I was thinking about a system of character customization that would allow for a lot of options but also make each option comparable, and allow for character development and a level of simplicity involving small numbers. The reasons behind wanting such a system are complex and not the issue of this thread.
What I came up with was using 6 fairly typical different attributes:
Strength (STR)
Dexterity (DEX)
Constitution (CON)
Willpower (WIL)
Intelligence (INT)
Charisma (CHA)
on a scale of 1 to 10.
Then I wanted to have them all start at 5, so:
STR5 DEX5 CON5 WIL5 INT5 CHA5
and then allow people to train up a skill someway to increase it by 1 while allowing another skill to decrease by 1, so for instance you could train your Willpower up and set your Intelligence as least important and after your training end up with:
STR5 DEX5 CON5 WIL6 INT4 CHA5
(this system is similar to Ultima Onlines stat system but with a hard lower limit that matches the higher limit)
Now, in this system 1 more point of WIL would be made to have comparable value to 1 more point of INT, and same for any attribute comparisons, which ideally would allow for many interesting combinations. How each of these attributes could be made to have interesting ingame effects and consequences is something I've put a fair bit of thought into and is very interesting in and of itself, and presents the problem of how these attributes could be made balanced with one another. But the balancing of the attributes is not the math problem which caused me to post after a night of pondering.
The problem I'm seeking a solution to is how many different options are there? Of secondary importance is if there is a nice tidy formula for the answer and if so what is it?
It's simple to show the max it can be, there are certainly no more than 1 million, since each of the six attributes has at most 10 choices, and 10 to the power of 6 is 1 million. But with the restriction that they all start at 5 and increasing one also decreases another it removes a lot of the million choices.
The simplest way to present the restriction is that the attributes must add to 30 and each attribute is between 1 and 10.
And there's definitely atleast 2100 choices, since you can ignore 3 attributes to allow them to compensate for whatever you do with the other 3 you pay attention to, and allow the 3 attributes you pay attention to to range from 1 to 10, 1 to 10, and 1 to 7, 700 choices, in 3 different ways (3 possibilities of which one you make the max of 7), making 2100. There are certainly more than 2100 though, since each of these 2100 choices gives varying wiggle room for the ignored 3 attributes.
The brute force method using a program would be to just output the 1 million options and remove any where the 6 attributes don't add to exactly 30. Not very neat but it'd get the job done.
Another brute force method would be to start each attributes at 1 and then distribute 24 points amoung them (similar to point-buy in DnD), of which there are 6 to the power of 24 possible ways. Needless to say this number is much larger than 1 million since it ignores the max of 10 for any given attribute, I mention it only since the concept could be useful in a tidy formula. But after you had this giant number you could remove any choices where any attribute exceeds 10.
Anyways, I'm interested to read anyones thoughts on a the answer (or a smaller lower and upper limit than 2100 and 1,000,000 respectively) or a tidy formula that gives the answer (or barring a formula, any work done towards a formula).
What I came up with was using 6 fairly typical different attributes:
Strength (STR)
Dexterity (DEX)
Constitution (CON)
Willpower (WIL)
Intelligence (INT)
Charisma (CHA)
on a scale of 1 to 10.
Then I wanted to have them all start at 5, so:
STR5 DEX5 CON5 WIL5 INT5 CHA5
and then allow people to train up a skill someway to increase it by 1 while allowing another skill to decrease by 1, so for instance you could train your Willpower up and set your Intelligence as least important and after your training end up with:
STR5 DEX5 CON5 WIL6 INT4 CHA5
(this system is similar to Ultima Onlines stat system but with a hard lower limit that matches the higher limit)
Now, in this system 1 more point of WIL would be made to have comparable value to 1 more point of INT, and same for any attribute comparisons, which ideally would allow for many interesting combinations. How each of these attributes could be made to have interesting ingame effects and consequences is something I've put a fair bit of thought into and is very interesting in and of itself, and presents the problem of how these attributes could be made balanced with one another. But the balancing of the attributes is not the math problem which caused me to post after a night of pondering.
The problem I'm seeking a solution to is how many different options are there? Of secondary importance is if there is a nice tidy formula for the answer and if so what is it?
It's simple to show the max it can be, there are certainly no more than 1 million, since each of the six attributes has at most 10 choices, and 10 to the power of 6 is 1 million. But with the restriction that they all start at 5 and increasing one also decreases another it removes a lot of the million choices.
The simplest way to present the restriction is that the attributes must add to 30 and each attribute is between 1 and 10.
And there's definitely atleast 2100 choices, since you can ignore 3 attributes to allow them to compensate for whatever you do with the other 3 you pay attention to, and allow the 3 attributes you pay attention to to range from 1 to 10, 1 to 10, and 1 to 7, 700 choices, in 3 different ways (3 possibilities of which one you make the max of 7), making 2100. There are certainly more than 2100 though, since each of these 2100 choices gives varying wiggle room for the ignored 3 attributes.
The brute force method using a program would be to just output the 1 million options and remove any where the 6 attributes don't add to exactly 30. Not very neat but it'd get the job done.
Another brute force method would be to start each attributes at 1 and then distribute 24 points amoung them (similar to point-buy in DnD), of which there are 6 to the power of 24 possible ways. Needless to say this number is much larger than 1 million since it ignores the max of 10 for any given attribute, I mention it only since the concept could be useful in a tidy formula. But after you had this giant number you could remove any choices where any attribute exceeds 10.
Anyways, I'm interested to read anyones thoughts on a the answer (or a smaller lower and upper limit than 2100 and 1,000,000 respectively) or a tidy formula that gives the answer (or barring a formula, any work done towards a formula).
