So… if my brain is reading that chart right, there seemed to be more progress spent on the systems at lower percentages?
I’m not sure if that’s the correct interpretation. Annoys me a bit, because I’d like to get it right.
Exactly right! Lower weekly activity incurs the 33% Thursday reinforcements, potentially multiple times, such that more
total activity is needed for one victory. The activity totals are simply 100% plus one-third per extra week required, although the weekly activity amounts where those increases occur are not so linear, and follow instead those equations earlier.
As an example, consider a 3-week victory; setting
n = 3 and using weekly activity (2 +
n)/3
n, we get 5/9 (~55%). One can verify this with the one-third reinforcement written as 3/9; the first week yields 5/9 reduced to 2/9, then 7/9 reduced to 4/9, then finally 9/9 for a victory.
The way to
arrive at such an expression is to give a meaningful name to the number we are trying to find, write a little model for how we know it works, then solve it! Let
x be the mystery weekly progress rate and let
n be the number of weeks involved, such that the total activity is
nx. We need to reach
100% progress of course, but each
additional week costs an extra
third, so we can write:
nx = 1 + (n − 1)/3
Solving for the weekly progress
x is then just a case of dividing it all by
n, although we can tidy it up a bit first; write the 1 as 3/3 and combine it with the fraction:
nx = (n − 1 + 3)/3 = (n + 2)/3
Now divide by
n:
x = (n + 2)/3n
Other targets such as 85% can be obtained by changing that original
1 to be
0.85. If it turns out that the reinforcements are actually 33.0% rather than a third, something I admit I never checked, use
0.33 (
n − 1). Hopefully those colours help to associate each part of that first expression with each concept at hand, and again at the end for making adjustments!
