Trivendrum Padmanabhaswamy temple math puzzle.

[P.J.]

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Trivendrum Padmanabhaswamy temple math puzzle.


The temple has 100 doors and they could be opened or closed only by elephants. To keep the elephants from getting bored, the mahoots had invented a game.


Every night all the 100 doors were closed. There were 100 elephants. Each elephant is instructed to open a closed door or close an open door.


The first elephant went by each of the closed door and opened them.


The second elephant went by every other door and closed them starting with the second door.


The third elephant went every third door and either opened a closed door or closed an open door, starting with the third door.


The fourth elephant went by every fourth door and so on until the hundredth elephant had to deal with just the 100th door.


When the hundredth elephant was done what was the state of each of the hundred doors? Which ones were open and which ones closed?


What if the temple had 1000 doors and 1000 elephants?


No search for answers on the internet please.

Have fun over the weekend.


SOURCE: FB

 

[Raji Ram]

If the elephants were ordered to do as mentioned in the OP,

surely at least the first few will punish the mahouts for the tiring work!! :whip:

 

[Raji Ram]

My mind was working on this problem, while I was preparing breakfast! :hungry:

The pattern is fascinating.

All the doors with numbers which are squares of digits from 1 to 10 will be closed!

Other 90 doors will be open!

The same can be extended to 1000 doors also. It will end with the square of 31. :thumb:

P.S: May be Prof. M S K Sir can get some formula to find this out!

 

[Raji Ram]

When I gave the answer for the 'extra man' puzzle, no other reply was there!

So, I expect :tape: in this thread also!

 

[yesmohan]

In Kerala, individuals own elephants. The more number of elephants a person owns is a symbol of how much wealth he owns. One of my friends in KOttayam who owns three estates is having 4 elephants of his own.

 

[mskmoorthy]

RR madam is correct (of course). Below is an explanation (highlight/select the region to read it) - For others who want to continue to think, please do so,

This is a problem on the parity of number of factors in disguise. Each elephant is a potential factor. The elephant numberred i opens/shuts the door i,2i,3i , ... etc. (We can save elephants walking - by the elephants operating a on/off remote switch to open/close doors )
All square numbers have odd number of factors. Or odd parity. So if all the doors are open at the beginning, at the end all the doors whose numbers are squares will be open.

To see why square numbers have odd number of factors, the factors come in pairs, but for when the number is asquare number. e.g. 6 (not a square number) 1x6, 2x3 - so 4 factors. For 36 (square number) 1x36, 2x18,3x12,4x9,6x6 has 9 factors (odd).

 

[mskmoorthy]

If all the doors are closed at the beginning, then all the doors with square numbers will be open for the reasons given earlier.

I am unable to edit my posts.

 

[Raji Ram]

Dear Prof. Sir,

Thank you very much. I could guess it but wanted you to confirm!

 

[zebra16]

My mind was working on this problem, while I was preparing breakfast! :hungry:

The pattern is fascinating.

All the doors with numbers which are squares of digits from 1 to 10 will be closed!

Other 90 doors will be open!

The same can be extended to 1000 doors also. It will end with the square of 31. :thumb:

P.S: May be Prof. M S K Sir can get some formula to find this out!

Without getting into the mumbo jumbo of maths formula, odd numbers added to the square of a number, in the ascending order, gives out the next square of a number.

1^2 plus three (the first odd number after 1 that is) = 4 = 2^2

2^2 plus five (the next odd number to 3) = 9 = 3^2

3^2 plus seven (the next odd number) = 16 = 4^2

4^2 plus nine ( ....... ditto ............) = 25 = 5^2

5^2 plus eleven (......ditto.........) = 36 = 6^2

and so on......

 

[sarang]

The difficulty is how to conclude that this logic works for all numbers. It needs a very sharp and clever mathematician to find the limit and the proof.

Without getting into the mumbo jumbo of maths formula, odd numbers added to the square of a number, in the ascending order, gives out the next square of a number.

1^2 plus three (the first odd number after 1 that is) = 4 = 2^2

2^2 plus five (the next odd number to 3) = 9 = 3^2

3^2 plus seven (the next odd number) = 16 = 4^2

4^2 plus nine ( ....... ditto ............) = 25 = 5^2

5^2 plus eleven (......ditto.........) = 36 = 6^2

and so on......

 

[Raji Ram]

Dear Narayan Sir,

Prof M S K also gave a simple solution.

Numbers are fascinating !

And there are more than one way of getting the solution.

Even in your solution, there is a formula:

If n is a whole number > 0, then

n square = (n-1) square + (2n - 1)

It is very easy to find the square of 11 with the above formula!! :becky:

 

[Raji Ram]

The difficulty is how to conclude that this logic works for all numbers. It needs a very sharp and clever mathematician to find the limit and the proof.

Check out the formula given in my previous post, Sarang Sir! It works.

 

[zebra16]

The difficulty is how to conclude that this logic works for all numbers. It needs a very sharp and clever mathematician to find the limit and the proof.

Mine is a Deshi paddhathi. I have worked out only till the required number. If it is useful and can be patented, I can think about establishing its proof

When I used to drive on two wheeler on long journeys to office, observing the licence numbers of vehicles moving ahead of me and overtaking me gave me this idea.

 

[Raji Ram]

This logic is simple if we take three category of numbers (viz) Primes; Squares; Other numbers.

1. Prime numbers: No elephant will meddle with it, after the first one! So, door has to be opened.

2. Squares: The factors, other than one and itself, will be in odd numbers only! So, door has to be closed.

3. Other numbers: The factors other than one and itself, will be in even numbers only! So, door has to be opened.

Hence, ONLY the doors with square numbers will be closed; others will be open.

Same logic extends to any number of doors.

 

[sarang]

Mine is a Deshi paddhathi. I have worked out only till the required number. If it is useful and can be patented, I can think about establishing its proof

When I used to drive on two wheeler on long journeys to office, observing the licence numbers of vehicles moving ahead of me and overtaking me gave me this idea.

If you are able to understand relationship between numbers by just seeing them, you are obviously in a different league. Salute.
Numbers are really fascinating. Since only we had the concept of very large numbers and names for them, let us hope knowledge we lost is regained at least in parts.

 

[Raji Ram]

Dear Sarang Sir,

(a-b) whole square formula is taught in school.

Now check the formula here:

...... Even in your solution, there is a formula:

If n is a whole number > 0, then

n square = (n-1) square + (2n - 1) ...