by Dave Youngs.
This month’s Puzzle Corner and Maximizing Math activities both deal with palindromes. Since Palindromic Puzzlers is an introductory activity for students who may not be familiar with palindromes, it should be done first.
Palindromes are words, phrases, or numbers that read the same from left to right and right to left. Examples of some palindromic words are mom, dad, and noon. Some of the more common palindromic phrases are “Madam, I’m Adam,” “Too hot to hoot,” and “A man, a plan, a canal, Panama.” (Note that in palindromic phrases it is permissible for the spacing, punctuation, and capitalization to change in the right-to-left reading.) While there is a finite number of palindromic words and phrases, the set of palindromic numbers is infinite. Our current year, 2002, is just one example of a palindromic number. By the way, there was a very significant palindromic event that took place this past February in many other parts of the world, but not in the United States. This event occurred at two minutes after eight on the evening of the 20th. Can you figure out what this palindromic event was? Try to do so before reading the answer that appears below.*
In this activity, students are first asked to brainstorm a list of palindromic words and numbers to get them used to the concept of palindromes. After this warm-up they are reminded that 2002 is a palindromic year and challenged to find the previous five palindromic years and the next five palindromic years. When they have done this, they are asked to describe any patterns they notice in this sequence of palindromic years.
Let’s look at some of the patterns in the sequence of palindromic years (1551, 1661, 1771, 1881, 1991, 2002, 2112, 2222, 2332, 2442, and 2552). A study of this sequence shows that the difference between any two of these palindromes (except for 1991 and 2002) is 110. (The difference between 1991 and 2002—the two palindromic years that span the millennial change—is 11.) The palindromes in the second millennium all start and end with ones with double numbers in between while those in the third millennium start and end with twos. The doubled numbers in the middle of the second millennial palindromes start with 55 and ascend by 11 each time until 99 is reached. The doubles in the third millennium start with 00 and ascend by 11 each time to 55. If any of these palindromic years is divided by 11, the resulting quotient is also a palindrome (1551/11=141, 1661/11=151, 1771/11=161). Making a list of these smaller palindromes also produces some interesting patterns as the example above shows. (Your students may find patterns other than the ones listed above. If they do, I’d love to hear from them.)
It is my hope that your students find this activity interesting and that it
might spark some of them to seek and play around with the many palindromes they
encounter in daily life. (For example, they might try to find all the palindromic
numbers that appear on a digital clock.) For those of you who remember to renew
your subscription, there’ll be another puzzle for you in the next issue.
In the meantime, if you have any questions or comments, please feel free to
contact me. My email is dyoungs@aimsedu.org.
Student Activity Page | Maximizing Math Activity (179k PDF file)
* This particular time and date, when written in the (24 hour) time and (day of the month/month) dating systems used in many countries was 20:02, 20/02, 2002! In the United States this same brief moment in time was the fairly pedestrian 8:02 p.m., 2/20, 2002.