The Bus Number

    While visiting a small town in the United States, I lost my overcoat in a bus. When I reported the matter to the bus company I was asked the number of the bus. Though I did not remember the exact number I did remember that the bus number had a certain peculiarity about it.  The number plate showed the bus number was a perfect square and also if the plate was turned upside down. The number would still be a perfect square – of course it was not?

   I came to know from the bus company they had only five hundred buses numbered from 1 to 500. From this I was able to deduce the bus number. Can you tell what the number?

Answer :  By experiment we find that the only numbers that can be turned upside down and still read as a number are 0, 1, 6, 8 and .

The numbers 0, 1 and 8remain 0, 1 8 when turned over, but 6 becomes 9 and 9 becomes 6. Therefore the possible numbers on the bus were 9, 16, 81, 100, 169 or 196. However the number 196 is the only number which becomes a perfect square when turned over because 961 is the perfect square of 31

Therefore 190 is the correct answer.