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## #1 2020-07-03 01:41:18

Bob
Registered: 2010-06-20
Posts: 8,914

### puzzle by Xodus

Let m,n be positive integers. When m/n is expressed as a decimal, three consecutive digits after the decimal point are 1,6, 7 in order. Find the smallest possible value of n.

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

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## #2 2020-07-03 14:31:10

pi_cubed
Member
From: A rhombicosidodecahedron
Registered: 2020-06-22
Posts: 63

### Re: puzzle by Xodus

Since 0.167=167/1000 and that is in the lowest terms, the smallest value of n must be 1000...

Last edited by pi_cubed (2020-07-03 14:32:19)

e to the i pi plus 1 is zero -Leonhard Euler | a squared plus b squared equals c squared -Pythagoras | Energy equals mass times the speed of light squared -Albert Einstein | i² = -1 | x^2+1=0 x^2=-1 x=i

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## #3 2020-07-03 20:50:00

Bob
Registered: 2010-06-20
Posts: 8,914

### Re: puzzle by Xodus

hi pi_cubed

Remember this isn't my problem.

It says '3 digits are'; it's unclear if that means the first three or any three.  It doesn't say the decimal is just 0.167  There may be more digits after that.

Assuming it's the first three, I have found an 'n' that is lower than 1000.  I got 0.167989....

If the answer may be 0.dddd167dddd then I think I could get an even lower n.

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

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## #4 2020-07-03 21:13:38

Bob
Registered: 2010-06-20
Posts: 8,914

### Re: puzzle by Xodus

401/491= 0.816700....

I am not yet claiming that is the lowest 'n'.  Still working on it.

B

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

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## #5 2020-07-03 22:11:25

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,653

### Re: puzzle by Xodus

bob bundy wrote:

Assuming it's the first three, I have found an 'n' that is lower than 1000.  I got 0.167989....Bob

Hi Bob,

For 167dddd..., I got m=22, n=131, m/n=0.1679389...

Did you omit the '3' that's in my quotient, or did you find a different n than 131?

Here's the complete list of lowest n I found:
167dddddd... : m=22,   n=131,  m/n=0.167938931...
d167ddddd... : m=107, n=131,  m/n=0.816793893...
dd167dddd... : m=50,   n=131,  m/n=0.381679389...
ddd167ddd... : m=5,     n=131,  m/n=0.038167938...
dddd167dd... : m=66,   n=131,  m/n=0.503816793...
ddddd167d... : m=59,   n=131,  m/n=0.450381679...
dddddd167... : m=19,   n=131,  m/n=0.145038167...

And for all such {167dddddd,d167ddddd,...dddddd167}, I found that the next-to-lowest n is 137.

Last edited by phrontister (2020-07-06 11:18:16)

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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## #6 2020-07-03 23:40:34

Bob
Registered: 2010-06-20
Posts: 8,914

### Re: puzzle by Xodus

hi Phro,

Great to hear from you.

Just knocked together a short prog. to crunch this.  Couldn't do better than 131.

Is there a way to get this analytically I wonder?

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

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## #7 2020-07-04 01:07:53

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,653

### Re: puzzle by Xodus

Hi Bob;

Nice to have another go at something.

I see that your last post came before I finished an edit to my previous post (#5), and so you might like to have a look at that one.

I wrote a small, clunky, BASIC program to get my results, and can't think of a cleverer approach.

It's interesting that all my results for the lowest n have n=131...which looks like a pattern to me.

But it all seems rather frivolous, and so I think I'll leave it there.

Last edited by phrontister (2020-08-05 21:52:24)

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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## #8 2020-08-05 21:11:57

Raveeshjji
Member
Registered: 2020-08-05
Posts: 1

### Re: puzzle by Xodus

I am not yet claiming that is the lowest 'n'.  Still working on it.

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