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Topic: Crossed Ladders Problem (again)
Replies: 6   Last Post: Dec 29, 2012 2:37 AM

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John Conway

Posts: 2,238
Registered: 12/3/04
Re: Crossed Ladders Problem (again)
Posted: Aug 24, 2001 10:04 AM
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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 GP-Pari 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 back-to-back, 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





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