Mom, wow! D. Leela and classmates "What are you drawing, Maya?" asked Nivedita, entering the room one dull Saturday afternoon. Leela looked at the picture that Maya was drawing and giggled. "Why have you called her `MOM'?" she asked. "It looks like a girl, with long hair and a frock!" "It's not Mom," said Maya, indignantly. "Can't you see?" "Yes, it is Mom," insisted Leela. "Look!" She picked up the paper Maya was drawing on, and then said, "Oh!" She had read the writing up-side down and when she looked at it the right way, it now said, "WOW." "It's raining; she's looking at the rainbow and saying `Wow', explained Maya. It was Nivedita's turn to look curious. "Where's the rainbow?" she asked. "I haven't yet drawn it!" said Maya, at which all the children laughed. Nivu took the drawing from Leela and turned it around and around. "Wow, Mom, Wow, Mom," she kept saying, as she turned it over and over. "What a nice word, you can read it both ways. "There are not many English letters that can be read upside-down. But 'M' and `W' are special. All other letters look like themselves, upside-down. Like, `O', I mean. I don't think there are any others which can also be read like a letter upside-down." Maya said, "There's 'X'. It's like `O', the same, whichever way you read it. And `H' and `I' as well." Leela said,"Maya, you found those letters easily because they look the same forward and backward too. They are nice and symmetric letters. But you'll see that `S' and `N' also look the same upside-down. But not backwards." Maya thought for a while. "That's right! I never realised it." Then she added, "But you can only read the letters upside down. Not all the words come out right, like `OX' or `SIX'. The words make no sense when you read them upside down. They are not like `WOW' and `MOM'." Leela was mumbling, "a,b,c... Yes," she said. If you are looking at small letters, `n' and `p' can also be read as `u' and `d' upside-down. But not as capital letters." The next few minutes were spent in trying to make words out of these special letters, that could also be read upside down. At that time, Amma came in. "Children, aren't you going down to play? It may rain soon. Why don't you go down before that?" "Amma! shouted Maya. "Amma!" "Yes?" asked Amma. "Amma!" shouted Maya again. And before Amma could show her impatience, she burst out, "It's like Mom, but different!" "Yes," said Nivedita. "`Amma' can be read both forwards and backwards." "And `Mom' can be read upside down." said Leela. "Yes," said Amma, smiling. "Amma is special, don't you know?" Then, she added, more seriously, "These words are called palindromes. They are not easy to find: `madam' and `level' are words that can be said both forwards and backwards. But people mostly make sentence palindromes." "What's that, Amma?" asked Maya. Amma reached out for a pencil and wrote, `Step on no pets.' "Can you read that backwards?" she asked. "Step on no pets," read out Maya, painstakingly. "Hey, that's the same sentence, backwards." "It's funny," said Leela. "Today we learned how to make palindrome numbers in school. Numbers like 383 or 2662 that are the same forwards or backwards," she explained to Maya. "That's easy," said Maya, and began writing a whole lot of palindromic numbers on the paper. "No," said Leela. "We were taught how to make them. Come, I'll show you." She looked at her mother. "Amma, we'll play downstairs later," she said. "Fine," said Mother. "Go on." "Think of a number," said Leela. "Write it backwards. Then add the two numbers. You'll get a palindrome number. Nivu, tell me a number." "227," said Nivedita promptly. "You, know, 22-7. 22nd July, the date of the solar eclipse," she explained. "Oh," said Leela. She wrote down the number on a sheet of paper. Below that, she wrote it backwards, and then added the numbers, like this. 227 +722 ---- 949 ---- "949 is a palindrome," shouted Maya. "Yes," said Leela. "That's what I just learned. Let's try 279." "Aha!" said Maya. "I know, that's your birthday, right." Leela didn't reply. She was looking, puzzled, at the paper. 279 +972 ---- 1251 ---- "1251 not a palindrome," she said. "I don't understand it. This is the method we were taught." Nivedita said, "You had a carry-over. The other sum didn't. Maybe that spoils things." She thought for a while. Then she smiled. "I've got it," she said. "You have to continue the process if it doesn't work out in one step." She wrote the number 1251 on the paper, reversed it, and added the numbers. 279 1251 +972 +1521 ---- ----- 1251 2772 ---- ----- "See, 2772 is a palindrome number. Some numbers aren't palindromes when you add their reversed numbers, but if you repeat it, you'll get a palindrome number after the second step. Let's see if we can get a 3-step palindrome." The children sat down busily adding various numbers. Soon they had a 3-step palindrome. 587 1372 4103 +785 +2731 +3014 ---- ----- ----- 1372 4103 7117 ---- ----- ----- "I wonder whether you can go on and find a 4-step palindrome number, and so on, or whether it stops somewhere," said Maya. Leela said, "If you take bigger and bigger numbers, I am sure you can. Let's stick to only three-digit numbers like we have tried. Perhaps then we can get palindromes with only 4 or 5 steps." Nivedita said, "Every time there is a carry-over, I think you will not get a palindrome in that step. That's because the carry-over adds to the number on the left, but not on the right. That's because you add from right to left," she added, when she saw Maya look puzzled. "That's right," said Leela. "I noticed that only when you stop getting a carry-over in that step is the number a palindrome." "So you must find a number with the largest digits: easy, that's 999!" said Maya. "Silly, 999 is already a palindrome," giggled Leela. "Maya must be right," said Nivedita. "Let's try 998 as the next largest number. I wonder how many steps it will take to get a palindrome number from it." Can you find out? 998 1897 9878 18667 95348 +899 +7981 +8789 +76681 +84359 ---- ----- ----- ------ ------ .... 1897 9878 18667 95348 179707 ---- ----- ----- ------ ------