A Mathematical Application to... Scrabble?

[Pen2014]

I'm currently looking for a science project topic, preferably in the mathematics field. Perhaps not completely mathematical, just something managable for around my level (PreCalculus, can manage Calculus) with a mathematical base.

After reading http://www.usc.edu/CSSF/History/2007/Projects/S1313.pdf , I've come up with an idea regarding Scrabble and mathematic probabilities. Perhaps calculating the mathematic value of each letter/piece, taking into account their point values, frequencies in the English language, placement of the tiles, number of tiles of each letter included, probability of getting other letters which could make a word, and such. However, it just doesn't seem appropriate enough for my level - in other words, it's too easy.

What could I expand on, or include as a variable, or calculate for instead of the mathematic value of each letter/piece?
Also, how could I form a question/hypothesis from this?
I'd like some opinions on the difficulty of this idea as well.

I'm completely open to other suggestions as well, although I cannot use topics that are simply repeats of what has been done (i.e. proving a known theorem, etc.)
All suggestions are greatly appreciated.
Thank you for your time!

note: I am aware this is a very sketchy idea, and I apologize

[deleted-71712]

Hi Pen2014,

I'm sorry that we've taken so long to get back to you. Hopefully some other experts will chime in soon, but here's a partial response...

How about mathematically evaluating some different *strategies* for playing Scrabble? For example, I've noticed that certain letters (E, S, T, R -- letters that show up in prefixes and suffixes) lend themselves very well to playing all 7 tiles at once, which yields a 50-point bonus. So if I have a lot of these letters in my rack, I will often make a few low-scoring plays in order to get rid of the letters that aren't as versatile for forming 7-letter words, with the hope of ending up with a net higher score when I'm able to make a 7-letter word later. So I would be really interested in a probabilistic evaluation of that particular strategy, and I'm sure you could think of a few more!

Amanda

[deleted-71349]

It would seem that you'd want to compute some value for a player's hand, that is, the collective values of the tiles. This could be based on inherent values for each tile, as you discuss, but also adding in valuation of that particular combination. I don't think that this topic would be in any way "too easy", but the question would be how useful any probabilistic valuations might be. The big added complication is that the actual value of a hand varies depending on the state of the game, which gets you into the strategic aspects that Amanda has mentioned. But there you could have a hypothesis to test. In experiments, do a valuation of players' hands as a game progresses, and then see how whether these cumulative valuations somehow converge and help to predict who wins.

In your discussion of possible ways of valuing tiles, several of the attributes mentioned are correlated: point value, language frequency, and numbers of tiles, for example. So these can be combined for valuation purposes. (Side note: Scrabble varies from language to language - maybe you could compare your approach for a non-English language version as well.)

Aside from basic valuations or strategic considerations, there are other factors like the extent of the players' vocabularies (assuming no dictionary cheating used), so maybe some value could be attached to that and used as a weighting factor.

So basically, what you could look at is whether there is an "easy" valuation model that gives a reasonable chance of predicting the outcome of a "complex" game. This is a very different matter from the project itself being too easy! I think it could have a lot of potential.

Hope this helps,
Gordon.

[deleted-71588]

Even if one has a large vocabulary and a set of letters that might generate a good score on a turn, the problem of whether the word will fit on the board is usually the biggest factor. The strategy of what to put down and where to put it down on any given turn involves the existing board state and what is available in terms of places to put things. It also involves thinking about whether you want to put it down and open a high scorring possibility for the next player (assuming they have the letters to take advantage of your play). This is NOT a simple game to model.

[Pen2014]

I've been convinced that this project will be challenging. I'm up for this challenge, but before I start, I would like to know if there's any guidelines or if it would be viable for me to develop my own formulas for each variable I consider. Basically, my instructor needs to make sure this is something I can pull off, and the methods for a project like this are vague. If anyone has any ideas, I'd love to hear them.

Thanks for your time,
Pen2014