The Bus Number
I lost my cellphone 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.
I came to know from the bus company they had only five hundred buses numbered from 5 to 504 excluding 100. What was the bus number?
The answer is 196.
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1 solution
The only 4-digit number that has this property is 8 3 2 = 6 8 8 9 , which also happens to be the same number upside-down. The only such 5-digit number is 2 6 3 2 = 6 9 1 6 9 , which again is the same number upside-down. I haven't checked this out for 6-digit numbers yet.
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But the Bus Number is a perfect square , 83 and 263 are not perfect squares.
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Yes, I know. The numbers of interest are 6 8 8 9 and 6 9 1 6 9 , which are both perfect squares right side up and upside down.
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@Brian Charlesworth – Okay ! I thought you were doubting the solution so I was trying to explain.
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@Nashita Rahman – No doubts about your solution at all. I was just extending your problem to 4-digit and 5-digit numbers. :)
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@Brian Charlesworth – That's Nice!! :) Keep it up!!
@Brian Charlesworth – Hey, are you doing it at random , that is selecting numbers at random and checking or you are finding it out by someway. If so, please share with us!
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@Md Zuhair – Actually, I just quickly scanned through a list of perfect squares up to 1 0 0 0 0 0 , looking for those with digits 0 , 1 , 6 , 8 , 9 only, and then checking if they were also squares upside-down. There aren't that many that fit these parameters so it only took a few minutes. :)
It is the bus number which is turned upside down and still remains a perfect square. I hope you got my point!
What about 1 or 100? These both satisfy the conditions as well!
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100 cannot be because when turned upside down it gives 001 which cannot be the bus number as in the question it is mentioned that the bus company had buses numbered from 5 to 500. You are right that 1 can be possible so I have edited the question and made bus numbers from 5 to 500. Sorry for the mistake!! Thanks for your feedback , it did help a lot!!
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And 961 is between 5 and 500?
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@Dan Ley – How come 961 is between 5 and 500? It is not possible!
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@Nashita Rahman – That's the point! So can't 100 be the bus number, even though 001 is outside the range? 169 is the answer, even though 196 is outside the range:) It may be better to specify that no leading zeroes are allowed.
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@Dan Ley – If you turn 196 upside-down , you get 961 and not 169! So 169 cannot be the answer.
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@Nashita Rahman – I agree with what you are saying here, but why does that mean that 100 cannot be the answer?
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@Dan Ley – 100 cannot be the answer because when turned upside down , it becomes 001 and the bus company did not have that bus number as mentioned in the question. I hope it is clear now :)
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@Nashita Rahman – 196 cannot be the answer because when turned upside down, it becomes 961 and the bus company did not have that bus number as mentioned in the question. This conclusion comes from your statement!
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@Dan Ley – You didn't read the question properly , I suppose . The question says the bus number is a perfect square when turned upside down. The bus company had buses numbered from 5 to 504 . Its only 196 that satisfies this so it's the answer and 961 cannot be because the bus company did not have that bus number.
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@Nashita Rahman – I suggest you to read the question again.
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@Nashita Rahman – I've read it three times:) Why can't the numberplate be 100?
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@Dan Ley – I can understand that there is a confusion with 100 as the bus number. So to clear the confusion I have excluded 100 from the question , I have edited it. Just read the question again , I hope it is fine now:)
@Dan Ley – Because there is no bus number as 001 . The bus company have buses numbered from 5 to 504.
@Dan Ley – No , 196 is inside the range , it does lie between 5 and 504.
Wow, Liked the question. Good problem made, but you could give it in logic as there is more logic than number theory
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Thank You! I think you are right it's more of logic so I have changed the topic of the problem to logic.
Do you mean to say that there are 500 buses from 5 to 500, or that there are 496 total?
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Oh sorry , actually I edited it previously and changed 1- 500 to 5-500 and I forgot to edit this. Thanks for pointing it out:)
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No worries, nice problem.
I have just edited it , I think now it's fine!! And sorry for the mistake.
By experiment we find that the only numbers that can be turned upside down and still be read as a number are 0 ,1,6,8 and 9.
The numbers 0 , 1 & 8 remain the same when turned over but 6 becomes 9 and 9 becomes 6. We know that the bus number was a perfect square.Therefore , the possible numbers on the bus were 9,16,81,100,169, 196. We see that only 196 when turned upside down remains a perfect square.
196 = 14x14 and 961 = 31x31.
The bus company had buses numbered from 5 to 504 so 961 cannot be the answer and so the correct answer is 196.