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Train track shapes (Posted on 2011-08-01) Difficulty: 3 of 5
I have a toy train set with lots of curved pieces. If you put any 8 of these pieces together you can make a circle with radius r.

Ignoring the width of the track, find the area (in terms of r) enclosed by each of the following configurations:

1) As the train goes around the track clockwise it goes 5 turns right then 1 turn left. This pattern continues twice.

2) 6 right 2 left, twice.

3) 5 right 1 left 1 right 1 left, twice.

4) 3 right 1 left, four times.

5) 4 right 1 left 2 right 1 left, twice.

6) 5 right 1 left 2 right 1 left 5 right 2 left.

Bonus: Are there any other configurations using 16 or fewer of these pieces?

See The Solution Submitted by Jer    
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Some Thoughts A start, with 1 and 2 (spoilers for these), and a thought | Comment 1 of 2

In a circular setup, each section of track subtends a sector of area pi*r^2 / 8.

1) There are two lobes of area 5*pi*r^2 / 8 plus a "waist" area connecting them. This waist area consists of a rhombus of side 2*r with two of the above-mentioned sectors cut out.  The vertices of the rhombus are the centers of the convex arcs (5/8 circles) and the centers of the concave arcs (1/8 circles). The base of the rhombus is 2*r and its height is 2*r*cos(45) = r*sqrt(2).

The total area in case 1 is therefore:

10*pi*r^2/8 + 2*sqrt(2)*r^2 - 2*pi*r^2 / 8

= (pi + 2*sqrt(2))*r^2

2) This time there are two 3/4 circles plus a square of side 2*r minus two quarter-circles:

6*pi*r^2 / 4 + 4*r^2 - pi*r^2 / 2

= (pi + 4) * r^2

Interesting aside: case 1 has an area smaller than two circles, while case 2 has an area larger than two circles. If fractional curved pieces could be used, there'd be some point at which the area would be that of exactly 2 circles. What would be the shape of such a track?

  Posted by Charlie on 2011-08-01 14:00:06
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