The magic and meaning of palindromes
Rajesh Kumar Thakur
The word ‘palindrome’ is derived from two Latin words – palin and dromos. Palin means again and dromos means way or direction. In simple words, a palindrome is a number, word, or sentence that reads the same forwards or backwards without any change in its meaning. Let’s begin with two everyday words – MOM and DAD. Read them left to right and right to left. Did you find any change? The same is true for numbers like 11, 121, 34543, etc.
Word palindromes
In the beginning of the 1^{st} century, Jews and Christians began to play with letters making a cross to show their respect to Jesus. The ROTAS – SATOR square pattern shows the first word palindrome written in a 5 x 5 square.
Image courtesy: www.commons.wikimedia.org
In Figure (a) – the central cross marked in RED colour – TENET – is associated with Judaism and Christianity. The + symbol represents Christianity, while the x symbol (the cross seen from another angle) represents Judaism. Figure (b) is a ROTAS – SATOR square found at Dura, Europos in Syria, while Figure (c) shows a ROTAS – SATOR palindrome in a mirror image format.
Number palindromes
While there is no record of who started number palindromes and when, in India, around the 9^{th }century, a Jain mathematician named Mahaviracharya in his book Ganit Sara Samgraha tried different number combinations to make a number necklace to offer to the prince as a noble gesture. In chapter 2, shlokas 1020, he described different combinations of numbers.
 Write the number 12345679 and multiply it by 9 to get a necklace of number
12345679 x 9 = 11,11,11,111  Write 3, six times, 6, five times and 7 in ascending order and multiply by 33 to get a necklace.
333333666667 x 33 = 1,10,00,01,10,00,011
He also mentioned few other numbers in the same shlokas,
142857143 x 7 = 1,00,00,00,001
37037037 x 3 = 11,11,11,111
11011011 x 91 = 1,00,20,02,001
Mahaviracharya identified a method to make number palindromes. He wrote, “Take a (two digit) number, reverse it. Add the original number and reverse the number. Repeat the process and you will finally end with a palindrome number.”
Suppose we take the number 23, its reverse is 32. Their sum 23 + 32 = 55 is a palindrome. This method is true for most numbers, but the number of steps may vary. Let’s take another example. If we take the number 89, we will end up with a palindrome in 24 steps.
Palindrome in a poem
The first palindromic sentence in English is believed to have been written by the poet John Taylor in 1614 – Lewd did I live & evil I did dwel. Another beautiful mirror/palindrome poem by Memory Trace–INSPIRATION – reads as follows
“Open floodgates,
once restrained tightly,
suddenly form rippled waters,
expressive thoughts flowing freely,
by frightful heart attending faithfully
INSPIRATION
faithfully attending heart frightful by
freely flowing expressive thoughts,
waters rippled form suddenly,
tightly restrained once,
floodgates open.”
In India, a poem called RamaKrishna Viloma Kayavam, written by the poet Surya Pandita in the 14^{th} century,has 36 verses following the Palindrome pattern. This poem is also unique because the first line of each verse in the poem is about Rama and the Ramayana and the second line is about Krishna and the Mahabharata. In one of the palindromic verses, the poet praises Sita who loves her son Luv, while the reverse words forming the next line speak about the teaching of the Bhagvad Gita by Lord Krishna.
Palindrome and spirituality
Palindromic numbers encourage us to find balance in our lives and to accept the dual elements within each of us. They act as a reminder that there are opportunities as well as light in every dark place and hardship. Palindromic numbers were considered auspicious in ancient Chinese belief systems. Palindromic numbers are revered and frequently connected to spiritual enlightenment in Hinduism. Greeks saw palindromic numbers as having a divine quality and serving as a sign of harmony and balance.
Properties of palindromes
 Generally, every number is a palindrome in some base or other. In the decimal system with base 10, we have 11, 22, 121, 1331, 14641 – as palindromes. 21 is not a palindrome in base 10; but if we express it in base 2 – the binary system – we get 10101, which is a palindrome.
 In base 10, any number can be expressed as the sum of at most three palindromes. In a researchpublished in 2017, it was proven that any positive integer with a base of 5 or more can be expressed as the sum of three palindromes.
 The square and cube of a number can also be a palindrome.
11×11=121
101×101=10201
10101×10101=102030201
11111×11111=123454321
1010101×1010101=102030403020
 Palindrome can be expressed in a magic square number. Here is a 4 x 4 palindromic magic square as well as an 8 x 8 magic square pattern.*

 Look at the following beautiful number patterns whose product in reverse order also gets reversed.
12×102=1224 and 21×201=4221
112×112=12544 and 211×211=44521

 A H Beiler in his book Recreations in the Theory of Numbershas shown a remarkable pattern involving palindromes.
1221+5445+6996 = 2112+3663+7887
1221^{2}+5445^{2}+6996^{2} = 2112^{2}+3663^{2}+7887^{2}
If this has piqued your curiosity, what are you waiting for? Keep diving into the ocean of palindromes and find the pearls to make a beautiful palindromic necklace.
*The magic constant of 4 x 4 magic square is 2002, which is a palindrome. The sum of any row, column, or diagonal is 2002. The 8 x 8 magic square has all palindromic numbers and the sum of its rows, columns, and diagonals is 4884, which is also a palindrome.
The author is Assistant Professor, DIET Daryaganj, New Delhi. He has authored 106 books and 700 articles on mathematics. He can be reached at rkthakur1974@gmail.com.
The magic and meaning of palindromes
Rajesh Kumar Thakur
The word ‘palindrome’ is derived from two Latin words – palin and dromos. Palin means again and dromos means way or direction. In simple words, a palindrome is a number, word, or sentence that reads the same forwards or backwards without any change in its meaning. Let’s begin with two everyday words – MOM and DAD. Read them left to right and right to left. Did you find any change? The same is true for numbers like 11, 121, 34543, etc.
Word palindromes
In the beginning of the 1^{st} century, Jews and Christians began to play with letters making a cross to show their respect to Jesus. The ROTAS – SATOR square pattern shows the first word palindrome written in a 5 x 5 square.
Image courtesy: www.commons.wikimedia.org
In Figure (a) – the central cross marked in RED colour – TENET – is associated with Judaism and Christianity. The + symbol represents Christianity, while the x symbol (the cross seen from another angle) represents Judaism. Figure (b) is a ROTAS – SATOR square found at Dura, Europos in Syria, while Figure (c) shows a ROTAS – SATOR palindrome in a mirror image format.
Number palindromes
While there is no record of who started number palindromes and when, in India, around the 9^{th }century, a Jain mathematician named Mahaviracharya in his book Ganit Sara Samgraha tried different number combinations to make a number necklace to offer to the prince as a noble gesture. In chapter 2, shlokas 1020, he described different combinations of numbers.
 Write the number 12345679 and multiply it by 9 to get a necklace of number
12345679 x 9 = 11,11,11,111  Write 3, six times, 6, five times and 7 in ascending order and multiply by 33 to get a necklace.
333333666667 x 33 = 1,10,00,01,10,00,011
He also mentioned few other numbers in the same shlokas,
142857143 x 7 = 1,00,00,00,001
37037037 x 3 = 11,11,11,111
11011011 x 91 = 1,00,20,02,001
Mahaviracharya identified a method to make number palindromes. He wrote, “Take a (two digit) number, reverse it. Add the original number and reverse the number. Repeat the process and you will finally end with a palindrome number.”
Suppose we take the number 23, its reverse is 32. Their sum 23 + 32 = 55 is a palindrome. This method is true for most numbers, but the number of steps may vary. Let’s take another example. If we take the number 89, we will end up with a palindrome in 24 steps.
Palindrome in a poem
The first palindromic sentence in English is believed to have been written by the poet John Taylor in 1614 – Lewd did I live & evil I did dwel. Another beautiful mirror/palindrome poem by Memory Trace–INSPIRATION – reads as follows
“Open floodgates,
once restrained tightly,
suddenly form rippled waters,
expressive thoughts flowing freely,
by frightful heart attending faithfully
INSPIRATION
faithfully attending heart frightful by
freely flowing expressive thoughts,
waters rippled form suddenly,
tightly restrained once,
floodgates open.”
In India, a poem called RamaKrishna Viloma Kayavam, written by the poet Surya Pandita in the 14^{th} century,has 36 verses following the Palindrome pattern. This poem is also unique because the first line of each verse in the poem is about Rama and the Ramayana and the second line is about Krishna and the Mahabharata. In one of the palindromic verses, the poet praises Sita who loves her son Luv, while the reverse words forming the next line speak about the teaching of the Bhagvad Gita by Lord Krishna.
Palindrome and spirituality
Palindromic numbers encourage us to find balance in our lives and to accept the dual elements within each of us. They act as a reminder that there are opportunities as well as light in every dark place and hardship. Palindromic numbers were considered auspicious in ancient Chinese belief systems. Palindromic numbers are revered and frequently connected to spiritual enlightenment in Hinduism. Greeks saw palindromic numbers as having a divine quality and serving as a sign of harmony and balance.
Properties of palindromes
 Generally, every number is a palindrome in some base or other. In the decimal system with base 10, we have 11, 22, 121, 1331, 14641 – as palindromes. 21 is not a palindrome in base 10; but if we express it in base 2 – the binary system – we get 10101, which is a palindrome.
 In base 10, any number can be expressed as the sum of at most three palindromes. In a researchpublished in 2017, it was proven that any positive integer with a base of 5 or more can be expressed as the sum of three palindromes.
 The square and cube of a number can also be a palindrome.
11×11=121
101×101=10201
10101×10101=102030201
11111×11111=123454321
1010101×1010101=102030403020
 Palindrome can be expressed in a magic square number. Here is a 4 x 4 palindromic magic square as well as an 8 x 8 magic square pattern.*

 Look at the following beautiful number patterns whose product in reverse order also gets reversed.
12×102=1224 and 21×201=4221
112×112=12544 and 211×211=44521

 A H Beiler in his book Recreations in the Theory of Numbershas shown a remarkable pattern involving palindromes.
1221+5445+6996 = 2112+3663+7887
1221^{2}+5445^{2}+6996^{2} = 2112^{2}+3663^{2}+7887^{2}
If this has piqued your curiosity, what are you waiting for? Keep diving into the ocean of palindromes and find the pearls to make a beautiful palindromic necklace.
*The magic constant of 4 x 4 magic square is 2002, which is a palindrome. The sum of any row, column, or diagonal is 2002. The 8 x 8 magic square has all palindromic numbers and the sum of its rows, columns, and diagonals is 4884, which is also a palindrome.
The author is Assistant Professor, DIET Daryaganj, New Delhi. He has authored 106 books and 700 articles on mathematics. He can be reached at rkthakur1974@gmail.com.