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math question involving letter tile distribution

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ACG
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Joined: 12/31/1969

Hi! I'm trying to make a card game (spinoff of the Junkyard Wars game) and I'm trying to determine the number of points I should give each letter.

The premise is this. The number of points for each letter depends on the tile distribution. If there are N of each letter in the deck (well, I'm saying N+1 since there are two blanks and I'm figuring each player will get one), the expected number of tiles we need to draw until we get one of them is 1 / (N+2) since there are N+2 gaps, presumably of equal size on the average, between the ends of the tile distribution and the letters drawn. So far so good.

Now here's the tricky part. Suppose there's are T (T's) and H (H's). The goal is to to get a T and an H. The upper limit is the sum of the expected values for the T and the expected value for the H (which is what I'm using right now -- I'm assuming it's slightly more difficult than the actual calculation because a blank may only be used once). So far so good.

Welll -- I'm now thinking of adding dipthongs. Say, "TH".

There are now TWO ways to get "TH" in your hand: T and H, or TH. The TH would do no good if you just want a T in your hand (which is probably the majority of the cases, so the formula to score the T won't change). But what should the score of the "TH" be?

Thanks in advance,

ACG

Sebastian
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Joined: 07/27/2008
Re: math question involving letter tile distribution

ACG wrote:
There are now TWO ways to get "TH" in your hand: T and H, or TH. The TH would do no good if you just want a T in your hand (which is probably the majority of the cases, so the formula to score the T won't change). But what should the score of the "TH" be?

It's impossible to say, because you haven't told us what we're doing with the tiles. If we're trying to create strings of letters in alphabetical order, then TH will be worthless and so the score should be zero. If we're trying to create dipthongs, then the score should be exactly T plus H, possibly with a bit extra because it's easier to find a combined set. If were trying to create english words, then you probably want to sit down with an electronic dictionary and grep.

ACG
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Joined: 12/31/1969
math question involving letter tile distribution

There are a bunch of words with T (CAT, for instance), some with H (HORSE)'s, and a few with TH's (THAN). The goal is to spell out the words with the stuff on the cards (which may include dipthongs).

Infernal
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Joined: 12/31/1969
math question involving letter tile distribution

To get a ratio of letter distrabutions (and work out a point distrabution for them), you could get a newspaper and count the number of times a letter is used (on a single page, or to be more thourogh the entire thing). This will be easier if you use an online document (afaik: they did this for scrabble).

ACG
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Joined: 12/31/1969
math question involving letter tile distribution

That would make sense, but in this case there is a forced word list of 70ish words. Each player draws eight letters and two word cards. The letters are placed face up and the words are face down. The goal is to play (in effect) gin rummy with letter cards to make one of the words in your hand. The point score of each word would be, in effect, the number of times you have to draw a card to get all the letters for that word given the tile distribution (which is effectively Scrabble's distribution until I add the dipthongs).

Here's a good example. Using Scrabble tiles, there are two H's and six T's. There are also two Y's.

Suppose the user had the words EARTH and HEART in his hand. You'd have expected that both would require the same number of letters to draw since they are anagrams. This would be correct if there were no dipthongs. However, with dipthongs, EARTH would be easier to make than HEART (and therefore fewer points) because it would be able to use the TH tile while HEART would not be.

ACG

Epigone
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Joined: 12/31/1969
math question involving letter tile distribution

So here's the scenario, then. You have a set of words to be scored, based on the expected number of tiles to be drawn before getting that word, given some distribution of word substrings (letters, dipthongs, maybe 'STR', etc.). If that's right (or if it's wrong!) you can PM me, I'll write up a monte carlo sim to tell you the expected numbers.

ACG
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Joined: 12/31/1969
math question involving letter tile distribution

That's it. Basically it's a Scrabble set with 106 tiles (extra AI, ST, CH, TH, RT, and TR tiles). One of the most intriguing possibilities is the word EARTH. That's got three ways to make it: E A R TH, E A RT H, E A R T H.

ACG

jwarrend
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Joined: 08/03/2008
math question involving letter tile distribution

A correction: what you're calling a diphthong is, I'm pretty sure, more correctly called a digraph. A diphthong is two vowels. From wikipedia:

In phonetics, a diphthong (Greek, "diphthongos", literally "with two sounds") is a vowel combination in a single syllable involving a quick but smooth movement from one vowel to another, often interpreted by listeners as a single vowel sound or phoneme

Links for the interested:

Diphthongs:
http://en.wikipedia.org/wiki/Diphthong
http://www.bartleby.com/68/55/1855.html
Digraphs:
http://www.bartleby.com/68/44/1844.html

You may now resume the game balancing discussion.

-Jeff, the grammar Nazi

Infernal
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Joined: 12/31/1969
math question involving letter tile distribution

You could not just stick to counting the letters, but also the dipthongs and diagraphs as well. Then you would have a complete list of ratios to use.

ACG
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Joined: 12/31/1969
math question involving letter tile distribution

I'm not sure the ratios are what we need here.

Here's the story. You have a word with TH in your hand. There are TH tiles, T tiles, and H tiles . The probability of drawing a T is X. The probability of drawing an H is Y. The probability of drawing a TH is Z. How many tiles will you have to draw until you get the letter combination "TH" in your hand?

That's it in a nutshell.

FYI, as it turns out the other diagraphs in play at the moment are TR, RT, CH, AI, and ST. It just so happened that I had picked words where those made sense :)

ACG

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