Perfect Run hypothetical
Imagine you are given a number of attempts to beat The Perfect Run (SMG2 final level). This number is of your own choosing.
Successfully beating the level within the chosen number of attempts earns you a cash prize. Failure to do so results in death (or lifelong imprisonment).
You choose your number of attempts before your first attempt. After this, the number cannot be changed. The cash prize starts at $1 billion USD for choosing 1 attempt and succeeding, and is halved by each additional attempt you chose to give yourself.
You must start playing the level immediately, with no warmup.
How many attempts would you choose to give yourself?
For reference, this chart says what your cash prize would be if you choose to have n attempts (n from 1 to 30).
Attempts chosen | Cash Prize
1 | $1 b
2 | $500 m
3 | $250 m
4 | $125 m
5 | $62.5 m
6 | $31.3 m
7 | $15.6 m
8 | $7.81 m
9 | $3.91 m
10 | $1.95 m
11 | $977 k
12 | $488 k
13 | $244 k
14 | $122 k
15 | $61 k
16 | $30.5 k
17 | $15.3 k
18 | $7,629
19 | $3,815
20 | $1,907
21 | $954
22 | $477
23 | $238
24 | $119
25 | $59.60
26 | $29.80
27 | $14.90
28 | $7.45
29 | $3.73
30 | $1.86