Difference between revisions of "User:Texaslax/Koins"
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| − | Koins ([[File:Coin.gif]]) are the currency of Texaslax. To abbreviate 10 Koins, you say [[File:Coin.gif]]x10. This page was last updated April | + | __NOTOC__ |
| + | <div style="background-color: #ffe0e0; border: 2px solid #ff4d4d; padding: 10px; text-align: center; font-weight: bold; color: #a60000; margin-bottom: 1em;"> | ||
| + | ⚠️ '''Important Announcement:''' | ||
| + | The Koins system contains a significant issue! | ||
| + | Due to 10% monthly compound interest on user Koins, the total Koins held by users will eventually exceed the Bank’s Koins — despite the Bank's monthly increases. | ||
| + | See the '''[[User:Texaslax/Koins#Mathematical_Proof:_Exponential_Growth_of_Users.27_Koins_Exceeds_Linear_Growth_of_the_Koins_Bank|Mathematical Proof]]''' section below for details. | ||
| + | </div> | ||
| + | |||
| + | |||
| + | |||
| + | == Koins == | ||
| + | Koins ([[File:Coin.gif]]) are the currency of [[User:Texaslax|Texaslax]]. To abbreviate 10 Koins, you say [[File:Coin.gif]]x10. This page was last updated April 11, 2013. | ||
| + | |||
== Their Looks == | == Their Looks == | ||
| − | Koins' pictures are from the coins in [ | + | Koins' pictures are from the coins in [//en.wikipedia.org/wiki/Super_Mario_Bros. Super Mario Bros.] They're are in this wiki as a .gif added by Texaslax. [[:File:Coin.gif|File:Coin.gif]] is the file page for Koins. [[File:Coin.gif]] |
== Conversion of Koins == | == Conversion of Koins == | ||
| Line 8: | Line 20: | ||
== The Koins Bank == | == The Koins Bank == | ||
There is a bank of Koins. It holds all Koins that are not in circulation. The Bank started with 10,000 Koins, and increases 500 Koins every month on the 1'st of the month. Note: Koins' lowest denomination is 1 Koin. Koins are completely integral.[[File:Coin.gif]] | There is a bank of Koins. It holds all Koins that are not in circulation. The Bank started with 10,000 Koins, and increases 500 Koins every month on the 1'st of the month. Note: Koins' lowest denomination is 1 Koin. Koins are completely integral.[[File:Coin.gif]] | ||
| + | |||
== A Koins Bank Account == | == A Koins Bank Account == | ||
Every Koins user has a 4-digit bank account number. This must be remembered. 'Bank account number' is abbreviated to BAN. All Koins transactions require a BAN. A BAN is provided for new Koins users upon their first transaction.[[File:Coin.gif]] | Every Koins user has a 4-digit bank account number. This must be remembered. 'Bank account number' is abbreviated to BAN. All Koins transactions require a BAN. A BAN is provided for new Koins users upon their first transaction.[[File:Coin.gif]] | ||
| + | |||
== How to Get Koins == | == How to Get Koins == | ||
To get Koins, you should: | To get Koins, you should: | ||
| − | * Comment often in my blog. | + | * Comment often in my [//{{SERVERNAME}}/community/c2405 blog]. |
* Play in my games. (And do well.) | * Play in my games. (And do well.) | ||
| Line 19: | Line 33: | ||
* Give me your currency... | * Give me your currency... | ||
| − | *Every month on the 1'st of the month you gain 10% interest on all of your Koins. | + | * Every month on the 1'st of the month you gain 10% interest on all of your Koins. |
[[File:Coin.gif]] | [[File:Coin.gif]] | ||
| Line 27: | Line 41: | ||
== How Many Koins Do I Have? == | == How Many Koins Do I Have? == | ||
| − | '''Bank Owns: 11, | + | '''Bank Owns: 11,296 out of 14,000 Koins''' |
| + | |||
| + | [[User:Texaslax/Koins by BAN|Koins by BAN]] | ||
| + | |||
| + | [[User:Texaslax/Koins by Amount|Koins by Amount]] | ||
| + | |||
| + | [[User:Texaslax/Koins by the Alphabet|Koins by the Alphabet]] | ||
| + | |||
| + | |||
| + | |||
| + | <div style="background-color: #ffe0e0; border: 2px solid #ff4d4d; padding: 10px; text-align: center; font-weight: bold; color: #a60000; margin-bottom: 1em;"> | ||
| + | ⚠️ '''Important Announcement:''' | ||
| + | The Koins system contains a significant issue! | ||
| + | Due to 10% monthly compound interest on user Koins, the total Koins held by users will eventually exceed the Bank’s Koins — despite the Bank's monthly increases. | ||
| + | See the '''[[User:Texaslax/Koins#Mathematical_Proof:_Exponential_Growth_of_Users.27_Koins_Exceeds_Linear_Growth_of_the_Koins_Bank|Mathematical Proof]]''' section below for details. | ||
| + | </div> | ||
| + | |||
| + | |||
| + | |||
| + | == Mathematical Proof: Why Users' Koins Will Eventually Outgrow the Koins Bank == | ||
| + | |||
| + | === Abstract === | ||
| + | |||
| + | This section shows that because users earn compound interest on their Koins, their total will eventually be more than the Koins held by the Bank — even though the Bank adds a fixed amount each month. The users' Koins grow exponentially thanks to a 10% monthly interest, while the Bank's Koins only grow by a set 500 Koins each month. | ||
| + | |||
| + | === 1. Introduction === | ||
| + | |||
| + | Koins are the currency used in Texaslax. The Bank and the users’ Koins grow differently: | ||
| + | |||
| + | * The '''Koins Bank''' starts with 10,000 Koins and adds 500 Koins every month. | ||
| + | * '''Users''' also start with 10,000 Koins but get 10% interest every month, which compounds. | ||
| + | |||
| + | Here, we’ll show that after some time, the users will have more Koins than the Bank, even though the Bank keeps adding Koins regularly. | ||
| + | |||
| + | === 2. Setting up the Math === | ||
| + | |||
| + | Let's define: | ||
| + | |||
| + | * '''B(t)''': Koins held by the Bank after '''t''' months. | ||
| + | * '''U(t)''': Koins held by users after '''t''' months. | ||
| + | * '''T(t)''': total Koins in circulation at month '''t''' (Bank + users). | ||
| + | * '''P''': initial Koins for both Bank and users = 10,000. | ||
| + | |||
| + | The growth formulas: | ||
| + | |||
| + | * Bank grows linearly: | ||
| + | <cmath> | ||
| + | B(t) = 10,000 + 500t | ||
| + | </cmath> | ||
| + | |||
| + | * Users grow exponentially because of interest: | ||
| + | <cmath> | ||
| + | U(t) = 10,000 \times (1.1)^t | ||
| + | </cmath> | ||
| + | |||
| + | Total Koins is just the sum: | ||
| + | <cmath> | ||
| + | T(t) = B(t) + U(t) = (10,000 + 500t) + 10,000 \times (1.1)^t | ||
| + | </cmath> | ||
| + | |||
| + | === 3. When Do Users Overtake the Bank? === | ||
| + | |||
| + | We want to find when: | ||
| + | <cmath> | ||
| + | U(t) > B(t) | ||
| + | </cmath> | ||
| + | |||
| + | Plug in the formulas: | ||
| + | <cmath> | ||
| + | 10,000 \times (1.1)^t > 10,000 + 500t | ||
| + | </cmath> | ||
| + | |||
| + | Divide both sides by 10,000: | ||
| + | <cmath> | ||
| + | (1.1)^t > 1 + 0.05t | ||
| + | </cmath> | ||
| + | |||
| + | We want to know for which '''t''' this becomes true. Once it does, users will always have more Koins than the Bank, because the exponential growth keeps accelerating. | ||
| + | |||
| + | === 4. Exponential vs. Linear Growth === | ||
| + | |||
| + | The left side, <math>(1.1)^t</math>, grows faster and faster every month (it’s exponential). | ||
| + | |||
| + | The right side, <math>1 + 0.05t</math>, just grows by the same amount every month (linear). | ||
| + | |||
| + | The key idea: exponential growth eventually beats linear growth — even if it starts slower. | ||
| + | |||
| + | So, after some point: | ||
| + | <cmath> | ||
| + | (1.1)^t > 1 + 0.05t | ||
| + | </cmath> | ||
| + | and from then on, | ||
| + | <cmath> | ||
| + | U(t) > B(t) | ||
| + | </cmath> | ||
| + | |||
| + | === 5. Rough Estimate === | ||
| + | |||
| + | If we check numerically, the users' Koins pass the Bank's Koins after around 70 months (a bit less than 6 years). After that, the users pull ahead faster and faster while the Bank keeps increasing at the same slow pace. | ||
| + | |||
| + | === 6. Conclusion === | ||
| + | |||
| + | This shows that because of compound interest, users' Koins will eventually be more than the Bank’s Koins, despite the Bank’s steady monthly additions. The exponential growth of users' Koins wins out over the Bank's linear increase. | ||
| − | + | This highlights how powerful interest can be over time, even starting from the same amount. | |
| − | + | === References === | |
| − | [[ | + | * [//en.wikipedia.org/wiki/Compound_interest Compound Interest Formula] |
| + | * [//en.wikipedia.org/wiki/Exponential_growth Exponential Growth] | ||
Latest revision as of 20:14, 25 June 2025
⚠️ Important Announcement: The Koins system contains a significant issue! Due to 10% monthly compound interest on user Koins, the total Koins held by users will eventually exceed the Bank’s Koins — despite the Bank's monthly increases. See the Mathematical Proof section below for details.
Koins
Koins (
) are the currency of Texaslax. To abbreviate 10 Koins, you say x10. This page was last updated April 11, 2013.
Their Looks
Koins' pictures are from the coins in Super Mario Bros. They're are in this wiki as a .gif added by Texaslax. File:Coin.gif is the file page for Koins.
Conversion of Koins
Koins are currently not able to be exchanged or converted into any other AoPS currencies.
The Koins Bank
There is a bank of Koins. It holds all Koins that are not in circulation. The Bank started with 10,000 Koins, and increases 500 Koins every month on the 1'st of the month. Note: Koins' lowest denomination is 1 Koin. Koins are completely integral.
A Koins Bank Account
Every Koins user has a 4-digit bank account number. This must be remembered. 'Bank account number' is abbreviated to BAN. All Koins transactions require a BAN. A BAN is provided for new Koins users upon their first transaction.
How to Get Koins
To get Koins, you should:
- Comment often in my blog.
- Play in my games. (And do well.)
- Give me your currency...
- Every month on the 1'st of the month you gain 10% interest on all of your Koins.
What to do With Koins
You buy things! (Obvious.) Or bribe me. (Not Obvious.) Go to the Koins Store.
How Many Koins Do I Have?
Bank Owns: 11,296 out of 14,000 Koins
⚠️ Important Announcement: The Koins system contains a significant issue! Due to 10% monthly compound interest on user Koins, the total Koins held by users will eventually exceed the Bank’s Koins — despite the Bank's monthly increases. See the Mathematical Proof section below for details.
Mathematical Proof: Why Users' Koins Will Eventually Outgrow the Koins Bank
Abstract
This section shows that because users earn compound interest on their Koins, their total will eventually be more than the Koins held by the Bank — even though the Bank adds a fixed amount each month. The users' Koins grow exponentially thanks to a 10% monthly interest, while the Bank's Koins only grow by a set 500 Koins each month.
1. Introduction
Koins are the currency used in Texaslax. The Bank and the users’ Koins grow differently:
- The Koins Bank starts with 10,000 Koins and adds 500 Koins every month.
- Users also start with 10,000 Koins but get 10% interest every month, which compounds.
Here, we’ll show that after some time, the users will have more Koins than the Bank, even though the Bank keeps adding Koins regularly.
2. Setting up the Math
Let's define:
- B(t): Koins held by the Bank after t months.
- U(t): Koins held by users after t months.
- T(t): total Koins in circulation at month t (Bank + users).
- P: initial Koins for both Bank and users = 10,000.
The growth formulas:
- Bank grows linearly:
![\[B(t) = 10,000 + 500t\]](http://latex.artofproblemsolving.com/c/4/5/c459906c79113fee8d3a1825b9d530133be6e147.png)
- Users grow exponentially because of interest:
![\[U(t) = 10,000 \times (1.1)^t\]](http://latex.artofproblemsolving.com/1/3/d/13d4d85ee6493e07bb52a8bf7cb9fb272b2930a4.png)
Total Koins is just the sum:
![\[T(t) = B(t) + U(t) = (10,000 + 500t) + 10,000 \times (1.1)^t\]](http://latex.artofproblemsolving.com/6/5/8/658787a7cb589c96feba412175df1f39f41b7caf.png)
3. When Do Users Overtake the Bank?
We want to find when:
![\[U(t) > B(t)\]](http://latex.artofproblemsolving.com/d/9/c/d9c8efa6cab7deeb3c63609bd2274bd97c9d20bf.png)
Plug in the formulas:
![\[10,000 \times (1.1)^t > 10,000 + 500t\]](http://latex.artofproblemsolving.com/9/7/2/9720ab2c546de732af24143812b62d8c3ba53153.png)
Divide both sides by 10,000:
![\[(1.1)^t > 1 + 0.05t\]](http://latex.artofproblemsolving.com/f/e/0/fe0b877370d03373cfd5187033b1cce308314cf3.png)
We want to know for which t this becomes true. Once it does, users will always have more Koins than the Bank, because the exponential growth keeps accelerating.
4. Exponential vs. Linear Growth
The left side, 
, grows faster and faster every month (it’s exponential).
The right side, 
, just grows by the same amount every month (linear).
The key idea: exponential growth eventually beats linear growth — even if it starts slower.
So, after some point:
and from then on,
5. Rough Estimate
If we check numerically, the users' Koins pass the Bank's Koins after around 70 months (a bit less than 6 years). After that, the users pull ahead faster and faster while the Bank keeps increasing at the same slow pace.
6. Conclusion
This shows that because of compound interest, users' Koins will eventually be more than the Bank’s Koins, despite the Bank’s steady monthly additions. The exponential growth of users' Koins wins out over the Bank's linear increase.
This highlights how powerful interest can be over time, even starting from the same amount.
