The Goat on a Rope

What’s going on

A goat eats every blade of grass it can reach. So the way you tie it up decides the shape of the bare patch it leaves. In the scene above, the shaded patch is what the goat can reach. Drag the pegs, the rail ends or the goat, change a rope with the slider, or pick a challenge.

We treat the goat as a dot, the ropes as unstretchable, and we let the goat step over ropes without getting tangled. Then there are a few basic ways to tie it:

• A peg. A rope of length R on a peg: the goat reaches a disc of radius R. Ten metres of rope gives about 314 m².
• A ring on a rope. Stretch a rope between two pegs and tie the goat to a ring that slides along it. Now the goat reaches every point within R of the stretched rope: a rectangle with a half-disc on each end, called a stadium. With a 20 m rail and a 5 m rope that is 20 × 10 + π × 5² ≈ 279 m².
• A wire loop. Let the ring slide round a closed loop of wire, and the goat grazes a band of width R on both sides of the wire.

The trick: several ropes at once

Now tie the goat with two ropes, each to its own peg or rail. The goat can only stand where both ropes reach. So its patch is the overlap of the two patches. Choose “Rectangle”: a horizontal rail with a 5 m rope gives a strip 10 m tall, a vertical rail with an 8 m rope gives a strip 16 m wide, and the goat ends up in a 16 × 10 m rectangle. Tick “Show each rope’s reach” to see the two strips as dashed outlines.

So every extra rope can only take grass away, never add it. A wide strip works like a knife: one of its straight edges cuts through the patch, while the other edge stays out of the way.

• Three ropes on three rails make a triangle.
• Four ropes, two of them slanted, make the arrow in the list.
• A peg with a 12 m rope makes a disc; a wide strip whose edge passes through the peg cuts it into a semicircle.

In this way you can make any polygon without dents, one rope per side.

A fence is a rope too

The last challenge uses a round fence. A goat inside a round fence of radius 10 m behaves exactly as if it were tied with a 10 m rope to the centre. So “inside the paddock and tied to a post on its fence” is again an overlap of two discs.

Try it yourself

1. A triangle of grass

How would you tie a goat so that it eats exactly a triangle? Then try it in the scene with “Add rail”.

Answer

Use three rails, one for each side. Put each rail parallel to its side, on the inside, at a distance R from it, and tie the goat to it with a rope of length R. Make R large enough for the strip to cover the whole triangle (at least half of the triangle's height on that side), and the rails long enough. Each strip then has one edge along a side of the triangle, and the overlap of the three strips is the triangle. The "Triangle" challenge is built this way.

2. Grass along a wire

A ring slides round a rectangular wire 16 m by 11 m, and the goat is tied to it with a 3 m rope. How much grass can it eat?

Answer

First count everything within 3 m of the rectangle, inside or out. That is the rectangle itself (176 m²), four strips 3 m wide along its sides (perimeter 54 m × 3 m = 162 m²) and four quarter-discs at the corners, which together make a disc of radius 3 (about 28.3 m²): 366.3 m² in all. But the goat cannot reach the middle: a 10 m × 5 m rectangle, more than 3 m from the wire. So the answer is about 366.3 − 50 ≈ 316 m². Pick "Wire loop" to check.

3. Half a round field

A round paddock has radius 10 m. The goat is tied to a post on its fence. How long should the rope be so that it can eat exactly half of the grass inside?

Answer

About 11.59 m, that is 1.1587 times the radius. A 10 m rope is too short: it reaches only about 39% of the paddock. To find the exact length you must solve an equation in which both an angle and its cosine appear, and there is no neat formula for that; you get closer and closer by trial. Pick "Half a round field" and move the rope slider to see the share climb to 50.0%.

Go deeper

• The half-a-field puzzle is very old: a version with a horse was printed in England in 1748, in an almanac called The Ladies’ Diary. In 2020 Ingo Ullisch finally wrote the answer as an exact (but complicated) formula.
• “Everything within R of a shape” is what engineers call an offset. Metal-cutting machines and robots planning a path round obstacles compute offsets all the time.