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 was the number?

Identification of the Unique Bus Number

The task is to identify a bus number within the range of 1 to 500 that is not only a perfect square but also remains a perfect square when the digits are viewed upside down. Following the clarified criteria:

Criteria for the Bus Number

• The number must be a perfect square.
• When turned upside down, the number must still represent a perfect square.
• The number must include only the digits 0, 1, 6, 8, and 9, as these are the digits that can represent other numbers or themselves when flipped.

Analysis

We consider the range of perfect squares from \(1^2\) to \(22^2\) as these are the perfect squares within the 1 to 500 range:

\[
1, 4, 9, \ldots, 484\ (22^2)
\]

Among these, the number that meets the specific requirement of being a perfect square that, when turned upside down, also becomes a perfect square, is 169:

– Original Number: \(169 = 13^2\), a perfect square.
– Upside Down: When 169 is flipped upside down, it becomes 961. Notably, 6 and 9 flip, while 1 remains the same, making the new figure, 961, which is \(31^2\), another perfect square.

Conclusion

Therefore, the bus number with the described peculiarity is 169. This number is unique in that it satisfies the condition of being a perfect square and also transforms into another perfect square (961) when viewed upside down.