A number theory problem by Chandrakant Pandey
J was an old friend of R and was coming to visit him from a small town. J came over late. When R asked him the reason, J told him that when he was coming to the city in the bus, he had placed his luggage under his seat. While getting off, he had forgotten to take back the luggage and by the time he had realized, the bus had left. So he went to a police station to report about it, which took him time. R inquired what happened next, to which J replied that he does not remember the bus number but there was a certain peculiarity about it – the bus number was a perfect square and when turned upside down, then it was also a perfect square! J requested R to help him. R quickly phoned the bus company and came to know they had only 500 buses numbered from 1 to 500. Using this information, R deduced the bus number within a minute. Can you tell what was the number?
The answer is 196.
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1 solution
We know that the only digits which can be turned upside down and still be read as a digit are 0, 1, 6, 8 and 9. The digits, 0, 1 and 8 remain 0, 1 and 8 when turned upside down, but 6 becomes 9 and 9 becomes 6. Therefore, the possible numbers on the bus are 9, 16, 81, 100, 169 and 196. Out of these, 196 is the only number which becomes a perfect square when turned over because 961 is the perfect square of 31. Hence, 196 is the bus number.