Wikipedia:Reference desk/Archives/Mathematics/2013 October 15

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October 15

The encrypted German Tank Problem

In German_tank_problem#Countermeasures, it describes a simple encryption where each digit is assigned a letter. Is it possible to arrive at a good estimate for the number of "tanks" if you know that such an encryption scheme is in use, but don't know the exact letter combinations (e.g. could you estimate the number of tanks from "XH, HI, IV, VL" etc.)? Obviously, there is an upper limit if all the numbers are n digit, or there is obvious zero padding (XXXXXHI), but how large a sample would you need to get close to the real value? MChesterMC (talk) 09:00, 15 October 2013 (UTC)

The Countermeasures subsection is badly off-topic. We do have an article on Cryptography however. Bo Jacoby (talk) 19:16, 15 October 2013 (UTC).

In your sample, just from the fact that there are 4 different starting digits you can deduce that there are at least 40 tanks, and from the fact the serial numbers have 2 digits you can deduce there must be less than 100. From the German point of view it would be better to just tack an extra random digit to the end of the serial number; this would have the added benefit of scaring the heck out the Allies. --RDBury (talk) 03:40, 16 October 2013 (UTC)

You can actually get only "at least 30" out of that sample, if we have HVIXL=01234. --Tardis (talk) 00:16, 18 October 2013 (UTC)

Bernoulli Numbers Help!

Can anyone explain 8:30-8:40 of this video?! I just don't get why that sum has to equal zero or how that leads to deriving the values 1, -0.5 etc. Someone else asked in the comments but I don't really understand the answers there either.

http://www.youtube.com/watch?v=XHQ0OzqTjd0 — Preceding unsigned comment added by 5.81.9.195 (talk) 15:39, 15 October 2013 (UTC)

You're equating two power series, meaning their coefficients need to be the same. Widener (talk) 06:18, 16 October 2013 (UTC)

Which power series? If the Bernoulli sum/expansion is one what is the other? — Preceding unsigned comment added by 86.147.189.134 (talk) 10:37, 16 October 2013 (UTC)

The other power series is the constant function $1$. The right hand side is the power series $\sum _{n=0}^{\infty }a_{n}s^{n}$ where $a_{n}={\frac {1}{(n+1)!}}\sum _{\mu =0}^{n}{\binom {n+1}{\mu }}\beta _{\mu }$. The left hand side is the power series $\sum _{n=0}^{\infty }b_{n}s^{n}$ where $b_{0}=1$ and $b_{n}=0$ for $n>0$. As these power series are equal, $a_{n}=b_{n}$ for all $n$. That is, ${\frac {1}{(n+1)!}}\sum _{\mu =0}^{n}{\binom {n+1}{\mu }}\beta _{\mu }=0$ if $n\neq 0$ (equivalently $\sum _{\mu =0}^{n}{\binom {n+1}{\mu }}\beta _{\mu }=0$). Also, ${\frac {1}{(0+1)!}}\sum _{\mu =0}^{0}{\binom {0+1}{\mu }}\beta _{\mu }=1\implies \beta _{0}=1$, and ${\frac {1}{(1+1)!}}\sum _{\mu =0}^{1}{\binom {1+1}{\mu }}\beta _{\mu }=0\implies \beta _{1}=-{\frac {1}{2}}$. Widener (talk) 13:04, 16 October 2013 (UTC)