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Re: Crossed Ladders Problem (again)
Posted:
Aug 24, 2001 10:04 AM


On 24 Aug 2001, Randall L. Rathbun quoted:
> > I wonder what the simplest rational solution is?
and wrote:
> Well, the simplest is for the width = 8, the first ladder = 17, the second > ladder = 10, and the height is 30/7. However I decided to use GPPari to > find integer solutions to this, which it did quickly  > > Here they are for all widths from 1 to 10,000: > > Double Ladders Problem  Integer Solutions > >  +  > sorted by width  sorted by height >    > width A B height  width A B height >    > 56 119 70 30  56 119 70 30
[and then many more, showing that this is indeed the simplest in every reasonable sense]
Thanks. Shortly after writing my last message, I proved a theorem to the effect that they are all obtained by putting two Pythagorean triangles backtoback, and used it to verify for myself that this was the simplest solution. [One only has to consider the first 4 or 5 Pythagorean triangles to do this.]
Very often (as here) there is a common factor of the two lengths that divides the width but not the height. It would be nice if it were the other way around, since the width is the unknown, but it seems that this never happens.
John Conway



