Question

The no. plate of a bus had peculiarity. The bus number was a perfect square. It was also a perfect square when the plate was turned upside down. The bus company had only five hundred buses numbered from 1 to 500. What can be the number?

• A. 169
• B. 36
• C. 196
• D. Cannot say

Answer 1

The Correct Option is A

To solve this problem, we must identify a bus number between 1 and 500 that is both a perfect square and remains a perfect square when read upside down. For clarity, let's explore this step-by-step:

• First, identify numbers that are perfect squares up to 500. Calculate the squares of numbers up to √500, which is approximately 22.36.
• Calculate: \(1^2 = 1\), \(2^2 = 4\), \(3^2 = 9\), \(4^2 = 16\), ..., \(22^2 = 484\).
• Next, consider the numeric symmetry when turned upside down. Certain digits transform into others or become illegible: 0 ↔ 0, 1 ↔ 1, 6 ↔ 9, 8 ↔ 8, 9 ↔ 6, 2, 3, 4, 5, 7 become illegible.
• We need a perfect square which remains another perfect square upside down. The candidate numbers are those containing only the digits 0, 1, 6, 8, 9.
• Examine options within our range:
  • 169 becomes 961 when flipped and \(961 = 31^2\), a perfect square.
  • 36 flips to an illegible combination (as it involves 2, 3, 4, 5, or 7).
  • 196 flips to a non-square (691 is not a perfect square).

Thus, the number is 169 as it's the only number that satisfies both conditions: it's a perfect square and turns into another perfect square (961) when flipped.