Math Puzzle: What are the palindrome numbers between 200 & 450?
Solving:
A palindrome number is one which reads the same reading from left to right as from right to left. As an example, 242, 484, 363 are all palindrome numbers since it’s the same number no matter which way you read it. However, 241, 487, 365 are not palindrome numbers.
Let us arrange the clues in a logical order:
- The palindrome numbers would be all 3 digit since they start from 200 & end at 450, both these starting & ending numbers are 3 digits
- The 1st & 3rd digit will be identical
- Possible digits that can be used in any palindrome number range from 0 to 9 – so that is 10 numbers
- The number must be more than or equal to 200
- The number must be less than or equal to 450
Let us look at the 3 places where the digits can be placed & name them A, B & C:
| A | B | C |
Let us look at the position A & C since the digits here will be identical.
1A. Since the range of numbers is >= 200,
- Position A can have the possible values of > = 2
- Position C can have the digits possibly > = 2
1B. Since the range of numbers is =< 450,
- Position A can have digits =< 4
- So, Position C can have digits =<4
Combining 1A & 2B, Position A can have digits >=2 & =<4 & thats 2,3 & 4.
Let’s take the 3 digit palindrome numbers starting with 2 & ending with 2:
| 2 | 0 – 9 | 2 |
| A | B | C |
2A. Thats 10 possibilities with B varying from 0 through 9 (10 digits, including 0) & starting & ending with 2
Using a similar logic, when the 1st & 3rd digits are 3:
| 3 | 0 – 9 | 3 |
| A | B | C |
2B. Thats 10 possibilities with B varying from 0 through 9 (10 digits, including 0) & starting & ending with 3
When the 1st & 3rd digits are 4, since the number =<450, the middle digit B must be less than 5, since 454 would be greater than the limit of 450.
| 4 | 0 – 4 | 4 |
| A | B | C |
2C. Thats 5 possibilities with B varying from 0 through 4 (5 digits, including 0) & starting & ending with 4
Adding up 2A, 2B & 2C, the total count of 3 digit palindrome numbers between 200 & 450 are 10+10+5 = 25
Answer: 25 numbers are palindromes between 200 & 450
You will find these sort of math puzzles in various math competitions like Perennial Math, Math Olympiad, Noetic Learning Math Contest, MathCounts, MOEMS Math Olympiad, U.S.A Mathematical Talent Search (USAMTS), Purple Comet Math, Math League, American Regions Mathematics League (ARML) & others.