WSJ weekend puzzle

[Quix0te]

Cornfield Planning

A farmer has a square plot of land that measures 100 feet on each side. She plans to grow corn in the plot, and she will install a fence around the corn. Fencing is expensive, so she wants to grow the corn in a shape that will maximize the ratio of the area of the cornfield to its perimeter.

What shape should the cornfield be?

 

[mashnut]

Cornfield Planning

A farmer has a square plot of land that measures 100 feet on each side. She plans to grow corn in the plot, and she will install a fence around the corn. Fencing is expensive, so she wants to grow the corn in a shape that will maximize the ratio of the area of the cornfield to its perimeter.

What shape should the cornfield be?

This doesn't seem to be a well-defined problem. It's well-proven that for a given perimeter, area is maximized when the shape is a circle, but you didn't specify whether we're trying to to maximize for a given fence length or for a given area. Either way the shape answer is a circle, but then the rest of your statement is fairly irrelevant.

 

[Quix0te]

This doesn't seem to be a well-defined problem. It's well-proven that for a given perimeter, area is maximized when the shape is a circle, but you didn't specify whether we're trying to to maximize for a given fence length or for a given area. Either way the shape answer is a circle, but then the rest of your statement is fairly irrelevant.

Right, that one was easy. Here's the tough puzzle:

At an annual party for a group of math professors, a couple tells a guest, “We have been married 10 years. One year ago we gave Jones the sum and product of the ages of our three children, but Jones didn’t get their ages right. Smith missed the problem earlier tonight, having heard the current sum and product of our children’s ages.” Assume all children’s ages are whole numbers less than 10.

How old are the couple’s children?

 

[iu_a_att]

This doesn't seem to be a well-defined problem. It's well-proven that for a given perimeter, area is maximized when the shape is a circle, but you didn't specify whether we're trying to to maximize for a given fence length or for a given area. Either way the shape answer is a circle, but then the rest of your statement is fairly irrelevant.

Spoiler: not sure but what about this

Well, I thought the problem was to maximize area/perimeter. If a circle then the ratio is r/2...but the maximum inscribed circle in a 100x100 square has radius 50...so the maximum ratio would be 25. But, suppose you just put a fence around the 100x100 square. That gives you a ratio of area/perimeter = (100x100)/(4*100)=25...the same as the totally inscribed circle. Intuitively, it seemed to me that you might get more area for perimeter by expanding the size of the circle beyond radius 50 but then since you only own 100x100 lot you will have to essentially end up with a shape that is like a bar of soap. Would have to write the formula to see though. Was too lazy to do it.

 

[i'vegotwinners]

Right, that one was easy. Here's the tough puzzle:

At an annual party for a group of math professors, a couple tells a guest, “We have been married 10 years. One year ago we gave Jones the sum and product of the ages of our three children, but Jones didn’t get their ages right. Smith missed the problem earlier tonight, having heard the current sum and product of our children’s ages.” Assume all children’s ages are whole numbers less than 10.

How old are the couple’s children?

 

[Aloha Hoosier]

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Your best post ever. Congratulations! I mean that. Sincerely.

 

[TheOriginalHappyGoat]

Spoiler: not sure but what about this

Well, I thought the problem was to maximize area/perimeter. If a circle then the ratio is r/2...but the maximum inscribed circle in a 100x100 square has radius 50...so the maximum ratio would be 25. But, suppose you just put a fence around the 100x100 square. That gives you a ratio of area/perimeter = (100x100)/(4*100)=25...the same as the totally inscribed circle. Intuitively, it seemed to me that you might get more area for perimeter by expanding the size of the circle beyond radius 50 but then since you only own 100x100 lot you will have to essentially end up with a shape that is like a bar of soap. Would have to write the formula to see though. Was too lazy to do it.

This was what I found, as well. The ratio of the area of a circle to the circumference of a circle is pi * r^2 / pi * 2r, which simplifies to r^2 / 2r. The ratio of area to perimeter for the same subscribing pair is d^2 / 4d. For any value r, and any value d=2r, those two ratios are identical. In fact, the ratio is also identical for a regular hexagon arranged so that the inradius of the hexagon is identical to the radius of the circle (i.e., two opposites sides of the hexagon are pressed against two sides of the original square property). I suspect we will find the answer is the same for numerous regular polygons, perhaps any regular polygon with an even number of sides.

Edit: I can confirm it does not work with pentagons, at least. The largest pentagon that can fit in a square has one vertex on the square's diagonal, and the other four on each of the square's sides. The side of the pentagon is ~.62573786 times the side of the square. At this size, when the ratio of area/perimeter is 25 for the square, circle and hexagon, it is only ~21.531357 for the pentagon.

 

[TheOriginalHappyGoat]

Right, that one was easy. Here's the tough puzzle:

At an annual party for a group of math professors, a couple tells a guest, “We have been married 10 years. One year ago we gave Jones the sum and product of the ages of our three children, but Jones didn’t get their ages right. Smith missed the problem earlier tonight, having heard the current sum and product of our children’s ages.” Assume all children’s ages are whole numbers less than 10.

How old are the couple’s children?

Is this a typo? There are many, many answers to this question, unless Jones and Smith are the same person. Edit: Even if Jones and Smith are the same person (or otherwise communicated), this question still can't be answered. Almost every possible age combination fits the criteria.

 

[Quix0te]

Is this a typo? There are many, many answers to this question, unless Jones and Smith are the same person. Edit: Even if Jones and Smith are the same person (or otherwise communicated), this question still can't be answered. Almost every possible age combination fits the criteria.

The cornfield is a circle. Here's the answer to the other puzzle from WSJ:

"The children are 4, 4, and 9 so their sum is 17 and their product is 144. Smith incorrectly guessed (3, 6, 8) for the ages. One year ago, the ages were (3, 3, 8) and Jones incorrectly guessed (2, 6, 6)."

I believe they are saying that 4, 4 and 9 is the only possible combination of digits under 10 that give the same unstated sum and product as another set of three numbers under ten AND, if you subtract one from each digit, provide yet another combination that also has the same unstated sum and product as a different set of three numbers.

 

[TheOriginalHappyGoat]

The cornfield is a circle. Here's the answer to the other puzzle from WSJ:

"The children are 4, 4, and 9 so their sum is 17 and their product is 144. Smith incorrectly guessed (3, 6, 8) for the ages. One year ago, the ages were (3, 3, 8) and Jones incorrectly guessed (2, 6, 6)."

I believe they are saying that 4, 4 and 9 is the only two possible combination of digits under 10 that give the same unstated sum and product as another set of three numbers under ten AND, if you subtract one from each digit, provide yet another combination that also has the same unstated sum and product as a different set of three numbers.

See our discussion above about shapes. I believe WSJ is wrong about ages, as well, but will double check my work and provide examples later if possible.

 

[Quix0te]

FYI, here's the answer to last week's puzzle:

Room for One More
As illustrated above, it is easy to place 2n circles, each with a diameter of 1, in a 2 × n rectangle.

What is the smallest value of n for which you can fit 2n + 1 such circles into a 2 × n rectangle?

ANSWER: n = 164, there is just enough room for 329 circles.

The circles are packed as shown below with spacing between circles as indicated. There are 7 circles on each end, with 105 sets of 3 circles in the middle. The smallest rectangle found so far containing 329 circles has 13 circles on each side, with 101 sets of 3 circles in the middle, and is 2 by 163.9973967…

 

[TheOriginalHappyGoat]

The cornfield is a circle. Here's the answer to the other puzzle from WSJ:

"The children are 4, 4, and 9 so their sum is 17 and their product is 144. Smith incorrectly guessed (3, 6, 8) for the ages. One year ago, the ages were (3, 3, 8) and Jones incorrectly guessed (2, 6, 6)."

I believe they are saying that 4, 4 and 9 is the only possible combination of digits under 10 that give the same unstated sum and product as another set of three numbers under ten AND, if you subtract one from each digit, provide yet another combination that also has the same unstated sum and product as a different set of three numbers.

The problem with the cornfield question is that a circle is the most efficient perimeter for a given area, but the problem doesn't ask for a given area. It asks for a shape that can fit in a 100x100 square. A 100x100 square has 10,000 square units area, but a circle with 10,000 square unites area cannot fit inside that circle. Only a circle with pi * 2500, or about 7,854, square units can. While the 10,000 square unit circle would be more efficient than the square, the 7,854 square unit circle is exactly as efficient as the 10,000 square unit square in which it is inscribed.

I no longer have a problem with the second question. I ignored the fact that the couple did not ask which child was which age, meaning children of ages 1, 3, and 5 would be identical to children of 5, 3, and 1. Stupid mistake.

 

[Quix0te]

The problem with the cornfield question is that a circle is the most efficient perimeter for a given area, but the problem doesn't ask for a given area. It asks for a shape that can fit in a 100x100 square. A 100x100 square has 10,000 square units area, but a circle with 10,000 square unites area cannot fit inside that circle. Only a circle with pi * 2500, or about 7,854, square units can. While the 10,000 square unit circle would be more efficient than the square, the 7,854 square unit circle is exactly as efficient as the 10,000 square unit square in which it is inscribed.

I no longer have a problem with the second question. I ignored the fact that the couple did not ask which child was which age, meaning children of ages 1, 3, and 5 would be identical to children of 5, 3, and 1. Stupid mistake.

I nonconcur, the lowest area/perimeter ratios is a circle and the lowest volume/surface area is a sphere.

Whatever, I am posting an old WSJ puzzle from several weeks ago that is really quite simple but it drove me nuts for days. I thought I had the answer for sure but I goofed.

 

[TheOriginalHappyGoat]

I nonconcur, the lowest area/perimeter ratios is a circle and the lowest volume/surface area is a sphere.

Again, that's only true for a given area. Because your problem requires us to fit a cornfield inside of a 100x100 square, that adds the wrinkle that both I and att noticed. As I mentioned above, I discovered that a regular hexagon could fit in that square which also matches the same efficiency. I suspect, but cannot prove, that any regular polygon with an even number of sides can fit within that square with equal efficiency.

None of them are as efficient as a circle with the same area, but the circle doesn't have the same area, because a circle with the same area would not fit within the plot of land posited by the question.

Again, see att's post above, as well as my detailed one from yesterday. We go through the math that seems to prove we are correct and the WSJ flubbed this one.

 

[Quix0te]

Again, that's only true for a given area. Because your problem requires us to fit a cornfield inside of a 100x100 square, that adds the wrinkle that both I and att noticed. As I mentioned above, I discovered that a regular hexagon could fit in that square which also matches the same efficiency. I suspect, but cannot prove, that any regular polygon with an even number of sides can fit within that square with equal efficiency.

None of them are as efficient as a circle with the same area, but the circle doesn't have the same area, because a circle with the same area would not fit within the plot of land posited by the question.

Again, see att's post above, as well as my detailed one from yesterday. We go through the math that seems to prove we are correct and the WSJ flubbed this one.

Your next challenge is posted.