Upper bonus

Dealing with the Upper Section bonus points

Fig11If a player scores a total of at least 63 points in the Upper Section, a bonus of 35 points is added to the upper section score. Though 63 points corresponds to three-of-a-kind for each of the six die faces, a common way to get the bonus is rolling four (or five, often using a "Yahtzee as a joker") of a larger number so that fewer of the smaller numbers are needed (a player can earn the bonus even if a he or she scores a "0" in an upper section box).

These bonus points are the thing that really complicate our life when playing the game. Just imagine: you rolled at the end of the first turn 6,6,6,6,5. Great combination! Boxes “Sixes” and “Four of a Kind” are both unused. Where would you place your combination into?

  • By placing the combination into Sixes you will earn 24 points and a great chance to get the upper section bonus at the end of the game.
  • By placing the combination into Four of a Kind you will earn 29 points, but your chances to get the 35 bonus points at the end for the Upper Section would be less in comparison to the previous case.

Note: the placement of the combinations highly depends on the points/score already achieved in the Upper Section.

The Upper Section bonus modifies in this regard the probabilistic values of the unused boxes. I hope you still remember: the probabilistic value of Ones was about 2.10. But if there is only the Ones box left/unused in the game and you haven’t reached 63 points in the Upper Section yet, the probabilistic value of this box could be more than 2.10, simply because we have to add to the value of the box the probabilistic value of the bonus too!!

There is the table of the probabilistic values of the box Ones depending on the upper section score missing from the min. 63 points:

Upper Score missing from the minimum of 63 points

Probabilistic value of the box Ones

0

37.10

1

34.83

2

26.56

3

14.52

4

5.76

5

2.57

6

2.10

7

2.10

8

2.10

...

...

63

2.10

The logic behind is simple: if the Upper score already reached 63 points we surely obtain the bonus (35 points) and the probabilistic value of the Ones (2.10). If we have to score at least 1 to obtain the bonus (to have 63 points in the Upper Section), the probabilistic value of the bonus+Ones = 34.83. Please note that the extra Yahtzee bonus was not taken into account.

Coming back to 6,6,6,6,5 and the two unused boxes left.

This is the way how P-Yahtzee solves this problem: it chooses the best out of the following two options:

  • The value of choosing the Sixes = 24 points + the combined probabilistic value of all remaining unused boxes
  • The value of choosing the Four of a Kind box: 29 points + the combined probabilistic value of all remaining unused boxes.

Important to mention: “the combined probabilistic value of all remaining unused boxes” is different in the two cases since in the first case you have chosen the Sixes and in the second case you have chosen the Four of a Kind. This value also depends on the points already achieved in the Upper Section.

So where you would put 6,6,6,6,5 then? The exact answer is:

At the beginning of the game it is wise to place this combination into Sixes box. This choice is better with about 3.6 points than choosing the Four of a Kind. As the game goes on, it may happen that you manage to score many points in the Upper Section ( for ex. more than 52), so there is no urgent need to place this combination into the Sixes box, since the Upper Section bonus will probably be obtained. Thus the Four of a Kind overcomes the Sixes. Suppose you have already scored 63 in the Upper Section, the Four of a Kind value of 6,6,6,6,5 is greater with about 7 points than the value of the Sixes box.

If only the Four of a Kind and Sixes remain unused at the end of the game, and you need the 24 points to obtain the Upper Score bonus, the Sixes box becomes significantly a better choice (with about 23 points).

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