Any mathematicians out there??

It can’t eat exactly half if it is tethered on the edge of a circle, even if the rope is the length of the radius.

a) just making the rope half the radius doesn't make an inner circle of the same area as the doughnut that is left. It is more complicated to work out how long the rope has to be to let Buttercup reach exactly half the area (but still possible)

Yes, I realised that, that's why I didn't attempt to post an equation .

I do have a friend (ex-colleague) who just might be able to solve it mathematically but he's not a farmer so wouldn't be factoring in the cow and her moods.

Anyway, some of the grass might be longer than in other areas.
😀

I spent a while establishing the equation to do just that (it is a long time since I did this kind of sum), before I realised that

a) just making the rope half the radius doesn't make an inner circle of the same area as the doughnut that is left. It is more complicated to work out how long the rope has to be to let Buttercup reach exactly half the area (but still possible)

b) more importantly, I think an important part of the puzzle is that she isn't tethered at the centre, but on the edge of the circle, so her "circle of influence" isn't even a whole circle. If you ask me, whoever invented this instrument of mental torture only did it to look down from a cloud and laugh at the poor saps trying to solve it.

At that point I decided to leave it to the experts and stay with lateral thinking and alternative solutions.

He could divide the area into two circles, each half the same area, tether the cow in the centre with the rope at the correct length to wander around the inner circle. When the grass there is all eaten then lengthen the tether so the cow eats the grass in the outer circle.
Of course, the grass in the inner circle could start growing again so the cow could wander back and munch that as well.

Or just leave the cow to please herself and the farmer could go to the pub.

I gave up and googled it. It's a long-standing mathematical problem, and even though I have a maths degree, I can't explain the answer!

Keeper1

Get a smaller field

Buy another cow and let them share the grass?

Posted one paragraph twice - if the farmer follows that suggestion twice he will have a very fat cow.

I am stuck on the equations, calculus etc, but I can tell the farmer how to make sure the cow eats half the grass without doing any sums at all.

1) If he was the one who marked out that non-standard field, he should know where the centre is, as he must have used a post with a rope round it.

2) He could either: -
a) build a straight fence or wall across the circle going through the centre point, and untie the cow to eat the grass in the half she is loose in, or:-

b) do as in 3) to 7) (this is a much more elegant solution) :-

3) Mark a place on the each side of the circumference of the circle, directly opposite each other.

4) Using that same length of rope that he used to mark the original circular field, attach it to one of the marks, and draw a semicircle from the centre point of the circle to the circumference.

5) Attach the rope to the other mark and draw a matching semicircle from the centre point to the other side of the circumference.

6) Build a fence or wall along each semicircle, untie the cow, and put the cow into one of new fields.

7) He now has two Yin-yang shaped fields which are not only the same mathematical area but are also exactly balanced aesthetically and symbolically, so the chances of his cow eating exactly half the grass are three times as good as with just a boring straight fence.

Alternatively, if he doesn't have wood for a fence or stone for a wall, he could put the cow into the barn and wait until the grass has turned into hay, then feed half of it to the cow. Meanwhile it can eat silage from last year's grass crop, made more exciting by potato peelings and apple cores from the farm kitchen.

Get a smaller field

If the farmer puts the cow into the barn and waits until the grass has turned into hay, he could feed half of it to the cow. Meanwhile it can eat silage from last year's grass crop, made more exciting by potato peelings and apple cores from the farm kitchen.

M0nica

The formula for the area of a semi circle is; πr2/2

Yes, but that is a semicircle with the non-semicircular edge flat. This one has the "other" edge as part of the circumference of the circular field, therefore it has two curved edges, neither of them defined by their radii, so there are (at least) two unknowns in the equation (or equations, as I suspect we have to do more than one operation)

I got as far as working out that you need to first calculate the area of the semi-circle, as mentioned by M0nica, then find out how to calculate the size of an oval that had the same area as the semi-circle ie. half the area of the circle. (The curve of the oval must fit into the circle and leave a negative space in the shape of a crescent that has the same area as the semi-circle) Then measure the diameter at the widest point to give the length of the rope. I did not have a clue how to do all this so I went off and got on with some gardening.

Aaargh! Calculus!

Grannynannywanny

You’ve got me stumped. I think I’d leave the cow in the adjoining field. Cut the grass on half the field, rake it up and throw it over the hedge 😀

The formula for the area of a semi circle is; πr2/2

A chum of mine had a goat tied to a stake hammered into the ground just near enough to the fence that the stupid creature (the goat, not my friend - or maybe...) jumped over and hanged itself on the rope over the fence.

Your task is to calculate the statistical odds of that happening...

I'm now reliably informed that there's advanced calculus used in this..........😱😱

My DH prides himself on his mathematical knowledge (and indeed has taught some maths in years gone by) and loves such puzzles. He has spent an hour on it so far with no luck and has gone to do something else. Never known this before!

For myself, I USED to be a wizard at Maths and got a Grade 1 in my O-level at age 14. I too am stumped, but not yet giving up!

Elegran

I'll send this puzzle to MY brother. He doesn't have enough to keep his brain exercised since he retired.

I'm not going to ask DH.
I could be there all day as he explains this, complete with diagrams - then asks me which farmer would be daft enough do this 🐄

My brain hurt just thinking about it.

Elegran

Now I have finished my coffee, I can see that my equation only gives you an area covered by a circle nearest the post which is clearly not as big as the area of the rest of the round field outside that circle.

I give up.

Yes, the first thing that struck me was the the cow is tethered to the edge of a circular field.

Logically
A) I've rarely seen a circular field
B) I've never seen a cow tied up in a field
C) The grass always looks greener on the other side of the fence so the cow might try to get through to another field.

You’ve got me stumped. I think I’d leave the cow in the adjoining field. Cut the grass on half the field, rake it up and throw it over the hedge 😀

This mathematical problem has been discussed for almost 300 years featuring tethered horses, cows, goats and caged birds.

www.quantamagazine.org/after-centuries-a-seemingly-simple-math-problem-gets-an-exact-solution-20201209/

Whats it fer ??
🤣🤣🤣🤣🤣🤣🤣

I googled this and still don’t understand the answer 😀