[Guest]

Hi All,

I've been an avid browser of this site for two years or so, and now I wish to start posting. I always like the idea that problems that may be encountered be real life problems. So this is the first in what I will hope to be a series of real life problems. That having been said, this also means that I most likely won't know the answer. But since there are many bright people on this site, I have faith that someone will be able to meet the challenges. The gauntlet has been thrown down!!

This is my first post, so any pointers you can give on better posting will also be appreciated. With that out of the way here is the problem:

IMob APP problem

I use a popular iPhone app called imob. In this app one of the goals is to earn money as a mobster. I wish to quickly earn 1 trillion dollars. For simplicity sake, and in case anyone else has the app in question, I will only list the property that is worthwhile to invest in, along with a couple other assumptions that will make the problem more straightforward.

You earn money by investing in property (which I will refer to as a "building"). Each property must be built on land (hereafter "land") that is purchased separately. The land price effectively never changes. However, the building price increases after each land purchase by 10% of the original price. Each building gives a return on your money that is static for each building you own. This return is compounded in the game every 1.0 hours.

Shopping Plaza - initial cost is $750,000 (each increase is $75,000) - land cost is $1,000,000 (and never changes) - return on money is $100,000 per building per hour

Night club - initial cost is 3,000,000 (each increase is $300,000) - land cost is $4,000,000 (and never changes) - return on money is $150,000 per building per hour

Casino- initial cost is 50,000,000 (each increase is $5,000,000) - land cost is $1,000,000 (and never changes) - return on money is $250,000 per building per hour

Assume that you start the game with $1,750,000, enough for one shopping plaza. You will note that your second shopping plaza will cost $1,825,000, and take something like 19 hours to achieve, your third one will cost $1,900,000 but only take something like 10 hours to achieve, etc.

Assuming that these questions can be answered at all, my questions are therefore:

1. What is the fastest time it will take to achieve $10,000,000 (10 million)?

2. How many of each building will you have in your inventory?

And for the harder challenge:

3 What is the fastest time it will take to achieve $1,000,000,000 (1 trillion)?

4. How many of each building will you have in your inventory?

Please assume that you will re-invest your income when you have enough, but only if it is worthwhile (in other words you need to stop investing at some point in order to achieve the 1 trillion). Also you will probably quickly come to the idea that the profit margins fall off precipitously and reach an asymptote.

[CaptainEd]

I don't mean to be picky, but I want to be sure of the numeric goals.

In U.S., 10^9 = $1,000,000,000 is one billion, not one trillion (and in U.K. I believe it is one thousand million, not one trillion)

In U.S., 10^12 = $1,000,000,000,000 is one trillion.

In U.K., I believe, 10^15 = $1,000,000,000,000,000 is one trillion

Is the goal for questions 3 and 4 based on 10^9, 10^12, or 10^15?

[Guest]

I don't mean to be picky, but I want to be sure of the numeric goals.

In U.S., 10^9 = $1,000,000,000 is one billion, not one trillion (and in U.K. I believe it is one thousand million, not one trillion)

In U.S., 10^12 = $1,000,000,000,000 is one trillion.

In U.K., I believe, 10^15 = $1,000,000,000,000,000 is one trillion

Is the goal for questions 3 and 4 based on 10^9, 10^12, or 10^15?

Oops, you are absolutely correct and good catch!!! I misplaced a few zeros there.

I mean 1 trillion (US annotation) that is:

$1,000,000,000,000 = 10^12

Which I had gotten wrong in the original post. I apparently cannot edit the original post (right?).

Edited June 11, 2009 by brothers

[Guest]

Hi All,

I've been an avid browser of this site for two years or so, and now I wish to start posting. I always like the idea that problems that may be encountered be real life problems. So this is the first in what I will hope to be a series of real life problems. That having been said, this also means that I most likely won't know the answer. But since there are many bright people on this site, I have faith that someone will be able to meet the challenges. The gauntlet has been thrown down!!

This is my first post, so any pointers you can give on better posting will also be appreciated. With that out of the way here is the problem:

IMob APP problem

I use a popular iPhone app called imob. In this app one of the goals is to earn money as a mobster. I wish to quickly earn 1 trillion dollars. For simplicity sake, and in case anyone else has the app in question, I will only list the property that is worthwhile to invest in, along with a couple other assumptions that will make the problem more straightforward.

You earn money by investing in property (which I will refer to as a "building"). Each property must be built on land (hereafter "land") that is purchased separately. The land price effectively never changes. However, the building price increases after each land purchase by 10% of the original price. Each building gives a return on your money that is static for each building you own. This return is compounded in the game every 1.0 hours.

Shopping Plaza - initial cost is $750,000 (each increase is $75,000) - land cost is $1,000,000 (and never changes) - return on money is $100,000 per building per hour

Night club - initial cost is 3,000,000 (each increase is $300,000) - land cost is $4,000,000 (and never changes) - return on money is $150,000 per building per hour

Casino- initial cost is 50,000,000 (each increase is $5,000,000) - land cost is $1,000,000 (and never changes) - return on money is $250,000 per building per hour

Assume that you start the game with $1,750,000, enough for one shopping plaza. You will note that your second shopping plaza will cost $1,825,000, and take something like 19 hours to achieve, your third one will cost $1,900,000 but only take something like 10 hours to achieve, etc.

Assuming that these questions can be answered at all, my questions are therefore:

1. What is the fastest time it will take to achieve $10,000,000 (10 million)?

2. How many of each building will you have in your inventory?

And for the harder challenge:

3 What is the fastest time it will take to achieve $1,000,000,000 (1 trillion)?

4. How many of each building will you have in your inventory?

Please assume that you will re-invest your income when you have enough, but only if it is worthwhile (in other words you need to stop investing at some point in order to achieve the 1 trillion). Also you will probably quickly come to the idea that the profit margins fall off precipitously and reach an asymptote.

The quickest way to reach 10 Million is to buy only 4 shopping plazas as you get the money. If you stop buying after 4, you will reach 10 mil at the 60th hour. This is assuming you bought your first property at zero hour. At this low of a goal, it makes no sense to purchase a higher valued property. I haven't done the second part yet.

[Guest]

For the 10 million mark, you need 4 shopping plazas and you will have 10 million at the end of 61st hour.

For 1 trillion mark, the least number of hours needed is 260. You buy 83 shopping plazas (SP) and 17 night clubs (NC). Don't buy any casinos as they dont give a very good return (not atleast if until the point of making 1 trillion... for 10 trillion, they would come in handy ).

You buy your last property on the 167th hour which is a shopping plaza.

You buy property considering when the return on investment for one is higher than the others and not just based on price.

For example, a NC becomes profitable only after having purchased 39 SP. The 40th SP is not as attractive as the 1st NC. However, the 40th, 41st and 42nd SP are more attractive than the 2nd NC.... and so on

I did this using excel simulation in the attached file. However, due to heavy formulas used the file size was more than 1 MB so I converted all formulas to values... You can see when to buy which property.

Sim.xls

Edited June 17, 2009 by DeeGee

[Guest]

For the 10 million mark, you need 4 shopping plazas and you will have 10 million at the end of 61st hour.

For 1 trillion mark, the least number of hours needed is 260. You buy 83 shopping plazas (SP) and 17 night clubs (NC). Don't buy any casinos as they dont give a very good return (not atleast if until the point of making 1 trillion... for 10 trillion, they would come in handy ).

You buy your last property on the 167th hour which is a shopping plaza.

You buy property considering when the return on investment for one is higher than the others and not just based on price.

For example, a NC becomes profitable only after having purchased 39 SP. The 40th SP is not as attractive as the 1st NC. However, the 40th, 41st and 42nd SP are more attractive than the 2nd NC.... and so on

I did this using excel simulation in the attached file. However, due to heavy formulas used the file size was more than 1 MB so I converted all formulas to values... You can see when to buy which property.

Very clever! Very good indeed! I wish I could see your formulas!! Did you use a checksum or if/then formula, perhaps to determine the best % yeild then integrate that into the next line of fields?

And how did you figure out when to stop buying? By simulation? Perhaps you could add another line of fields with a simple:

=(1,000,000,000 divided by the income (which can also be calculated using columns D-F)) + (sum of hours it took to get here (Bx of the line that you are on))

Then have the computer pick the lowest value. Excell 2007 has a conditional formatting tool that is useful for this, I'm not sure if it is in previous versions.

As stated in an above post, we also want to achieve the greater value of 10^12. I assume that it would (simply?) require additional modeling with excell, though you might run out of fields (or maybe not, since your income is ever increasing, it might not take that much more).

Finally, I am curious as to whether anyone thinks that this could be formulated this into a linear calculation or somehting along those lines? Perhpas we can make the assumption that the hourly increase in money doesn't increase only on the hour, but increases fractionally during the hour (I am not sure, but I think this will help the managibility of the calculation).

Edited June 23, 2009 by brothers

[Guest]

I've tried to replicate DeeGee's simulation in the attached file, but I've made some changes to accommodate my formulas. You'll have to copy the last row and paste it further down to see the full results. As you can then see my method offers a solution of 5,231 hours to accumulate $1,000,000,000,000.

Game.xls

Edited June 25, 2009 by hookemhorns

[Guest]

I've tried to replicate DeeGee's simulation in the attached file, but I've made some changes to accommodate my formulas. You'll have to copy the last row and paste it further down to see the full results. As you can then see my method offers a solution of 5,231 hours to accumulate $1,000,000,000,000.

So very elegantly and cleanly formulated. Bravo!

[Guest]

Wait, I've got a question:

Can you buy multiple buildings each hour? Or only one per hour?

The excel results above assume only one building purchased per hour. If this isn't the case, then it can likely be achieved sooner as there's no reason to sit on the cash when the formula says buy... I think it would push the "stop buying now" tipping point later, but the rate of return after that, due to the additional buildings, would be overpoweringly higher.

PiEater

[Guest]

Wait, I've got a question:

Can you buy multiple buildings each hour? Or only one per hour?

The excel results above assume only one building purchased per hour. If this isn't the case, then it can likely be achieved sooner as there's no reason to sit on the cash when the formula says buy... I think it would push the "stop buying now" tipping point later, but the rate of return after that, due to the additional buildings, would be overpoweringly higher.

PiEater

That's an excellent question, PiEater - one that I obviously hadn't considered. I've made some modifications to my formulas in the attached workbook (again, due to attachment size limitations you'll need to copy and paste the last row). The spreadsheet now has multiple rows for any hour during which a purchase occurs. However, I know that this solution is no longer optimal although I believe it to be very close.

For instance look at row 724 (hour 351). The formula is saying not to buy because the most optimal buy is of building 3. After the next hour we will buy an instance of both building 1 and building 3. But we have the cash right now to buy building 1 and doing so will not prohibit us from buying building 3 next hour, so we are missing out on $100,000.

Even though I'm able to identify the problem I'm not able to come up with a solution yet.

Game.xls