To show that there are no identisquares with more than one digit, we need to prove that no square number with more than one digit can have all of its digits the same. Here's the approach:

Step 1: Represent a potential identisquare

Let’s suppose an $n$-digit identisquare exists where all the digits are the same. This implies the square number must be of the form:

$\underbrace{d d d \dots d}_{n \text{ digits}} = d \times (10^{n-1} + 10^{n-2} + \dots + 1) = d \times \frac{10^n - 1}{9}$ where $d$ is the repeating digit (a digit from 0 to 9), and $n \geq 2$ (since we are looking for numbers with more than one digit).

Step 2: Check possible values of $d$

For $d$ to be a square number, the possible values for $d$ are limited to digits that are perfect squares themselves: $d = 0, 1, 4, 9$.

So, we need to check if any square numbers of two or more digits can have all digits equal to 0, 1, 4, or 9.

Step 3: Analyze square numbers

Next, we analyze the properties of square numbers.

• Identisquare made of 0's: The only square number composed entirely of 0's is $0$, which is a single-digit identisquare.

• Identisquare made of 1's: If a number consists of all 1’s, then the possible candidate is $111 \dots 1$. However, no number of the form $111 \dots 1$ is a perfect square, as squares grow much more slowly compared to numbers made of identical digits. For example:

  • $1^2 = 1$,
  • $11^2 = 121$ (not all digits are 1),
  • $111^2 = 12321$ (again, not all digits are 1),
  • and so on.

• Identisquare made of 4's: Similarly, if a number consists of all 4's (e.g., 44, 444), squaring such numbers does not yield numbers with all digits equal to 4. For example:

  • $4^2 = 16$,
  • $44^2 = 1936$,
  • $444^2 = 197136$, and so on.

• Identisquare made of 9's: Lastly, if a number consists of all 9’s, squaring it produces results where the digits are not all 9’s. For example:

  • $9^2 = 81$,
  • $99^2 = 9801$,
  • $999^2 = 998001$, and so on.

Step 4: Conclusion

From this analysis, we observe that no square number with more than one digit can have all of its digits the same. Therefore, there are no identisquares with more than one digit.

Relative Questions:

1. What is the definition of a perfect square?
2. How do we find the square of a number using long multiplication?
3. Can a number with more than one digit have all identical digits and still be a cube?
4. Why do squares grow faster than numbers composed of repeated digits?
5. What are the properties of numbers formed by repeating the same digit?

Tip:

Remember that perfect squares have unique digit patterns and grow significantly faster compared to numbers made up of repeated identical digits.