This article is the third post in my series “No Tile Left Behind” focusing on leaves. For a list of articles on this topic, click here.

The value of leave L with elements {E(1), . . . E(k)} (expressed by the notation V{E(1)… E(k)} is equal to the value of the mean of all leave (L+1) with elements {P(1)E(1),…P(k+1)E(k+1} (where V represents the value function and P represents the probability function)

In English, this means that the worth of a leave is the average of that leave plus one other tile included after we take into account the probability of drawing each tile. This is shown by the following chart, which can be confirmed by the program Quackle:

Letter + 1 with AERT +:

A: 8.7 | B: 16.2 | C: 22.1 | D: 19.9 | E: 12.8 | F: 12.1 | G: 11.9 | H: 19.6 | I: 16.4 |

J: 4.4 | K: 9.5 | L: 19.3 | M: 18.7 | N: 20 | O: 10.5 | P: 19.6 | Q: 2 | R: 12.7 |

S: 34.8 | T: 13 | U: 8.5 | V: 11.5 | W: 9.8 | X: 12 | Y: 9 | Z: 14.2 | ?: 45.4 |

Average of AERTx = 15.4 (consistent)

Standard deviation: 7.36

% of tiles higher than the mean: 41%

This sort of analysis has one major practical application: during exchanges. During exchanges, we often have to decide whether an extra marginal tile is worth keeping. In an attempt to ask ourselves this question, we should create and try to approximate tables like the one above if we want to figure out whether or not to keep a certain tile on an exchange. Often times this is not so simple, as these numbers are not universal for each position: in certain cases they need to be adjusted. Adjusting these values will be covered in my next article.