Introduction / Context:
This puzzle asks you to identify a number whose square is the striking palindromic value 12345678987654321. A palindromic number reads the same forwards and backwards, and this particular one increases from 1 up to 9 and then decreases back to 1. The question wants the whole number which, when multiplied by itself, produces this pattern. Such problems test your familiarity with special numerical patterns and your ability to reason about the approximate size of square roots and how they relate to the length of the resulting number.
Given Data / Assumptions:
- The target number is 12345678987654321.
- We are looking for a whole number n such that n * n equals this target.
- The answer choices are 11111111, 111111111, 1111111111, and 123456789.
- We assume normal rules of decimal multiplication.
- The structure of the target number suggests a pattern rather than a random square.
Concept / Approach:
The key idea is that repeating numbers like 11, 111, or 111111111 often produce attractive patterns when squared. For example, 11 squared is 121, and 111 squared is 12321. These squares show a growing sequence of digits and then a decrease. Extending that idea, 111111111 squared gives a much longer palindromic number, and in fact, it yields exactly 12345678987654321. Knowing this pattern or at least recognising that a nine digit string of ones is the right size helps to select the correct option without doing full long multiplication in the exam hall.
Step-by-Step Solution:
Step 1: Estimate the number of digits in the square root. The target has 17 digits, so the square root should have 9 digits because a 9 digit number squared typically has 17 or 18 digits.
Step 2: Among the options, 111111111 is the only 9 digit number.
Step 3: Recall related patterns: 11 * 11 = 121 and 111 * 111 = 12321. These show the same building then reversing structure.
Step 4: Extend the pattern mentally: 111111111 squared is known to create 12345678987654321, matching both the length and the digit pattern of the target.
Step 5: Rule out 11111111 (8 digits) and 1111111111 (10 digits), because their squares would not have exactly 17 digits.
Step 6: Rule out 123456789, because its square would be larger and would not follow the symmetric pattern of digits that the target has.
Verification / Alternative check:
If you wish to check further, note the general pattern: if you square a number made up of n ones, you get a palindrome that climbs from 1 up to n and back down, when n is 1 through 9. For n = 9, the square of 111111111 is 12345678987654321. Although doing full long multiplication is possible, recognising this well known pattern is faster and is often taught in number puzzle books. Additionally, the digit count argument reinforces that the correct root must have 9 digits, which uniquely points to 111111111 among the choices.
Why Other Options Are Wrong:
11111111 has only 8 digits, so its square would have about 15 or 16 digits, not 17. 1111111111 has 10 digits, so its square would have 19 or 20 digits, longer than the target. The number 123456789, while famous in its own right, does not have the simple all ones structure that produces symmetric palindromes when squared, and its square does not equal the given number. Thus, they cannot be correct roots for 12345678987654321.
Common Pitfalls:
A frequent mistake is to guess based on the familiarity of 123456789 without considering the number of digits in the square root. Another pitfall is to underestimate how useful pattern recognition is in contest style questions and attempt heavy calculations under time pressure. In puzzles involving palindromic numbers with neat sequences, it is often best to think about special forms such as strings of ones or repeated digits before resorting to brute force arithmetic.
Final Answer:
The number whose square is 12345678987654321 is
111111111.
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