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Two good reads.

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Christmas time is reading time… and I actually read two quite amazing books:

  1. Jacques Hadamard: The Psychology of Invention in the Mathematical Field
    A classic. Very nice introspection and discussion about “how” mathematics is done. In particular it discusses a lot of potential theories of mathematical innovation and invention (as one might have guessed from the title).
  2. Norbert Wiener: I Am a Mathematician
    Part two of his autobiography. A very interesting read with a lot of big names and a lot of interesting insights into various aspects in and around mathematics.  (thanks to Dave Goldberg for the pointer)

Written by Sebastian

December 29, 2011 at 10:44 pm

Mastermind – five questions do suffice

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Today I would like to talk about the Mastermind game and related (recreational?!) math problems – the references that I provide in the following are probably not complete. Most of you might know this game from the 70s and 80s. The first player is making up a secret sequence of colored pebbles (of a total of 6 colors) and the other player has to figure out the sequence by asking questions about the code by proposing potential solutions. The first player then indicates the number of color matches.

File:Mastermind.jpg

Mastermind (source: Wikipedia)

More precisely, Wikipedia says:

The codebreaker tries to guess the pattern, in both order and color, within twelve (or ten, or eight) turns. Each guess is made by placing a row of code pegs on the decoding board. Once placed, the codemaker provides feedback by placing from zero to four key pegs in the small holes of the row with the guess. A colored (often black) key peg is placed for each code peg from the guess which is correct in both color and position. A white peg indicates the existence of a correct color peg placed in the wrong position.

If there are duplicate colours in the guess, they cannot all be awarded a key peg unless they correspond to the same number of duplicate colours in the hidden code. For example, if the hidden code is white-white-black-black and the player guesses white-white-white-black, the codemaker will award two colored pegs for the two correct whites, nothing for the third white as there is not a third white in the code, and a colored peg for the black. No indication is given of the fact that the code also includes a second black.

Once feedback is provided, another guess is made; guesses and feedback continue to alternate until either the codebreaker guesses correctly, or twelve (or ten, or eight) incorrect guesses are made.

In a slightly more formal way, we have a string in \{1,...,6\}^4 and the “decoder” wants to reconstruct this string by inferring from the provided feedback. One of the natural questions that arise is of course how many questions do suffice. Knuth [Knuth76] then showed that five questions suffice to be able to always reconstruct the secret string. What is interesting about the proof is that it is a “table” – basically output of a computer program. This lookup table can be used so find a next question at any given point. The table is a greedy optimization in some sense: “Figure 1 [the lookup table] was found by choosing at every stage a test pattern that minimizes the maximum number of remaining possibilities, over all 15 responses by the codemaker”.

Later in 1983, Vasicek Chvátal dedicated a paper on the Mastermind game to Paul Erdős for his 70th birthday. Chvátal looked at generalized admissible Mastermind vectors denoted by V(n,k) of vectors of length n with k different colors. It is not too hard to see that the minimum number of questions f(n,k) needed to correctly identify any string in V(n,k) is bounded from below by

f(n,k) \geq \frac{n \log k}{\log \binom{n+2}{2}}

which arises from the fact that there are only \binom{n+2}{2} different answers and n^k different strings have to be distinguished. Complementing this bound, Chvátal showed that the number of questions needed to be asked without waiting for the answer (i.e., the questions are asked in one go, then the answers to all questions are provided at once, and then the code has to be uniquely identified) can be bounded from above as follows: the number of questions needed for this static case will be denoted by g(n,k) and for any \epsilon > 0 there exists n(\epsilon) so that for all n > n(\epsilon) and k < n^{1-\epsilon} we have

g(n,k) \leq (2+ \epsilon) n \frac{1+2 \log k}{\log n - \log k}

and clearly we have f(n,k) \leq g(n,k). The proof uses the probabilistic method in a nice way. Moreover, Chvátal also provides some upper and lower bounds for special cases. Those of you guys that know about my addiction to the Chvátal-Gomory closure and its friends might have already guessed that this is exactly how I came across the problem…

The latter problem where we do not wait for the answers is usually called the static mastermind problem whereas the classical version is called the dynamic mastermind problem. Later in 2003 and 2004 Goddard (see [Godd03,04]) provided optimal values for the minimal number of questions to be asked both in the dynamic as well as static case and also for the average number (denoted by r(n,k)) of questions needed whenever the secret string is uniformly picked at random. With the notation from above we have the following number of questions (tables taken from [Godd03,04]):

For the average number of queries needed (r(n,k)) we obtain:

Positions
2 3 4 5 6 7
Colors 2 –  2 2.250 2.750 3.031 3.500 3.875
3 –  2.333 2.704 3.037 3.358
4 –  2.813 3.219 3.535
5 –  3.240 3.608 3.941
6 –  3.667 3.954 4.340
7 –  4.041 4.297
8 –  4.438
9 –  4.790
10-  5.170

and similarly for the dynamic case we have the following minimum number of queries f(n,k):

Positions
2 3 4 5 6 7 8
Colors 2 –  3 3 4 4 5 5 6
3 –  4 4 4 4 5 <= 6
4 –  4 4 4 5 <= 6
5 –  5 5 5 <= 6
6 –  5 5 5
7 –  6 6 <= 6
8 –  6
9 –  7
10-  7

and for the static case g(n,k) we have  the following table. Note that in the table below the final “query” that states the recovered string is not counted as in comparison to the ones above. Therefore in order to compare the values with the ones above you need to add “1” to each entry.

Positions
2 3 4 5 6 7 8
Colors 2 –  2 2 3 3 4 5 5
3 –  2 3 3 4 4 <= 5
4 –  3 4 4 5
5 –  4 4 5
6 –  4 5 6
7 –  5 6 <= 7
8 –  6 7 <= 8
9 –  6 8
10 7 9

(there seems to be a typo for n = 2 and k = 3 in one of the tables, as the static case has a better performance than the dynamic case which is not possible).

In order to be able to actually check (with a computer) whether a certain number of questions suffices, we have to exclude symmetries in a smart way. Otherwise the space of potential candidates is too large. In this context, in particular the orderly generation framework of [McKay98] is very powerful. The idea behind that framework is to incrementally extend the considered structures in such a way that we only add a canonical candidate per orbit. Moreover, after having extended our structure to the next “size” we need to check whether it is isomorphic to one of the previously explored structures. In this case we do not consider it. For each candidate we check whether the number of distinct answers is equal to the total number of possible secret codes. In this case there is a bijection between the two and therefore we can decode the code. However it is not clear that this bijection needs to have a “nice” structure or that it is “compact” in some sense.

References:

  1. [Knuth76]: Knuth, D.E. 1976. “The computer as a master mind.” Journal of Recreational Mathematics. http://colorcode.laebisch.com/links/Donald.E.Knuth.pdf (Accessed June 9, 2011).
  2. [Chvátal83]: Chvátal, V. 1983. “Mastermind.” Combinatorica 3: 325-329.
  3. [McKay98]: McKay, B.D. 1998. “Isomorph-free exhaustive generation.” Journal of Algorithms 26(2): 306–324.
  4. [Good03]: Goddard, W. 2003. “Static Mastermind.” Journal of Combinatorial Mathematics and Combinatorial Computing 47: 225-236
  5. [Godd04]: Goddard, W. 2004. “Mastermind Revisited.”  Journal of Combinatorial Mathematics and Combinatorial Computing 51: 215-220

Written by Sebastian

October 13, 2011 at 4:20 pm

Cambridge Mathematical Tripos

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Timothy Gowers just started a new series of blog posts for first-year mathematics students. While the blog posts will be centered around Cambridge’s courses I am pretty sure that the discussed topics and hints will be valuable to other students as well. In fact, what I find most impressive is the goal of the series: to teach people how to do mathematics! We all learned what mathematics is and results have been presented to us in a nice, cleaned-up fashion. However only very few of us were taught how to solve/approach problems – most of us learned it the hard way at some point. It is as if you go to a restaurant to get great food: this does not teach you how to cook yourself! In particular it does not teach you that the nice result is a product of quite a mess in the kitchen. When doing math, everybody will reach her or his limit sooner or later (as compared to math in school which was easy for many math students) and it is precisely this point in time, when students start to doubt their own potential. In fact some kind of a bias is bound to take place: every math problem that can be solved is “easy” and every problem that is not solved is a small personal crisis – “am I good enough”? As you did not see the mess in the kitchen, one might think that things come easy in a nice form or not at all. In the end there is no positive feedback available anymore, only negative feedback.

I am very much looking forward to this series and I am sure that Tim has some valuable insights to share!

Written by Sebastian

September 25, 2011 at 9:15 pm

Does IBM care for CPLEX at all?

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I just got an email from IBM asking me to participate in the Academic Initiative Survey. I participated with the aim to address a few shortcomings with respect to IBM’s support for, and interest in their optimization products, e.g., that it is quite a hassle to download cplex as one has to go through an uncountably infinite number of pages before one actually reaches the download page – if one reaches it at all. Also there were a few other things that I wanted to address.

But guess what. One of the first question was to what academic field one belongs to. Operations Research? Mathematics? nada. That was already a bad omen. And in fact. There were only two references to optimization at all “Linear Programming” and “Integer Programming” in the courses that I teach / want to teach (out of a gazzillion listed including a lot of voodoo stuff). Effectively, optimization and the optimization products were virtually not present at all. Cplex, OPL, OPL Studio and none of the other optimization tools were even mentioned.

This apparent lack of interest raises serious questions about IBM’s future plans for cplex and their optimization products. In particular, questions about continuity and support. Who knows… 10 years ago I would have been really scared as cplex was the strongest industrial strength solver and therefore choice number one in many applications – however times have changed and fortunately there are alternatives now.

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March 11, 2011 at 5:02 pm

If you ever wonder why you need an ipad

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Written by Sebastian

May 19, 2010 at 12:34 pm

Posted in what else is out there

Tagged with

Bobby Tables

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This xkcd comic is rather old and most of you might have seen it already… Anyways, I found it pretty hilarious (yes, geek humor) and so I wanted to share it with you:

Exploits of a mom

Written by Sebastian

March 11, 2010 at 1:16 pm

European Capital of Culture: Ruhr.2010

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The European Capital of Culture for 2010 is Essen in Germany, on behalf of the Ruhr Area (“Ruhrpott” or “Ruhrgebiet” as we say) and its several cities (e.g. Bochum, Dortmund, Gelsenkirchen, Duisburg, Oberhausen, …). The official homepage can be found here [9]. As I grew up in Essen and practically lived there the first “part” of my life I thought I take the opportunity and provide you with some background information and links, just in case you are in Germany in 2010 and you have a chance to visit that diverse and really interesting area. Unfortunately, some of the articles are in German only, but I tried to provide English resources whenever possible.

According to wikipedia, the Ruhr Area is the fourth largest urban area in Europe with some 7.3 million inhabitants. What is special about this area is its unique industry culture and the transformation in the 90’s away from heavy industry to a more service oriented area (see [2], [3], [4], German only). As you can imagine, this transition brought a lot of initial problems and desperation to people heavily relying on heavy industry for their income and the fact that those people could not be easily moved to different jobs (see [1], German only). Now that this transition is (almost) complete, the Ruhr Area presents its distinguished face to the world as the European Capital of Culture. Also check out the special issue on Spiegel ([3], German only) about Ruhr.2010.

Events and must-sees:

An event planer with events scheduled in 2010 can be found in [8] and following the same link, you can also find the Top-10 attractions in the Ruhrpott (according to Spiegel and I kinda agree).

Impressions from the Ruhr Area:

In [5] you can listen to some “typical” sounds of the Ruhr Area. These sound recordings were taken by Richard Ortmann and represent some of the day to day noises you would have heard (and partly still would hear today). Also check out his sound archive with a more comprehensive collection of sound nostalgia [6]. Probably one of the most famous songs about the Ruhr Area is Herbert Grönemeyer’s “Bochum” (about one of the big cities in this area):

Right now, 09.01.2010 15:47, there is a live show about Ruhr.2010 as the European Capital of Culture on ZDF (one of the German main tv stations). You can find it also in the ZDF mediathek [7] together with many other shows and pictures about the Ruhr Area where you can watch it for free.

Here are a few pictures from various photographers depicting the Ruhr Area:

  1. Ruhr.2010
  2. Ruhr.2010 (2)
  3. Ruhr.2010 (3)

References:

[1]: “Wir sind die Dinosaurier – und wir wollen nicht aussterben” – Spiegel 11.02.2007 (German only)
[2]: Von der stinkenden Brühe zum Lebenselexier – Spiegel 29.08.2007 (German only)
[3]: Kulturhauptstadt Ruhr.2010 – Spiegel 08.01.2010 (German only)
[4]: Industrie in XXL – Spiegel 22.06.2009 (German only)
[5]: Der Pulsschlag aus Stahl verklingt – Spiegel 20.10.2007 (German only – but for the sounds it does not matter)
[6]: Richard Ortmann – Sound Archive (German only – but for the sounds it does not matter)
[7]: ZDF mediathek’s Ruhr.2010 collection (tv shows and pictures)
[8]: Spiegel Ruhr.2010 event planer including Top 10 attractions (German only – but rather self-explaining)
[9]: Ruhr.2010 – official homepage

Written by Sebastian

January 9, 2010 at 4:39 pm

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