PALINDROMES: A Teacher’s Guide
A number that remains unchanged in value upon writing it in reverse is called a palidromic number, or simply a palindrome. The idea is borrowed directly from a popular form of word play. Certain words, like dad, noon, radar, etc. possess this reversal invariant property. Even many sentences can be formed that also exhibit it. Examples: Madam, I’m Adam. and Rats live on no evil star. Of course, here we will mainly be concerned with the numerical version, though the use of palindromic words should not be ignored as a means of explaining this concept to students.
[Note: The word palindrome is derived from the Greek palíndromos, meaning running back again (palín = AGAIN + drom-, drameîn = RUN). A palindrome is a word or phrase which reads the same in both directions. (Source: What are palindromes?)]
The teacher has two possible methods of introducing palindromes to the student: the direct and the indirect. The former consists of a straightforward definition (“A palidrome is ….”), followed by examples and appropriate exercises.
However, this writer believes that the indirect approach provides an excellent opportunity to provide students with a moderately easy situation to test their observation skills. Our success with students at a wide variety of levels has led us to this conclusion. However, you, as the teacher, may devise your own method of presentation. Here is our basic method.
We begin by presenting a few examples of numbers that are palindromes, and asking if anyone sees what makes these numbers special. We start with three-place numbers and gradually present larger ones. Depending on the responses received, we at times present numbers that are NOT palindromes, for contrast. Other times, we give a number that meets the criteria of the students’ attempts of definition, yet is not a palindrome. By being patient, the proper definition usually is deduced with little help from us. Then we proceed to the reverse-and-add activity to be described below.
Once the definition has been “discovered”, we then introduce “things to do” with it. The one that provides the most interest is the Reverse-and-Add procedure. It is quite simple:
- Pick a number, any number.
- Reverse that number, writing it beneath the other.
- Add the two numbers.
- If a palindrome results, “Voila! Eureka!”; if not, reverse that sum underneath itself, and continue until such time as one is obtained.
Simple as it is — and it REALLY is — it is impossible in a practical sense to just say when, if ever, a palindrome will be encountered. Sometimes it occurs at the first or second addition, other times it requires lots more. For example, 89 does not yield one until the 24th addition, consisting of 13 digits. [To see an actual account of what happened one day in my class, click here.]
Another enigma is the number 196. Here it is literally not known whether a palindrome will appear. A computer has performed over 4000 additions on it [see update below], and none has shown up yet. However, this, like Goldbach’s Conjecture, does not prove the case. It just says the problem is beyond the paper-and-pencil method of solution.
But to return to our classroom work, we present a variety of examples in the following manner (with variations, of course):
38 85 475
+ 83 + 58 + 574
121 143 1049
+ 341 + 9401
From this point on, various lines are possible. One is to give the three numbers 195, 196, and 197 and ask the students to find palindromes. Usually 195 is done first, with a palindrome appearing at the 4th addition step. This is followed by 197; its palindrome appears on the 7th step. But as stated above, 196 is a mystery. So to spark interest and enthusiasm, we offer a “reward” to the first person who finds a palindrome correctly computed from 196. This never fails to get them working frantically. Of course, while this may seem to be an unfair problem, there are many lessons than can be learned in this sort of exercise. Invariably, students start announcing that “I’ve found one!” You know in advance that an error of some kind has been committed. With care, you will locate a basic fact error, or one involving improper carrying, or improper reversing, etc., etc. A bit of advice is in order here: prepare yourself with a list of the first 25 or so sums to make it more efficient in locating the point at which an error was made (it may not be the only one, or even the first). Also, take extreme care in making your list, lest you commit an error yourself. (It CAN and does happen, even to the best of us.)
Another approach we have taken, with equally interesting results, is to offer 25 cents to the first person who finds a palindrome based on 89, correctly computed. [Note: this article and work was originally done in the late ’70s, when a quarter was a bit more attractive. But, even then, 25 was an essential part of the activity.] This of course, is possible; and the payoff is “fair” — virtually 1 cent per addition. Yet, to discourage sloppy or careless work, we offer a counter-bet of 5 cents that the student must pay us IF an error can be found. Of course, we don’t take the students, as somebody either wins from the class, or the money is returned to them. This makes some students be a little more careful and aware that even in an apparently simple problem, one can make mistakes. As time passes and a few (2 or 3) losers have had to pay up, we give the hint that they had better do more than 20 addition steps, or refuse to look at their work (knowing they haven’t spent enough time on it and should look for their own mistakes). After this problem is conquered, we present the 196 case, with an appriately greater reward (e.g. $10). Other times we give them the choice of problems.
There are other types of palindrome processes. The reverse-and-add procedure can be modified to ask the question: If you keep on going with the R&A work, how many palindromes appear in 10 (or 15) addition steps? Other questions include: Does a palindrome yield a palindrome? How many palindromes can come up “in a row”? Can a palindrome only occur if there has not occurred a “carry”? (Can you think of your own?)
Palindromes can arise in other settings. For example, squaring, cubing, or finding other powers of certain numbers will yield palindromes. How many can you find?
Encouraging students to be on the lookout for strange or unique numbers ca have interesting ramifications. Ask them to be alert and see who can find a license plate number that is a palindrome. Or a phone number, a social security number, a house address, a zipcode. Since numbers are all around us in this modern world, we ought to make them our friends. In other words, “Let’s don’t fight it. Learn to recognize interesting numbers with strange properties. Then it is more like hunting for diamonds or nuggets of gold. We appreciate them more for their rarity or unique characteristics.”
[NOTE: (July 2001) I have recently been introduced to Patrick De Geest of Belgium. He has a marvelous website devoted largely to alindromes, called WORLD!Of Numbers. Go there soon.]
[UPDATE: (February 2002) The number of reversal steps has been increased to many millions. Click here to learn more.
Below is a sample of a worksheet that could be developed to give elementary students some nontrivial investigation experience. Whether calculators should be allowed on this work is an open question. But sometimes, as in the “196” and “89” problems, the display window of the calculator is too small to contain the sums; so there it is “back to basics”, I guess: good ol’ paper-and-pencil.
1. JUST FOR PRACTICE: make sure that you understand the “reverse- and-add” procedure by completing this table.
||No. of additions
2. in each pair below produce the same palindrome, but they don’t need the same number of additions. So, for each pair, (a) find the palindrome for each pair; and (b) state the number of additions for each number.
(a) 6228 & 2673 (b) 197 & 5873
3. The seed numbers in this group all produce palindromes that have a strange common property. Find the palindrome for each, then state what the strange thing is. (Can you find a seed number of your own that will do the same thing?)
(a) 338 (b) 676 (c) 726 (d) 7482 (e) 53877
4. Here the “problem numbers” tell you how many additions are required to find a palindrome.
(1) 47 (2) 238 (3) 86 (4) 78
(5) 6358 (6) 62354 (7) 118471 (8) 8779
Can you find a seed number that requires 9, 10 or more additions? There are many.
5. The seed numbers 428, 527, and 626 all yield the same palindrome with the SAME number of additions. Look at them closely to see the pattern; what is it? Then give another number (less than 428) that belongs to this family BEFORE trying it. Finally, prove your choice is correct. (Using that idea, can you give several seed numbers that produce 15851? HINT: See exercise #1 above.)
*6. Prepare a short report by investigating the patterns you find when you use the ten numbers in this sequence:
606, 616, 626, 636, … , 696.