09:48:00   Umum

In  a paper by S. W. Golomb and A. W. Hales, "Hypercube Tic-Tac-Toe" [It's about the generalization of Tic-Tac-Toe game to n dimensional cube] there is an interesting passage about the following equality

\[\left\lceil\frac{2}{2^{1/n}-1}\right\rceil = \left\lfloor \frac{2n}{\log 2} \right\rfloor \]

the authors said that this equality is false but they gave a miss-calculated counterexample (due to the error in the computer multi-precision system). 

The equality is indeed very interesting! 

In 18th century when the computer wasn't good enough, this might be a kind of problem that might fool those who rely only on "trial and error method" [some profession joke usually refer it to engineer :p ]

Let's use it on the following joke :D

Suppose that the energy explosion risk given by the $n$ connected reactors devices is given by the formula:

\[E_n := n^2 \left(\left\lceil\frac{2}{2^{1/n}-1}\right\rceil - \left\lfloor \frac{2n}{\log 2} \right\rfloor\right) \]

(In TJ)

Some amateur engineers who ignore the power of mathematical proof asserts that $E_n=0$, They conclude this after performing a trial method i.e  inputting several values of $n$. 

As they already check that $E_1=E_2= \cdots = E_{10^{13}}=0$. Quite convincing for the assertion that $E_n=0$ for all $n$.  

Furthermore, by direct computation $E_{n}=0$ for $10^{20}\leq n \leq 10^{23}$. So what else? 

By this observation, the engineer said that the device is safe because the risk is always zero, and so they proceed by activating the device.

Suppose that at a very unlucky day, only 777451915729369 reactors on this device are fully functional, nevertheless we still have $E_{777451915729369}=0$, and the risk is still zero.

But then there is a damned cat goes to the lab and accidentally malfunctioning one reactor, resulting only $777451915729368$ reactors are fully functional.

At that time the world will be destroyed by 

\[E_{777451915729368} =604431481271264322211417679424\]

TJ explosion. End Of The Story.

The nature of $E_n$ is very interesting, $E_n=0$ for a very long time  then it takes a nonzero value  instantly for the first time when $n=777451915729368$ (very big), after that  $E_n=0$ again for a very long time until it instantly nonzero again for the second time when $n=140894092055857794$ (and of course bigger). 

So be careful on the trial-error conclusion :p

The sequence $\frac{E_n}{n^2}$ numbers on OEIS is A1299935. See http://oeis.org/A129935