Unspeakable Words is a spelling card game with a Cthulhu mythos setting. You start the game with five little markers. You have a hand of seven cards, each with a letter of the alphabet and a score. (In service of the Cthulhu mythos, the score for a given letter matches the number of corners in the letter and your markets are little Cthulhu idols.) On your turn you spell a word of at least three letters, total up the score for the word, then add it to your total. Then, roll a twenty sided die; if you roll less than the word's total, you lose one of your markers. If you run out of markers, you lose. The first person to reach 100 points wins, or the last person to not lose wins.
My gut figured that it didn't matter how long of a word you spelled, your long term odds of reaching 100 before you lost were the same. So, I ran the numbers. Turns out I'm wrong. What I was overlooking is that if you were to entirely spell 1 point words, you'd be guaranteed to win in 100 turns; you always roll at least 1 and pass. As a result, there is an interesting tradeoff: go for small scoring words and have a high chance of making 100 points, or be aggressive and risk early failure for a chance to win early.
This table assumes that you always manage to play the score you are shooting for, excepting the last turn, in which case you score exactly the number of points necessary to reach 100. The chance to survive is expressed as both an approximate percentage and an exact fraction.
| Score | Turns to win | Chance to survive | |
|---|---|---|---|
| 1 | 100 | 100.0% | 1/1 |
| 2 | 50 | 89.6% | 25230943748850073255103306873244343737342901024615600687881120469/28147497671065600000000000000000000000000000000000000000000000000 |
| 3 | 34 | 77.0% | 76968213041700886847453925881949/100000000000000000000000000000000 |
| 4 | 25 | 68.2% | 114438559254596363471370378543091/167772160000000000000000000000000 |
| 5 | 20 | 63.0% | 60047937765376/95367431640625 |
| 6 | 17 | 59.6% | 640386405/1073741824 |
| 7 | 15 | 57.3% | 1145498996774531/2000000000000000 |
| 8 | 13 | 54.8% | 8975830177257983/16384000000000000 |
| 9 | 12 | 53.3% | 5202873/9765625 |
| 10 | 10 | 50.4% | 2582551509263/5120000000000 |
| 11 | 10 | 50.0% | 1/2 |
| 12 | 9 | 48.4% | 24762585249/51200000000 |
| 13 | 8 | 46.4% | 36248/78125 |
| 14 | 8 | 45.4% | 2326134419/5120000000 |
| 15 | 7 | 43.4% | 1735857/4000000 |
| 16 | 7 | 42.2% | 6907/16384 |
| 17 | 6 | 38.6% | 241/625 |
| 18 | 6 | 38.0% | 973101/2560000 |
| 19 | 6 | 34.4% | 3439/10000 |
| 20 | 5 | 22.6% | 723901/3200000 |
| 21 | 5 | 25.0% | 1/4 |
| 22 | 5 | 45.0% | 9/20 |
| 23 | 5 | 65.0% | 13/20 |
| 24 | 5 | 85.0% | 17/20 |
| 25+ | 4 | 100.0% | 1/1 |