Now that you know that on every first roll there are 252 possible outcomes, and you know the weight associated with each of these and their probability, it is then important to consider the variety of choices you may encounter as you play through the two re-rolls.

For example, let’s look at the first roll of 1-2-3-4-5. If you count the number of different strategies that may be adopted for this hand you must know how to categorize the strategies based on how many dice are held. Begin by holding all five dice. There are obviously no decisions in this case, since there is only one possible strategy. However, if you hold only four dice you can find five possible ways to proceed. Each possibility is created by the decision to re-roll a different die.

If you hold three dice, there are 10 ways to proceed because each die has 5 possibilities created. In the same way, you can also determine that holding two dice also yields 10 possibilities, holding one die yields five possibilities, and holding no dice at all has only one possibility. Therefore, there is a total of 1+5+10+10+5+1 which equals 32 possible strategies that may be adopted for the combination of 1-2-3-4-5 based on how many and which dice you decide to hold.

By repeating this exercise for each of the other possible 252 starting rolls, you can determine how many different ways the very first roll in a Yahtzee game can be played. That adds up to 4,368 different paths the game can take if you perform the calculation described above for each possible starting roll. Now that is a lot of different ways for Yahtzee to be played.

Extending the calculation to encompass the remainder of a Yahtzee game which is 13 hands of up to three rolls each, including placement in the scorecard and considering all the paths of rolls and strategies, you will find that there are something like 10 to the 135 which is 1 followed by 135 zeros. This is such an impressive number than even if all the zeros were shown to you wouldn’t even be able to grasp the size of this number. It is hard to understand the complexity of Yahtzee, but suffice it to say that you will never have the same game twice.

Yahtzee’s brilliant combination of simplicity in the rules, and variety of the game play is what makes it one of the world’s most popular games. Though it takes only a few minutes to learn, for the rest of your life you’ll never experience the same result twice. Given Yahtzee’s mind-boggling complexity, it may be difficult to realize that you can find an optimal way to play it, but you most certainly can.