Activity W 6.4 Pretty Palindromic Patterns
Engaging students in these puzzles will make them more observant about numbers.
Pretty Palindromic Patterns
Procedure
The numbers that can be read the same from left to right and from right to left are called palindromes or palindromic numbers.
Part I: Forming Palindromes from Given Digits
Given digits: 3, 4, 5
Palindromes you can form:
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2-digit: 33, 44, 55
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3-digit: 343, 454, 353, 535, 445, 544, 333, 444, 555
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5-digit: 34543, 35453, 34343, 44444, etc.
The game challenge:
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❗ Most palindromes formed → Winner
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❗ Longest palindrome formed → Winner
For example, 232, 444, 54645, etc.
1. Students may be asked to write all palindromes using certain number of digits.
For example, using the digits 3, 4, 5 we can form palindromes 343, 454, 34543, 333, etc.
Students may form as many palindromes using the given digits.
2. A game can be played, based on this.
The one who forms maximum number of palindromes using the given digits will be the winner.
Or
The one who forms the longest palindrome will be the winner, etc.
Procedure
1. Students may write a 2-digit number and reverse the order of the digits.
Add these two numbers.
Part II: Reverse and Add Process
2. They may check whether the addition is a palindrome.
If not, continue the process of reversing the digits and adding them.
3. They may check, if they get a palindrome at some stage or not.
For example,
Example 1:
Start with: 36 → 36 + 63 = 99
Palindrome in 1 step
36 + 63 = 99 (a palindrome!)
Example 2:
Start with: 39 → 39 + 93 = 132
132 + 231 = 363
Palindrome in 2 steps
39 + 93 = 132 (not a palindrome)
132 + 231 = 363 (a Palindrome!)
Example 3:
Start with: 89
89 + 98 = 187
187 + 781 = 968
968 + 869 = 1837
1837 + 7381 = 9218
9218 + 8129 = 17347
17347 + 74371 = 91718
91718 + 81719 = 173437
173437 + 734371 = 907808
907808 + 808709 = 1716517
Palindrome (10 steps!)
4. Students may be asked to do this for different numbers.
Students may explore and tell, for which numbers it took only one step, few steps or large number of steps.
5. Students may explore whether reversing and adding numbers repeatedly, starting with a 2-digit number, always give a palindrome?
Do All Numbers Reach a Palindrome?
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For 2-digit numbers, almost all lead to a palindrome within a few steps.
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For some large numbers (like 196), it’s unknown if they ever reach a palindrome — this is an unsolved math mystery.
Observation: Most numbers do reach a palindrome, but some take many steps
III. Procedure
tth th h t u
Write the number in words:
I am a 5-digit palindrome.
I am an odd number.
My ‘t’ digit is double of my ‘u’ digit.
My ‘h’ digit is double of my ‘t’ digit.
Who am I? _________________
Since it’s a palindrome, its structure is:
Let’s use:
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A = 1 (odd)
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u = 1 → t = 2 × 1 = 2
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t = 2 → h = 2 × 2 = 4
So we have:
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u = 1
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t = 2
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h = 4
Therefore, the number is:
→ A B C B A → 1 2 4 2 1
Answer: 12421
Teacher/students may create more such puzzles and give others to solve.
Palindromic Puzzles for Students
Puzzle 1:
I am a 3-digit palindrome.
I am less than 500.
My middle digit is the smallest possible odd number.
Who am I?
Answer: 101, 303, etc. (Middle digit = 1, smallest odd)
Puzzle 2:
I am a 4-digit palindrome.
The sum of my digits is 22.
My outer digits are the same and even.
What could I be?
Answer: 2662 (2 + 6 + 6 + 2 = 16), 4774, 6446 (check sums)
Puzzle 3:
Form the longest 7-digit palindrome using only two digits: 2 and 5.
Answer: 2552552 or 2225222, etc.
Puzzle 4:
Start with 65. Reverse and add.
Continue the process until you get a palindrome.
How many steps does it take?
65 + 56 = 121 ✅ 1 step only!
Puzzle 5:
I am a 5-digit palindrome.
The sum of all my digits is 25.
My digits include only 3 different digits.
What could I be?
Answer: 35853, 46964, etc.
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