## Teaser 3034: Reservoir development

**From The Sunday Times, 15th November 2020** [link]

A straight track from an observation post, O, touches a circular reservoir at a boat yard, Y, and a straight road from O meets the reservoir at the nearest point, A, with OA then extended by a bridge across the reservoir’s diameter to a disembarking point, B. Distances OY, OA and AB are whole numbers of metres, with the latter two distances being square numbers.

Following development, a larger circular reservoir is constructed on the other side of the track, again touching OY at Y, with the corresponding new road and bridge having all the same properties as before. For both reservoirs, the roads are shorter than 500m, and shorter than their associated bridges. The larger bridge is 3969m long.

What is the length of the smaller bridge?

[teaser3034]

## Jim Randell 5:44 pm

on13 November 2020 Permalink |We can solve this puzzle using applications of Pythagoras’ Theorem.

This Python program runs in 47ms.

Run:[ @repl.it ]Solution:The smaller bridge is 2304 m long.The layout looks like this:

And the puzzle can be solved by considering the right-angled triangles OYX’ and OYX.

The calculated distances are:

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## Jim Randell 2:29 pm

on14 November 2020 Permalink |Or, with a bit more analysis:

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## Tony Brooke-Taylor 11:34 am

on17 November 2020 Permalink |I first did this assuming that (using your notation) R+B also had to be a square. If I interpret your code correctly, I think that is also what you have done. I got the same value for R as your code produces. Following this through you get a unique solution for r. However, on reflection I don’t think the puzzle applies that constraint. In the puzzle’s notation, that would require OB and its equivalent to be squares, which I cannot see in the puzzle. If I relax the constraint I get three solutions. I am guessing that either the puzzle requires the addition of that constraint or I cannot read, but the more interesting question for me is whether or not it is more than a coincidence that the unique solution has that additional property.

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## Jim Randell 11:56 am

on17 November 2020 Permalink |@Tony: Thanks for your comment.

We don’t need to assume that

(R + B)is a perfect square (and the puzzle doesn’t tell us that), but it follows from considering the right-angled triangle OYX’ (where X’ is the centre point of the larger reservoir).We have (where integer

tis the length of the track OY):Now, we are told that

Ris a square number, and obviouslyt² is, so it follows that(R + B)must also be a perfect square.LikeLike

## Tony Brooke-Taylor 6:31 pm

on17 November 2020 Permalink |Thanks Jim. The constraint I had failed to apply was that t must be an integer.

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## Frits 9:20 pm

on17 November 2020 Permalink |LikeLike