Puzzle: Numbers without using '6'

[Eccentric]

Ok This is good one from TOI

All numbers less than 10,000,000 and not containing the digit "6" are arranged in descending order. What is the 66666th number on the list?

[DesiGuru]

i think i atleast know the method for this one...
To get to the number of such numbers use P&C logic of 10 possible digits and total arrangements possible without any 6 in them.
This total should be pretty close to 666666. that should lead to the answer.
But u know what P&C is something i despise and hate right now.

Probably cause i dont get its basic concept.

K

PS: Someone wants to volunteer to force the understanding of P&C and probability too on me... maybe a good explanation hidden here some where...

[GauravShah]

Originally Posted by Eccentric

Ok This is good one from TOI

All numbers less than 10,000,000 and not containing the digit "6" are arranged in descending order. What is the 66666th number on the list?

Finding total numbers between 10,000,000 - 9,900,000 not containing the digit "6".

Numbers are of the form 9,9xx,xxx ... where x = 0 to 9.

It is given by 1*1*9*9*9*9*9 = 59,049

Next, finding total numbers between 9,890,000 - 9,900,000 not containing "6".

Numbers are of the form 9,89x,xxx ... So its given by 1*1*1*9*9*9*9 = 6,561

Total till now = 59,049 + 6,561 = 65,610.

Next Range - 9,889,000 - 9,890,000 .... Form 9,889,xxx ... given by 1*1*1*1*9*9*9 = 729.

Total till now = 65,610 + 729 = 66,339.

Next Range - 9,888,900 - 9,889,000 ... Given by 9*9 = 81.
So from 9,888,600 - 9,888,900 = 81*3 = 273

So total now = 66,339 + 324 = 66,663.

i.e. there are 66,663 numbers between 9,888,600 - 10,000,000 not containing the digit "6".

So the 66,666 th number will be 9,888,597

Am getting a feeling that must have goofed up big time somewhere But hoping its correct

No doubt there will be a better method, but didn't feel like leaving it half way thru. So please come out with better solutions.

Enjoy

Gaurav.

PS: A new thread wasn't required for this i guess, mods do the needful .

[Govi]

hey eccentric

one small suggestion. I remember the earlier problem (about the numbers on black board one) you posted, as i said that too was from the column MINDSPORT by Mukul Sharma in Sunday Times. This problem too is from the same column.

Dude instead of coming out with a new thread everytime (adding to the complications of searching ) why not jus post all this discussion in jus one thread.


Cheers

Govi...........

[Eccentric]

Good work Gaurav
your and mine answer matches...

I'll post mine solution
If a digit is missing just treat the base as '9'. This will mean we are using digits 0-8 for our numbers. But as 6 is missing instead of 9, so finally we will replace 6 by 7, 7 by 8 and 8 by 9 in answer.

// Just a simple hint if we want to find 6 number from top when 1-99 (i.e. less than 100) are written in decending order, answer = 100 - 6 = 94 !

If we write 66666 number, in base 9, in accending order, then the 666666th number in base 9 = 111403. (convert 666666 to base 9)

When in decending order 66666 number from top = Subtaction in base 9 (10000000-111403) = 8777486
This is our answer in base 9.

convert this to base 10, so that 6 is missing (as specified above, 6 by 7, 7 by 8 and 8 by 9) Answer = 9888597.


Now if I post new question on the same thread, I think we lose control of what we are discussing. Too many posts and not able to figure out what was the last question.. Creating a new thread you keep track of that and additionally if you want to refer to ist problem disussed that would be much easier.
What more this doesn't even consume more db space and I feel the other way round that it makes search easier. Only problem is we will have too many threads on this forum, but for that I feel that we should have a saparate forum jsut for puzzles.

[GauravShah]

Originally Posted by Eccentric

Good work Gaurav
your and mine answer matches...

I'll post mine solution
If a digit is missing just treat the base as '9'. This will mean we are using digits 0-8 for our numbers. But as 6 is missing instead of 9, so finally we will replace 6 by 7, 7 by 8 and 8 by 9 in answer.

// Just a simple hint if we want to find 6 number from top when 1-99 (i.e. less than 100) are written in decending order, answer = 100 - 6 = 94 !

If we write 66666 number, in base 9, in accending order, then the 666666th number in base 9 = 111403. (convert 666666 to base 9)

When in decending order 66666 number from top = Subtaction in base 9 (10000000-111403) = 8777486
This is our answer in base 9.

convert this to base 10, so that 6 is missing (as specified above, 6 by 7, 7 by 8 and 8 by 9) Answer = 9888597.


Now if I post new question on the same thread, I think we lose control of what we are discussing. Too many posts and not able to figure out what was the last question.. Creating a new thread you keep track of that and additionally if you want to refer to ist problem disussed that would be much easier.
What more this doesn't even consume more db space and I feel the other way round that it makes search easier. Only problem is we will have too many threads on this forum, but for that I feel that we should have a saparate forum jsut for puzzles.

Cool Solution . Keep posting such questions with interesting solutions .

Gaurav.

[DesiGuru]

Originally Posted by Eccentric

If a digit is missing just treat the base as '9'. This will mean we are using digits 0-8 for our numbers. But as 6 is missing instead of 9, so finally we will replace 6 by 7, 7 by 8 and 8 by 9 in answer.

// Just a simple hint if we want to find 6 number from top when 1-99 (i.e. less than 100) are written in decending order, answer = 100 - 6 = 94 !

If we write 66666 number, in base 9, in accending order, then the 666666th number in base 9 = 111403. (convert 666666 to base 9)

When in decending order 66666 number from top = Subtaction in base 9 (10000000-111403) = 8777486
This is our answer in base 9.

convert this to base 10, so that 6 is missing (as specified above, 6 by 7, 7 by 8 and 8 by 9) Answer = 9888597.

Now that is a impressive way to get the solution.
Eccentric did you get it on your own or its the solution given in mindsport??
Unless ur a polymath, thinking like the above is rare...

Good Job!
K

[reachmonil]

Originally Posted by Eccentric

Good work Gaurav
your and mine answer matches...

I'll post mine solution
If a digit is missing just treat the base as '9'. This will mean we are using digits 0-8 for our numbers. But as 6 is missing instead of 9, so finally we will replace 6 by 7, 7 by 8 and 8 by 9 in answer.

// Just a simple hint if we want to find 6 number from top when 1-99 (i.e. less than 100) are written in decending order, answer = 100 - 6 = 94 !

If we write 66666 number, in base 9, in accending order, then the 666666th number in base 9 = 111403. (convert 666666 to base 9)

When in decending order 66666 number from top = Subtaction in base 9 (10000000-111403) = 8777486
This is our answer in base 9.

convert this to base 10, so that 6 is missing (as specified above, 6 by 7, 7 by 8 and 8 by 9) Answer = 9888597.

Hey.... dat was really smart work.!


well... so wat i can generalise from here on is that when such number identification problems are given.... the base conversion would work well! will ease a lotsa calculations! aint it!? keep it up mate!


well.. if anyone can locate any similar problems with the solution know... can u post it... we can work it out here itself n see if this method worrks or no! well... logicaly speaking... dont c y it shudnt!

[reachmonil]

Originally Posted by Eccentric

If we write 66666 number, in base 9, in accending order, then the 666666th number in base 9 = 111403. (convert 666666 to base 9)

When in decending order 66666 number from top = Subtaction in base 9 (10000000-111403) = 8777486
This is our answer in base 9.

convert this to base 10, so that 6 is missing (as specified above, 6 by 7, 7 by 8 and 8 by 9) Answer = 9888597.

HIe eccentric...

well dear.. had a doubt. dunno know if its really stuppid... coz i tried using the same trick on a smaller level of ns... 1-25... descending order.. n read the 10th no in the stack. n this method has worked rite.

but still... wanted to ask it out ! when we have converted the no we want in base 9 (66666 -> 111403). n then we subtract it form 10000000 (which is a base 10 no!!??!! ) so... could u explain ... is it fine!? n how?!

[Eccentric]

Originally Posted by reachmonil

HIe eccentric...

well dear.. had a doubt. dunno know if its really stuppid... coz i tried using the same trick on a smaller level of ns... 1-25... descending order.. n read the 10th no in the stack. n this method has worked rite.

but still... wanted to ask it out ! when we have converted the no we want in base 9 (66666 -> 111403). n then we subtract it form 10000000 (which is a base 10 no!!??!! ) so... could u explain ... is it fine!? n how?!

The question doesn't say that there are 10000000 numbers in the listy. It just say all number less than 10000000 not containing '6', hence 10000000 is already in base 9 form.
There are only 9^8 numbers in the list.!

The question asks for all numbers less that 10000000, (This is the number on top of list) Had we been using only two digits 0,1 (binary) the 2nd number from top would have been 1111111. If we are using base 9(9 digits 0 1 2 3 4 5 6 7 , 2nd number is 8888888. Both these are subraction of 1 from 10000000, in different bases. I hope you got that !

Now in case I need 2nd number from top, for binary I should subtract 10 from original number. That is why I subtract base9 from of 66666 from 10000000.