# A Very Stupid Math Question

I know I should be able to figure this out, but I can’t. Mock me mercilessly if you must, but if someone can help…

I play a game in which I attack other players. If my attack is successful, I get 15% of their remaining regions.

WHat I’m trying to figure out is what formula I could use to work out how many turns it would take to eliminate all of their regions (because I only get 7 turns a day, and I want to know whether they have a small enough number of regions that I could wipe them out in one day). Up until this point, I’ve just been breaking out the calculator, and counting how many times I hit “- 15%” until I reach a number low enough that I know that they’ll be eliminated with my next attack.

So say an opponent has 1,142 regions (number drawn completely outta nowhere). How can I figure out how many turns, assuming a successful attack every time, it would take to bring that 1142 down to, say, 1?

Initial number of regions: a
Final number of regions: b
Proportion gained per turn: r
Number of turns: n

The general formula is:

b = a * (1 - r) ^ n

To calculate n:

log b = log a + n * log (1 - r)

n = log (b / a) / log (1 - r)

In the example: a = 1142, b = 1, r = 0.15

n = log (1 / 1142) / log (0.85) = -7.0405 / -0.1625 = 43.32

You can use either natural or base 10 logs, the result will be the same.

How can you aquire all of something by taking a percentage of it each time? Seems impossible to me.

Well, once you reach a certain threshhold, it stops taking away a percentage and instead just takes away some set number (like 20, or something). Once it gets to that point, it’s only a couple of turns until the player is eliminated.

The problem is that every time you take away 15% of their regions, the 15% you will take away on their next turn is lower, so it’s an asymptotic curve converging on the limit of 1. I just ran a quick calculation (admittedly without rounding), and for you to get under 1.5 you would need 43 turns starting from 1142.

OK, I just did the same thing with rounding, and (surprise!!!) you will never get your opponent below 3 regions. This is because 15% of 3 is less than one, so you can’t take 15% away from them at that point. To get to three regions left over, it will take 37 turns when starting from 1142.

Here is a chart of how the number of regions plots against the number of turns (without rounding), and here is a chart of the number of regions left over after each turn (with rounding).

Thought so. My formula will only work if b is higher than this threshold, obviously.

Thanks everyone for the help! (And for not mocking me ) Both the formula and the chart are very helpful!

I just noticed the link to my first chart is messed up. It should go to here.

im sorry i cannot help with the op, but do you think i could check this game out? it sounds like a fun thing i could play with my mum, we used to play email chess but its getting rather tiresome

Sure. It’s a text-based game called “Barren Realms Elite” and I play it on at bbsmates.com. Think a very primitive version of Civilization-type games. You have to telnet into the sytem to play, but that’s pretty easy.

http://www.bbsmates.com/lord.asp

If you follow that link and click on the red graphic in the center of the screen, Telnet should launch and ask if you want to start a session. The rest should be self-explanatory (you’ll have to register a login name etc, but that takes maybe 10 seconds). There are two games, which can be reached by pressing either B (game one) or R (game two) from the main menu.

thanks tminc - it sounds fun, and something i could win at

Just as an FYI, you wouldn’t just be playing against your mother There are other players as well. But it’s a lot of fun.

Another game on the same system is Global War - a version of Risk. Highly recommended, especially if you like strategy type games. Again, there are other players, and each game requires at least 3 players (the exact number of players in a game are determined by the person who begins the game) but still a lot of fun.

If it helps, keep in mind that each time you take away 15% of their regions, that leaves them with 85% of what they previously had.

So, after the first turn, they have 85% of 1142: (.85)(1142)

After the second turn, they have 85% of that: (.85)(.85)(1142), or (.85)[sup]2/sup

After the third turn, they have 85% of that: (.85)(.85)(.85)(1142), or (.85)[sup]3/sup

After the nth turn, they have (.85)[sup]n/sup

To find how many turns it would take for this to reach a certain number A, you’d solve the equation

(.85)[sup]n/sup = A

for n.

(.85)[sup]n[/sup] = A/1142
n = log[sub].85[/sub] (A/1142)
= log (A/1142) / log(.85)