When you have eliminated the impossible, whatever remains, however improbable, must be the truth.
I recently discovered a neat problem while looking at job applications. From the Sparx Deck, the problem is as follows:
A basketball player decides to spend an afternoon practicing free throws and recording her performance. Over the first hour, her scoring percentage is less than 75%, but by the end of the session, her overall scoring percentage for the day is more than 75%. Is it necessarily true that at some moment during the session her scoring percentage was exactly 75%?
The answer to the above problem is yes. This may seem counter-intuitive but the reasoning will soon become clear as I was interested in solving the extension:
Extension: How special is the choice of 75% in this question? Does the same result hold for other percentages?
So let us restate the general problem:
A basketball player decides to spend an afternoon practicing free throws and recording her performance. Over the first hour, her scoring percentage is less than p%, but by the end of the session, her overall scoring percentage for the day is more than p%. Is it necessarily true that at some moment during the session her scoring percentage was exactly p%?
To do this we will prove that the alternative, that the percentage can skip over p%, must be false for some values of p and we will then investigate what decides these values.
First let us express the percentage as a fraction \(p=\frac{a}{b} \times 100\) subject to the constraint \(a<b\) as it is impossible for the scoring percentage to exceed 100%.
If we assume that it is possible to skip over p then there must be some point where the current percentage is less than p but by scoring just one more basket it will exceed p.
If we let B be the number of baskets scored and N be the total number of shots attempted then we can express these requirements as:
\[ \frac{B}{N}<\frac{a}{b} \quad \mathsf{ and } \quad \frac{B+1}{N+1}>\frac{a}{b} \]
These can be re-arranged to give: \[ bB < aN \] \[ b(B+1) > a(N+1) \]
The latter of which can be rearranged to: \[bB >aN + a - b\]
Remember we have the constraint that \(a<b\), hence: \[aN+a-b < aN\]
Therefore we require that:
\[ aN+a-b < bB < aN\]
As we also require that a, b, B and N are all integers then clearly this can only be satisfied when: \[ a-b < -1 \]
In any case that \(a-b = -1 \) the scoring percentage will always pass through p if it starts below it and ends up exceeding it.
For example this is the case for \(p=75\%\) as \(a=3\) and \(b=4\) but it is not the case for \(p=60\%\) where \(a=3\) and \(b=5\).
Now we can see why it must pass through 75% and have determined the other percentages for which this condition holds.