Observe the pattern in squares of numbers made of only 1’s:

$1^2 = 1$ → 1 digit in N, 1 digit in N² $11^2 = 121$ → 2 digits in N, 3 digits in N² $111^2 = 12321$ → 3 digits in N, 5 digits in N² $1111^2 = 1234321$ → 4 digits in N, 7 digits in N² $11111^2 = 123454321$ → 5 digits in N, 9 digits in N²

We see a clear pattern:

• The number of digits in N² = $2n - 1$
• The highest middle digit in N² = $n$

Given N² = 12345678987654321 → The middle and highest digit = 9 → So, $n = 9$

Hence, N = 111111111 (nine 1’s) and N has 9 digits.