Palindrome Week, Words, n-digit numbers, odometer readings

PALINDROME: In mathematics, a palindrome is a number that reads the same forward and backward. For example, 353, 787, 1331, 42724.

All numbers that have the same digits such as 4, 11, 55, 222, and 6666 are other examples of palindromes. 

This was a special week during 2019. This week is a Palindrome Week, (well, 10 days actually) when each of 10 consecutive dates can be read the same backward and forward. Palindrome Week is unique to the few countries, such as the USA, that write dates in the month-day-year format (9-10-19). September's series of palindromes only work if the date is written with the last two digits of the year and no zero before the month — 09-10-2019 isn't a palindrome, for example. Because much of the world formats dates as day-month-year, the mathematical oddity is mostly noted in the U.S. I asked them to record all 11 palindromic dates in September 2019. the 9th month using USA format of m-d-yy (only last 2-digits of year). Many of them realized that September 1st was also a palindrome 9119. We talked about whether there is a palindrome week during 2020 and found that it will start with February 1st, 2020, or 02120.

With 4th Grade and up, we used the month-day-year (all 4 digits) format and looked at how many days there are in the new century (January 1, 2000 to December 31, 2100). They multiplied 365 days by 100 and then added on another 25 days for leap days. Out of the 36750 days there are only 38 palindromic dates in the 21st Century. They were challenged to find all of them.

Then, I had the children think of 3 and 4-letter words that form a palindrome. I gave them all of the 5 and 6-letter words that are palindromes with definitions. My favorite palindrome is AIBOHPHOBIA, which is an 11 letter palindrome which means, “fear of palindromes.”

After exploring the wonderful words that are palindromes, we turned to numbers that read the same both forwards and backwards. I gave them a few digits like 1234 and asked them to create an odd number digit palindrome and an even number digit palindrome (so, 1234321 and 12344321). There is a great challenge page on the pdf for this exercise.

I then had them think about all 1-digit numbers that are palindromes and to record how many there are. It was easy enough for the to write them down starting with 0, 1, 2 .. 9, but the challenge was counting how many. Many said 9 but they were not focused on zero. So of course, the answer was 10. When I asked them to write down all of the 2-digit palindromes, they were curious as to whether 00 was a two digit number. It is not, zero is a one digit number so starting with 11, 22, 33 … 99, there are 11.

Then we explored three-digit palindromes and attacked a challenge that would determine the total number of 3-digit palindromes. They figured that using an organized system of numbers starting with the smallest, 101, until reaching the largest, 999.

Many of the Mathletes started with 101, 111,121,131,141,151,161,171,181,191, vertically and 101, 202, 303, 404, … 909 horizontally on top. Some realized that there are ten rows and nine columns so 10x9=90 3-digit palindromes. For 4th grade and up, they tackled the challenge of 4-digit palindromes 1001, 1111, 1221, 1331, 1441, 1551,1661, 1771, 1881, 1991, 2002.....  Surprisingly, the same pattern as 3-digits so a total of 90 numbers.

Then they were able to explore 5-digit palindromes starting with 10001, 10101, 10201, etc. Some noticed that there are additional 3-digit palindromes imbedded inside the 5-digits that start and end with zero, with a total of 100, but there are 9 separate thousands for each group of 100 palindromes, so a total of 900. See my algorithm for the number of palindromes for each number of digits. This uses the fundamental counting principle, so for 5-digit numbers, there can be 9 digits in the first digit of palindrome, 10 digits in the second position and 10 in the third position, so 9x10x10=900. The last two positions in a 5-digit palindrome just follow the first two digits.

So looking at this incredible sequence of numbers, for the following number of digits:

1-digit palindromes (1,2,3,4,5,6,7,8,9) 9

2-digit "     (11,22,33,44,55,66,77,88,99) 9

3-digit " 90

4-digit " 90 

5-digit " 900

6-digit " 900

7-digit " 9000

8-digit " 9000

9-digit " 90000

Finally, I had the 4-7th graders learn about odometer readings. My boys and I always looked at the odometer which tells you the number of miles. Then we determined how many more miles until the next palindrome. To this day, they are in their 20s and still take pictures of the odometer when it reads a palindrome. The last page in the pdf gives them six odometer readings and they are challenged to calculate how many miles to the next palindrome. For example, if there are 25,000 miles on the odometer, there are 52 miles until it reaches 25052.

This is a great way for your children and you to use the time in the car mathematically. It is not only challenging for them to find the next palindrome, but it then requires the to use subtraction to figure out how many miles remaining. Actually, I prefer to add from the current odometer reading until the palindrome (subtraction is really the reverse of addition).