## Teaser 3143: Pipe fittings

**From The Sunday Times, 18th December 2022** [link] [link]

A plumber had three thin metal pipes with square, rectangular and elliptical cross-sections. In order to fit them into his van, he slid the rectangular pipe inside the elliptical pipe and the elliptical pipe inside the square pipe, before placing the pipe assembly in the van. There are four points where the pipes all touch, as shown in the diagram. The maximum and minimum widths of the elliptical and rectangular pipes and the diagonal width of the square pipe were all even numbers of mm less than 1000, of which one was a perfect square.

What were the five widths (in increasing order)?

[teaser3143]

## Jim Randell 5:20 pm

on16 December 2022 Permalink |If we consider the diagram to be centred on the origin (0, 0), and the point where all four figures meet in the (+, +) quadrant is

(x, y), then the rectangle has dimensions(2x, 2y)and the diagonal of the square is2zwherez = x + y.And if the ellipse has semi-major axis

aand semi-minor axisb, then the equation of the tangent at the point(p, q)is:and this is the same as the line that defines the side of the square tube:

Hence:

Assuming

exactlyone of the required widths is a perfect square gives a unique answer to the puzzle.This Python program runs in 134ms. (Internal runtime is 76ms).

Run:[ @replit ]Solution:The widths are (in mm): 180, 300, 320, 400, 500.Swapping [[

`icount_exactly()`

]] for [[`icount_at_least()`

]] reveals 4 further solutions, so it is necessary to assumeexactlyone of the widths is a perfect square.LikeLike

## Jim Randell 6:09 pm

on16 December 2022 Permalink |Or we can use the [[

`pythagorean_triples()`

]] function from theenigma.pylibrary to get a more efficient program.This Python program runs in 58ms. (Internal runtime is 777µs).

Run:[ @replit ]LikeLike

## GeoffR 4:51 pm

on17 December 2022 Permalink |LikeLike