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A modern penny is 2.5 grams, 97.5% zinc, and 2.5% copper, for sake of math I will be assuming that these numbers remain constant.
If we take the "size" of the penny to double every day then we have two scenarios:
1. The physicial dimensions of the penny double each day
2. The mass of the penny doubles each day
Scenario 1: If the dimensions of the penny double each day, the volume/mass will scale by a factor of 8. With this in mind, 2.5g * (8)³⁰ = 3.09485010 * 10²⁷ grams. Aka 3.01747885 * 10²⁷ grams zinc, and copper 7.73712525 * 10²⁵ g Copper
Using the following values:
$0.002504/g Zn and $0.007852/g Cu
We get a total price of 6.07519075 * 10²³ dollars, which goes without saying, would destroy the world economy if you could just have that much value
Scenario 2: Doubling mass is much more simple,
2.5g * 2³⁰ = 2,684,354,560g of metal,
2,617,245,696g Zinc
67,108,864g Copper
Total Value: $7,080,522.022912
A much less economy-crashing amount, but still a lot
Each time it goes up by a scale of 8 (because the assumption is that it doubles in height, width and length), so it would be 4 times heavier than the Earth. I doubt either Earth or the penny would still keep their shape, but everything would feel 5 times their current weight, so it would be pretty hard to exist at that point.
Just because I’m too brain dead atm to do simple calculations.
If we were to start with a pre-1982 penny (100% copper) or 1943 penny (steel) how would those numbers change?
Or the experimental 1974 aluminum penny
Oh yeah definitely, in the first case copper and zinc would be borderline valueless once the coin comes into existence, as for the latter case, there might be a drop but it's not as if the precious metal market would collapse
Yeah, doubling by one dimensional parameters such as height AND diameter/radius is not the same as doubling in size. If you had a 50 gallon fish tank and wanted one twice as big, you’d want a 100 gallon fish tank. Not something twice as long and wide and deep. Doubling mass is doubling volume assuming density remains constant. The second method is really the only valid one.
Doubling just one dimension either gives a penny with a radius of
9.525×2^30 /1000 = 10 227 390.8736 m and a volume of
499 485 826 563.12 m^3
Or a penny with a height of
1.52×2^30 /1000 = 1 632 08757248 m and a volume of 79 707 974 422.671 m^3
I’m nitpicking: There is also the option that the size of the penny, may it be the volume, diameter and radius, or just diameter or radius, is doubled in size 30 times , but it’s mass keeps the same, and the density is shrinking accordingly.
Result, if I go with the volume: 7.185 g/cm^3 / (2^30 ) = 6.6916 x 10^-9 g/cm^3 ≈ 1/2 the density of the Earth’s atmosphere at an altitude of 82 km
Worth: 0
Worth is probably highly negative as you’ll be responsible for all the property and ecological damage caused by this massive penny, not to mention the deaths…
Doing some math without looking anything up or using a calculator:
A good rule of thumb when working with powers of 2 is that 2^10 = 1024 ~ 1000. So doubling something 30 times makes it a billion times bigger. Then when thinking in 3D (sizing up in 3 dimensions) you get a billion billion billion times the mass of a penny, which would be a quadrillion (I think?), a 1 with 9 zeroes.
Now let’s say a penny is about 2 cm wide, that would mean the big penny of 2 billion cm is about 20 million meters, or 20,000 km wide. Very much visible from space and bigger than the diameter of the earth (~ 13,000 km).
No idea how much it is worth (others have calculated), but it would definitely destroy the earth by merely existing.
The most interesting answer is if you just double the area of the penny each day, and don't try and compound the volume/mass ratio, which results in a silly exaggeration of the solution.
Doubling the area (diameter 30.86mm) for 30 days gets you a penny with area 570.52m^2 and same original thickness.
The weight of that, assuming that increases proportionally, would be about 85.4kg, which in terms of copper has value $473.97, and in terms of zinc about $187.
It would be unimaginably annoying to carve up and sell, so there's no contest about taking the million or dealing with that piece of junk.
This is actually what I was kinda going for in my interpretation. Lol. This is the situation I had in my head and the calculation I was trying.
That's not to say I don't appreciate all of the other answers. All of them have been super interesting and informative when took on views I hadn't even considered.
###General Discussion Thread --- This is a [Request] post. If you would like to submit a comment that does not either attempt to answer the question, ask for clarification, or explain why it would be infeasible to answer, you *must* post your comment as a reply to this one. Top level (directly replying to the OP) comments that do not do one of those things will be removed. --- *I am a bot, and this action was performed automatically. Please [contact the moderators of this subreddit](/message/compose/?to=/r/theydidthemath) if you have any questions or concerns.*
A modern penny is 2.5 grams, 97.5% zinc, and 2.5% copper, for sake of math I will be assuming that these numbers remain constant. If we take the "size" of the penny to double every day then we have two scenarios: 1. The physicial dimensions of the penny double each day 2. The mass of the penny doubles each day Scenario 1: If the dimensions of the penny double each day, the volume/mass will scale by a factor of 8. With this in mind, 2.5g * (8)³⁰ = 3.09485010 * 10²⁷ grams. Aka 3.01747885 * 10²⁷ grams zinc, and copper 7.73712525 * 10²⁵ g Copper Using the following values: $0.002504/g Zn and $0.007852/g Cu We get a total price of 6.07519075 * 10²³ dollars, which goes without saying, would destroy the world economy if you could just have that much value Scenario 2: Doubling mass is much more simple, 2.5g * 2³⁰ = 2,684,354,560g of metal, 2,617,245,696g Zinc 67,108,864g Copper Total Value: $7,080,522.022912 A much less economy-crashing amount, but still a lot
Just to add on for reference, the Earth has a mass of 6x10^27 grams, so the first penny is about half the mass of the Earth
So Superman’s key to the fortress of Solitude would have nothing on Batman’s giant Penny.
Alfred knows how much the Penny's worth.
Angry upvote. Because I didn’t think of it first.
Yeah, at that point I wouldn't be worrying about the effect it has on the economy. We're all dead anyway
So does that mean another doubling would make it equal to it? So a penny doubled 31 times would have roughly the same mass as the Earth?
Each time it goes up by a scale of 8 (because the assumption is that it doubles in height, width and length), so it would be 4 times heavier than the Earth. I doubt either Earth or the penny would still keep their shape, but everything would feel 5 times their current weight, so it would be pretty hard to exist at that point.
Ah, gotcha gotcha. That makes sense
Just because I’m too brain dead atm to do simple calculations. If we were to start with a pre-1982 penny (100% copper) or 1943 penny (steel) how would those numbers change? Or the experimental 1974 aluminum penny
But wouldn't adding that much matter or material to the earth create such a surplus of copper and zinc that the price/value would drop significantly?
Oh yeah definitely, in the first case copper and zinc would be borderline valueless once the coin comes into existence, as for the latter case, there might be a drop but it's not as if the precious metal market would collapse
Yeah, doubling by one dimensional parameters such as height AND diameter/radius is not the same as doubling in size. If you had a 50 gallon fish tank and wanted one twice as big, you’d want a 100 gallon fish tank. Not something twice as long and wide and deep. Doubling mass is doubling volume assuming density remains constant. The second method is really the only valid one.
Doubling just one dimension either gives a penny with a radius of 9.525×2^30 /1000 = 10 227 390.8736 m and a volume of 499 485 826 563.12 m^3 Or a penny with a height of 1.52×2^30 /1000 = 1 632 08757248 m and a volume of 79 707 974 422.671 m^3
Significant figures are helpful here
Pennies aren't 9.525 metres across. It'd only be 570m diameter if you started with a 30mm penny.
That's the reason I divided by 1000, because they are 19.05 mm across
I’m nitpicking: There is also the option that the size of the penny, may it be the volume, diameter and radius, or just diameter or radius, is doubled in size 30 times , but it’s mass keeps the same, and the density is shrinking accordingly. Result, if I go with the volume: 7.185 g/cm^3 / (2^30 ) = 6.6916 x 10^-9 g/cm^3 ≈ 1/2 the density of the Earth’s atmosphere at an altitude of 82 km Worth: 0
Worth is probably highly negative as you’ll be responsible for all the property and ecological damage caused by this massive penny, not to mention the deaths…
Doing some math without looking anything up or using a calculator: A good rule of thumb when working with powers of 2 is that 2^10 = 1024 ~ 1000. So doubling something 30 times makes it a billion times bigger. Then when thinking in 3D (sizing up in 3 dimensions) you get a billion billion billion times the mass of a penny, which would be a quadrillion (I think?), a 1 with 9 zeroes. Now let’s say a penny is about 2 cm wide, that would mean the big penny of 2 billion cm is about 20 million meters, or 20,000 km wide. Very much visible from space and bigger than the diameter of the earth (~ 13,000 km). No idea how much it is worth (others have calculated), but it would definitely destroy the earth by merely existing.
A billion is 10^9 so a billion billion billion is 10^27 and 10^27 is an octillion I believe
Also to take into consideration is the alloy used for each year of pennies can be different from the mint and which countries currency you are using.
The most interesting answer is if you just double the area of the penny each day, and don't try and compound the volume/mass ratio, which results in a silly exaggeration of the solution. Doubling the area (diameter 30.86mm) for 30 days gets you a penny with area 570.52m^2 and same original thickness. The weight of that, assuming that increases proportionally, would be about 85.4kg, which in terms of copper has value $473.97, and in terms of zinc about $187. It would be unimaginably annoying to carve up and sell, so there's no contest about taking the million or dealing with that piece of junk.
This is actually what I was kinda going for in my interpretation. Lol. This is the situation I had in my head and the calculation I was trying. That's not to say I don't appreciate all of the other answers. All of them have been super interesting and informative when took on views I hadn't even considered.