Here's the Kicker

Solution

Answer: 14

Let \(r\) be the number of seconds you rest between kicks. The accuracy rule is: - if \(r \ge 20\), your scoring probability is \(100\%\) - if you rest fewer than \(20\) seconds, the accuracy drops by \[ 1 + 2 + 3 + \cdots + (20-r) \] percent. That sum is a triangular number, so the accuracy for \(r \le 20\) is \[ p(r)=1-\frac{(20-r)(21-r)}{200}. \] So, for example: \[ r=17 \quad \Rightarrow \quad p(17)=1-\frac{3\cdot 4}{200}=1-\frac{12}{200}=0.94, \] which matches the \(6\%\) loss. Since the penalty becomes bigger than \(100\%\) when \(r\) gets too small, the realistic model is \[ p(r)=\max\!\left(0,\ 1-\frac{(20-r)(21-r)}{200}\right), \] with \(p(r)=1\) for all \(r\ge 20\). Now model how many kicks you get in \(100\) seconds. If you start with a kick at time \(0\), then keep waiting \(r\) seconds between kicks, the number of kicks is \[ N(r)=\left\lfloor \frac{100}{r} \right\rfloor + 1. \] So the expected number of goals is \[ E(r)=N(r)\,p(r) = \left(\left\lfloor \frac{100}{r} \right\rfloor + 1\right) \max\!\left(0,\ 1-\frac{(20-r)(21-r)}{200}\right). \] That is the model. Now the intuition is just a tradeoff: - larger \(r\) means fewer kicks, but each kick is more accurate - smaller \(r\) means more kicks, but your accuracy starts collapsing pretty fast because the penalty grows like \[ 1+2+\cdots+(20-r), \] not just linearly. Checking the sensible integer values: \[ r=20:\quad 6 \text{ kicks},\quad p=1.00,\quad E=6.00 \] \[ r=16:\quad 7 \text{ kicks},\quad p=0.90,\quad E=6.30 \] \[ r=15:\quad 7 \text{ kicks},\quad p=0.85,\quad E=5.95 \] \[ r=14:\quad 8 \text{ kicks},\quad p=0.79,\quad E=6.32 \] \[ r=13:\quad 8 \text{ kicks},\quad p=0.72,\quad E=5.76 \] So the best choice is \[ r=14 \text{ seconds}. \] At \(14\) seconds of rest, your expected number of goals is \[ E(14)=8 \cdot 0.79 = 6.32. \] So the optimal strategy is to rest \[ 14 \text{ seconds between kicks}, \] giving an expected total of about \[ 6.32 \text{ goals}. \]