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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 Worksheet | 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.
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