The Aliens and Presidents Puzzle

Solution:

If we let m= # of edible people and n= # inedible people, then total world population is n+m. We also have:

\begin{equation} {n \choose 2}/{n+m \choose 2}=\frac{1}{2} \end{equation}

From this we get: \begin{equation} n=\frac{1}{2}(2m+1+\sqrt{8m^2+1})\rightarrow 8m^2+1=p^2\rightarrow 8m^2=(p+1)(p-1)=2q*2(q-1) \end{equation} or \begin{equation} m^2=\frac{q}{2}(q-1). \end{equation} So, either \begin{equation} q=b^2,\phantom{bladida}\frac{q-1}{2}=a^2 \end{equation} with \begin{equation} 2a^2+1=b^2. \end{equation} Or, \begin{equation} q-1=b^2, q/2=a^2, \text{ and } 2a^2-1=b^2. \end{equation} So, $b^2=2a^2\pm 1$ (Pell’s Eq.), and $m=ab$. Conversely $b^2\rightarrow\sqrt{8m^2+1}$ is a whole number.

Difference Equation:

Let $b_0=b_1=1,$ $b_{k+1}=2b_k+b_{k-1}$ and let $a_0=0,$ $a_1=1,$ $a_{k+1}=2a_k+a_{k-1}.$

Lemma:

a and b satisfy Pell’s equation if and only if $a=a_k,$ $b=b_k$ for some $k\geq 0$.

Proof: A bunch of algebraic manipulation. The details are left as an excercise for the reader because I’m too lazy to typeset it.

Thus, any solution to Eq. 1 $\rightarrow m=m_k=a_kb_k$, for $k\geq1$, and so $n_k\approx$ $\frac{1}{2}(2m_k+1+2\sqrt{2m_k})=(1+\sqrt{2})m_k+\frac{1}{2}$ for large k.

The solution of the difference equation is given by $b_k=\frac{1}{2}(\beta^k+(\frac{-1}{\beta})^k)\approx\frac{\beta^k}{2},$ with $\beta=1+\sqrt{2}.$

Since $2a_k^2+(-1)^k=b_k^2\longrightarrow a_k\approx \frac{b_k}{\sqrt{2}}\approx \frac{\beta}{2\sqrt{2}}.$ This gives $m_k\approx\frac{\beta^{2k}}{4\sqrt{2}}$, and $n_k\approx\frac{\beta^{2k+1}}{4\sqrt{2}}+\frac{1}{2}.$ Now $T_k=m_k+n_k\approx\frac{\beta^{2k+1}}{4\sqrt{2}}+\frac{1}{2}.$ In fact, $T_k=\frac{(\beta+1)\beta^{2k}}{4\sqrt{2}}+\frac{1}{2}$ rounded to the nearest integer.

To be clear, the $T_k$ we have just produced is the generating sequence for numbers with the property $n+m=T_k$ and the exact probability condition for choosing two inedibles. That is, there are only some numbers (populations of the Earth) with these properties. For example, $T_{12}=927.5*10^6,$ $T_{13}=5.4*10^9,$ and $T_{14}=31.5*10^9.$ The world population since 1800 $> 10^9,$ and now it is $7.4*10^9.$ So, there is only one number in the generated sequence that corresponds to a valid Earth population during the period since 1800. Hence, the aliens showed up at $T_{13}$ in 1992 and George H. W. Bush was president.