Question 2
Will reversing and adding numbers repeatedly, starting with a 2-digit number, always give a palindrome? Explore and find out.*
- A palindrome is a number that reads the same forwards and backwards (e.g., , , ).
- The Reverse and Add process:
- Take a number.
- Reverse its digits to form a new number.
- Add the two numbers together.
- If the sum is a palindrome, stop. If not, repeat the process with the new sum.
- We want to explore whether this process always eventually produces a palindrome when starting with -digit numbers.
Step 1 · Explore with 56 (1 Step)
Start with :
- Reverse of
is a palindrome. It is reached in step.
Step 2 · Explore with 29 (1 Step)
Start with :
- Reverse of
is a palindrome. It is reached in step.
Step 3 · Explore with 84 (Multiple Steps)
Start with :
- Reverse of
Since is not a palindrome, repeat the process:
- Reverse of
is a palindrome. It is reached in steps.
Step 4 · General Conclusion
All -digit numbers eventually produce a palindrome using this reverse-and-add process (most take to steps, while and take the longest at steps).
Note: For some higher numbers (such as ), no palindrome has ever been found despite millions of iterations (called suspected Lychrel numbers).
Yes, starting with any -digit number and repeatedly reversing and adding digits will always eventually produce a palindrome.
- Stopping Too Early: Stopping after the first addition when the sum is not a palindrome, rather than repeating the reverse-and-add operation on the new sum.
- Reversal Errors: Incorrectly adding the digits themselves instead of reversing the place value order (e.g., reversing gives , not ).
More questions in A
Complete Table 2 with 5-digit numbers whose digits are ‘1’, ‘0’, ‘6’, ‘3’, and ‘9’ in some order. Only a coloured cell should have a number greater than all its neighbours.
The biggest number in the table is _________ . The smallest even number in the table is _________ . The smallest number greater than in the table is _________ . Once you have filled the table above, put commas appropriately after the thousands digit.
Will reversing and adding numbers repeatedly, starting with a 2-digit number, always give a palindrome? Explore and find out.*
Explore
Take different 4-digit numbers and try carrying out these steps. Find out what happens. Check with your friends what they got.
Explore
Carry out these same steps with a few 3-digit numbers. What number will start repeating?
Make some more Collatz sequences like those above, starting with your favourite whole numbers. Do you always reach ?
Do you believe the conjecture of Collatz that all such sequences will eventually reach ? Why or why not?
The first player says 1, 2 or 3. Then the two players take turns adding 1, 2, or 3 to the previous number said. The first player to reach 21 wins!
Play this game several times with your classmate. Are you starting to see the winning strategy? Which player can always win if they play correctly? What is the pattern of numbers that the winning player should say?
The first player says a number between 1 and 10. Then the two players take turns adding a number between 1 and 10 to the previous number said. The first player to reach 99 wins!
Play this game several times with your classmate. See if you can figure out the corresponding winning strategy in this case! Which player can always win? What is the pattern of numbers that the winning player should say this time?