Re: Pi day
- To: mathgroup at smc.vnet.net
- Subject: [mg108335] Re: Pi day
- From: Patrick Scheibe <pscheibe at trm.uni-leipzig.de>
- Date: Sun, 14 Mar 2010 05:13:50 -0500 (EST)
Hi,
imagine you would take two good rational approximations for Pi:
Rationalize[Pi, 10^-#] & /@ {11, 22}
gives
{833719/265381, 1299139324288/413528890451}
what you see is that the first approximation uses 6 digits and the last
one 13 at most.
Taking the numerators as x-values and the denominators as y-values and
you can think of this as a line between the points {833719,265381} and
{1299139324288,413528890451}.
All points "near" this line seem to be good approximations to Pi.
Furthermore, the numbers in the last points contain to much digits to
get a rational expression where every digit is different.
Now you could say, I'm going from the last point downwards on the line
and check the points near the line. If a point fulfills the constraint
that every digit in x and y value is unique, you probably have a good
solution.
I don't know what you have so far but I got (zeros included, since I
don't want to solve existing exercises):
8954632071/2850347916
which is 7.299378879110918*^-10 away from Pi.
Cheers
Patrick
On Sat, 2010-03-13 at 07:55 -0500, Tom wrote:
> Hello, I am a high school math teacher and the following puzzle was
> posed by a few math teachers I am in contact with.
>
> Create a fraction whose numerator has the digits 1 - 9 (used once)
> and whose denominator has the digits 1 - 9 (used one) .
>
> Which fraction has a value closest to the value of pi?
>
> I've worked on some "brute force" checks and managed to check all
> possible fractions with 2,3,4,5 and 6 digits. But after that, there
> are just too many possibilities.
>
> I don't have the programming ability to implement something elegant in
> Mathematica.
>
> Is there anyone who could suggest an approach to find the solution to
> this?
>
> Sincerely,
>
> Tom
>