Cute Maths Problem I stumbled on in New Scientist today...

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Pi*R^2=L^2 by area relations. Pi*diameter is the circumference of the circle in a length of line that divides the crop from field. I'm assuming the farmers are free to arrange field area to whatever they want to match circle area. Because Aliens... My issue is if the problem includes field area under crop area or not.

You could say L*W-Pi*R^2=brown field area.
Then Pi*R^2=Green crop area.
If set Green=Brown then L*W=2*Pi*R^2.
 
So here is the answer:

Any straight line that evenly divides a circle or a square will pass through the centre of said circle or square.

As such, any straight line that passes through both the centre of the circle and the centre of the square, will evenly cut both the field and the crop in half:

FieldCropProblemSolution.jpg

So to solve, you draw a line that goes from the centre of the circle to the centre of the square, then just extend it straight to the edges of the field, and there is your cut line...

KennB was the first person who PM'd me the correct answer...

To everyone else, I hope you had a little bit of fun thinking about it over the weekend...
 
Is the challenge to find a single line that will always work in any field, or a method to find the line that will work in a particular field but it would be different for the next field?

I'm pretty sure the latter is an easy solution but the former is impossible.

Is the challenge to find a single line that will always work in any field
Correct - the solution will work with any field / crop combination...

So you said, and then the answer you gave was not a single line that works in any field, but a method. I was excited to find the answer I thought to be impossible.
 
The given answer is a single line if you extend the "line connected the centers" to the edges of the square.

But, the answer violates one of the statements given in the puzzle. Using the centers of BOTH the circle and the square implies that you know "the crop’s position within the field".

My solution doesn't require knowing the center of the square. It's not mathematical, just a practical task the two farmers can do alone. Any takers?
 
Piece of cake!

I like to think of it as a cake with a single large frosting flower, for a birthday party for twins. You have to cut the cake so each twin gets the same amount of cake and flower...
 
Wait, wait. That solution is incorrect. I saw the diagonal line thing right away, but ONE solution didn't work for ALL scenarios. Also violates not knowing the position of the crop, which is necessary for drawing a line through it's center.

The problem, as written, does not have a solution. The author of the New Scientist article must have had a "momentary lapse of logic". Needs to be re-written.
 
Wait, wait. That solution is incorrect. I saw the diagonal line thing right away, but ONE solution didn't work for ALL scenarios. Also violates not knowing the position of the crop, which is necessary for drawing a line through it's center.

The problem, as written, does not have a solution. The author of the New Scientist article must have had a "momentary lapse of logic". Needs to be re-written.

I agree. My solution doesn't require knowing the position of the crop circle within the field or the center of the field. But, it does require knowing the center of the circle, which doesn't violate the puzzle statement. Since the crop circle is formed by the irrigation system, the center is always known.
 
My solution was the same as the the one snrkl posted as the correct answer. I do think the problem statement could be made a bit clearer, but it is not incorrect.

Two farmers inherit a square field containing a crop planted in a circle. Without knowing the exact size of the field or crop, or the crop’s position within the field, how can they draw a single line to divide both the crop and field equally?

The problem simply states that you do not know the position of the crop with respect to the field (i.e. the position of the centre of the crop with resect to the field). It does not say you are not able to identify the centre of the crop (or the field for that matter). Also the wording implies finding a method to draw a single line, not a single line that will work in all cases (i.e. "how can they draw...").
 
Apologies if my response was unclear... At least we can all count our blessings that it was only a cute maths problem and not an inches/mm snafu that prevented bits of the ISS fitting together... ;^)

So you said, and then the answer you gave was not a single line that works in any field, but a method. I was excited to find the answer I thought to be impossible.
 
Indeed, for this engineer that solution seems extremely hand-wavy to me on both finding the center of the square you don't know the dimensions of, and finding the center of the circle you don't know the dimensions of. :)

Provided you had a way to mark a point and a pair of ropes you knew were long enough to reach from corner to opposite corner of the square (someone else would have to provide those ropes I guess), AND you had the necessary means to pull those ropes straight (much easier said than done if we're talking about any reasonably-sized field), you could connect the 2 pairs of opposite corners of the square with the ropes and use the intersection to determine its center, then use those same two ropes to determine the center of the circle (one person stands anywhere along the edge, the other walks around the edge keeping the line tight until they don't need extra line (which would still be difficult to find the true center this way since there will probably be a decent length where the rope doesn't appear to change lengths at all, so how do you tell exactly when it was the longest? And I'm basically ignoring the weight of said rope and how much it might sag/stretch while doing this), then the first person stands at a reasonably distant spot and repeats the same procedure with the second line, intersection of the two lines is the circle's center.

But sure, once you managed to figure out what those two centers are, connecting the two centers would be quite a bit easier in comparison. Extending that line all the way to the ends of the field would be quite a bit harder again (not doing it with only two people, unless they could see when the line was over the two points, which would have to be a pretty small field).

I guess a possibly(?) "easier" way to find two lines that bisect the circle, provided the two people could walk in perfectly straight lines (makes extending the above line easier) AND could determine that they were walking away from each other (inside the field) at a perfectly 90° angle (OK, neither of those sound easier than the rope method.. :p ), if they drew a line between the points where they each hit the edge of the field, and did that twice from two separate starting points, the intersection of those lines would also mark the center of the circle.

If the center of the field was outside of the crop, there'd be another way to find the line that goes through the center of the circle without actually finding the center first (by sweeping the line around the field center to find the two points tangent to the field, then finding the middle of the line connecting those two points, then drawing a line from the field center through that center), but if the field center fell within the crop this wouldn't work (probably another way to handle that, just don't know if off the top of my head).

While in school, I used to joke about encountering a problem like "A ball rolls off the end of the table. How long does it take to hit the ground? Ignore physics." This problem certainly seems to have the "ignore physics" part implied. :)

It's funny though, I hadn't realized just how much I'd forgotten about trig and properties of regular shapes and such until I started 3D printing and modeling things, and had to dredge-up all of those old memories like SOHCAHTOA, etc. :p
 
It's funny though, I hadn't realized just how much I'd forgotten about trig and properties of regular shapes and such until I started 3D printing and modeling things, and had to dredge-up all of those old memories like SOHCAHTOA, etc. :p

I have another cute problem I am thinking about putting up later that uses the properties of triangle... the "elegant" solution is a long one (there are 54 ways to solve it apparently) I just need to figure out how to give it a rocket slant and decide whether the likely arguments that will inevitably crop up are worth the effort!! ;)
 
Do share.

If someone is still pondering my alternate solution, I'll wait a little longer. My babbling in this thread has some clues, but I'll repeat here for convenience: you know the center of the crop circle because it is formed by the irrigation system anchored at the center; you don't need to find the center of the field; other than knowing the center of the circle, you don't have to wander around the property marking any intermediate lines; two people can do this in a practical way. Another clue is that my solution assumes the field is flat and you can see across it.
 
Do you need to know where a corner of the field is?
 
OK, here's my alternate (practical) way to divide up the field and crop. I'll give the two-person solution, but it can be done by one person by estimating and correcting. The solution assumes it's fairly flat and you can see across it.

1) Go to the center of the crop circle and put a tall pole there with a flag on it. You know the center because the irrigation system is how the crop circle was formed.

2) One person walks to the closest corner of the field and the other person walk to the opposite far corner. Since you are inheriting the property, it must be staked out in its corners, at least.

3) The person on the closest corner begins walking along the perimeter of the property in the direction closest to the center flag. The other person walks along the perimeter in the opposite direction.

4) Choose a method to make sure you both measure off the same distance as you go. Equal length of rope, a measuring wheel, etc. Choose a method to communicate with each other (low-tech, primitive: semaphore flags; or radio/phones. Choose a method to see at least one of you as you go (flag on a pole, binoculars, depends on the size of the property).

5) Continue walking the same distance in opposite directions along the perimeter until the center flag in the crop circle lines up with the person on the opposite side. The dividing line is a single line the connects the three points, two on the perimeter and one at the center of the crop circle.
 
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