, 44, 77, 121, 343, 767, 949, 1221, 4554, 5445, 8778, 9009. Children can say such numbers and the teacher can write them on the board. All the numbers remain the same when flipped. However, you will find that some four-digit numbers change when flipped but do not yield the expected result. You can ask the children to find out why?

The teacher can introduce the term ‘palindromic numbers’ (numbers that read the same in both directions). I have deliberately connected the lesson on palindromic numbers with 1089 because 1089 very beautifully and naturally opens the door to palindromic numbers.

You can take examples from English language such as MOM, WOW, DAD, NOON, MADAM, RADAR, ROTOR, MADAM I’M ADAM, etc. These are palindromes (words, phrases, or sentences that read the same in both directions). The world of palindromic numbers is quite big and I will only write as much as I have worked out with the children of my class.

The teacher can ask some more questions about palindromic numbers like: Can you tell a year in the past that’s a palindromic number? (Ans: 2002, 1991,…) Do all palindromic numbers appear similar? Some palindromic numbers like 333, 66, 4444 have same digits but 191, 232, 4334 have different digits and are still palindromic numbers.

[Note that 0,1,2.3,4,5,6,7,8 and 9 are also palindromic numbers.]

You can write an interesting case of 11 (the first two-digit palindromic positive integer), 111 and 1111,…that always yields palindromic numbers.

111 = 11

112 = 121

113 = 1331

114 = 14641

115 = 161051

116 = 1771561

1111 = 111

1112 = 12321

1113 = 1367631

11111 = 1111

11112 = 1234321

You’ll get palindromic numbers till the fourth power of 11, third power of 111 and second power of 1111.

Now you can show students how a palindromic number can be generated from any given number. You need to continually add a number to its reversal (i.e., the number written in the reverse order of digits) until you get a palindrome.

Let’s take a number 21

Now reverse the digits, you’ll get 12.

Now add both numbers: 21 + 12 = 33, a palindrome.

Sometime you need to repeat the process until you reach a palindromic number.

Take the number 75.

Reverse the number: 57

Now add both number: 75 + 57 = 132, not a palindromic number.

Reverse the new number: 231

Adding both numbers: 132 + 231 = 363, a palindromic number.

For some numbers you might have to repeat the process many times till you get a palindrome. For the number 98, you might have to repeat the process for 24 times. Children can make palindromic numbers through this process. Now the teacher can ask if all numbers can be made palindromic through this process? Can there be a definite yes or no for this question?

You can even pose some fine questions like: Can you find the total number of palindromic numbers up to 1000? How many of them are prime and how many are composite?

The teacher can ask new and interesting questions to spark the imagination of children. It is also possible to relate palindromes with geometry by using symmetrical figures. These are called visual palindromes.

The world of mathematics is infinitely deep. Like a well, the more you dig, more knowledge comes to you.

**References**

1. Math wonders to inspire teachers and students by Alfred S. Posamentier

2. An investigation of palindromes and their place in mathematics by Ryan Andrew Nivens

3. How to be a mathemagician by Aditi Singhal by and Sudhir Singhal

The author teaches mathematics and languages to primary and high school children. He works with ‘Shiksharth’ an NGO working towards bringing contextually relevant and meaningful education to rural/tribal communities in Sukma, Chhattisgarh. When not teaching, he can be found in poetry sessions with his students. He can be reached at irockmad@gmail.com.