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When you are playing with optimal strategy you are maximizing your expect score. This is, if you could play the optimal strategy perfectly, you would obtain the highest possible average score per game. Learning these rules will help you achieve this each and every time, although the chances are slim that you will do this every game. It is far more likely that you will be known as a great Yahtzee player by your friends and family.

When you are playing with others your goal may be a little different than playing perfectly. Specifically your goal may be just to win the game. You may in fact, have little regard to your actual score, as long as it is higher than the score of your opponents. If you were to play a large number of games against your opponents, recording each score then adding them up to determine who was the winner, then the strategy presented would be the one to consider. However, the pay of a single game is subject to many fluctuations in outcome. For example, if you are lucky you may get a Yahtzee early on. If you are unlucky, you may fail to get a Large Straight or the Upper Section bonus, despite nearly perfect play.

It is a fact that when playing one or more opponents and trying to achieve the highest score in the group; your strategy is likely to deviate somewhat from the optimal strategy you have learned. There is a difference in your opponent’s styles, and ultimately their results will often force you to adopt a different strategy, one that maximizes your chance of achieving a new goal of simply catching up to your competitors.

In Yahtzee, there are times where you will naturally get exactly what you need no matter what you do. It is important that you don’t get too caught up in the excitement because this is rare. Most likely you will always be fighting for what you need, and therefore, the strategy you have learned will keep you ahead in the game. However, there are times where you will not roll anything of what you need and you will constantly be trying to keep up with your opponents. This is the time where you may just want to keep up with their progress and watch what they do. If you find yourself behind your opponents you need to depart from the basic strategy and aggressively try to make up the difference.

You don’t need to overcompensate for your bad dice dramatically if it is early in the game. If you continue to play according to basic strategy and secure the Upper Section bonus, it may become evident that your opponent will not achieve his bonus and you will have automatically taken a lead. However, if you are still behind in the later hands of the game, be prepared to do almost anything. No matter what you need, just go for it. Maybe you’ll get lucky and hit back to back Yahtzees or anything else you need to go on and win the game.

On occasion, the quest for the Upper Section bonus will have to be abandoned when you are late in the game. The exact point at which this occurs is difficult to pinpoint because every game is different. The exact timing is a sensitive reaction to the hands remaining and the scoring boxes left to be filled.

Because in principle we don’t want to stop our chances of getting the Upper Section bonus of 35 points, it is usually preferable, if we have to put a zero somewhere on the scorecard, that it be put in the Lower Section. However, there are two cases in which a zero may properly be put in the Upper Section. The first is if you are comfortably ahead or have already clinched your 63 points and the bonus. The second time is if you are way behind. In either case, it is correct to score Aces, or Twos as zero before the 4 of a kind, Full House, or Large Straight. Similarly, Threes may be scored as zero before the Full House or the Large Straight. In all other cases, including those in which there is still an outside chance of making the Upper Section bonus, if an entry must be filled with zero putting it in the Lower Section is preferred.

With this in mind, if late in the game there are two Upper Section categories unfilled, and you need three of one type and four of the other to reach the bonus, your chance of succeeding is slim. In general, it is safe to say that with just a few hands remaining, if you are still behind in the Upper Section, then you probably won’t be getting the 25 point bonus. Therefore, you are better off concentrating on completing the other portions of the scorecard to your best advantage. One additional consideration comes into play when you are playing against opponents. When your primary goal is to win the game, it will be at times necessary to deviate slightly from this advice in order to optimize your likelihood of victory.

When it comes to those zero entries, it is not uncommon for one or more boxes to be filled with a zero at the end of the game. This is not a sign of poor play. In fact, it is almost to be expected. Even when playing the best you can, it is very likely that one or more entries will be filled with a zero. Now that you understand the order in which you can afford to put those zeros you still are likely to win, providing your dice cooperate. So, don’t worry if you must fill one in, because it is something that is common in this game, and will happen to everyone.

There are many end game considerations to make actually. With one unfilled score box there is one hand left to be played and usually you are shooting for a particular outcome. For example, you may need a Large Straight or four 2’s. Strategically, the play of the hand becomes fairly apparent. Assume your score card is complete except for the Large Straight. That is, you have one hand left to play, and if you can make a Large Straight you will get 40 points. However, if you don’t make the hand, you’ll have to take the zero points for this last box, since the rest of your scorecard is already full. Of course, you won’t always make the Large Straight. So, on average how many points is this unfilled category really worth?

Through expert analysis, it is shown that the Large Straight will be made 27% of the time, assuming there is perfect play in trying to achieve it. It may surprise you to learn that an unfilled Small Straight, although only worth 30 points, actually has a higher expected value. In instances where it is the Small Straight or nothing, you will succeed about 62% of the time. This means that an empty Small Straight box is worth more than an empty Large Straight box. Specifically, it is worth about 8 extra points to your total score if you are looking at averages of all the points in all of your games. So, if you are presented with a situation in which only the two boxes of Small and Large Straights are unfilled, and you must choose one of the to fill with 0, do you know what you should do?

You need to consider the game in which you are playing. Let’s assume you have just finished playing the 12th hand and the final outcome was 3-3-4-4-5. This is neither a Small Straight nor a Large Straight, but one of those two boxes must be filled with zero. If you fill Small Straight with 0, then the expected value for the last hand is 10.6 points, since the Large Straight is left unfilled. However, if you fill the Large Straight with zero, then the expected value for the last hand is 18.5 points since you only have the Small Straight to fill in. To maximize your expected score, the correct answer is to fill the Large Straight with zero and hope to fill the Small Straight on the final hand.

Putting a zero in a box with a potential of 40 points as opposed to one with 30 points may seem difficult to do, but you are far more likely to get the Small Straight as opposed to the Large Straight. That is why it is more valuable to preserve the Small Straight than the Large Straight late in the game.

When you are late in the game, there really is an order in which you should be placing your zeros if you find yourself in that situation. The key order to remember is Yahtzee, then 4 of a kind, Full House, and then the Large Straight. This is the proper thing to do not just on the last hand, but throughout the entire end game.

The Yahtzee end game is more than just getting to the finish line. In fact, the last few hands of the game are highly strategic. First, you’ll need to understand that with only a few score boxes left unfilled, you will need to aggressively try to make these specific hands. Second, because only a few scoring boxes are left open, you may on occasion end up with a hand that does not correspond to any of the available categories.

When this happens you may be forced to fill in a zero somewhere on the score card. Knowing the proper order in which you should fill categories with the zero is crucial to you winning the game. In identical situations, with one or two hands left to play, a skilled end game player can wind up with 10 or more extra points than an unskilled player. The difference is knowing the proper strategy relative to your position.

You are generally playing the end game with the last few unfilled categories in mind. At this point, instead of taking what the dice give to you as you had played in the opening game, it is necessary to force the issue. Desperate measures will sometimes become necessary. At this point, you will have to scramble as necessary to optimize your last few rolls.

For example, if you have only the Large Straight and 4 of a kind left to fill, and the first roll is 1-2-4-5-6 you should keep the 2-4-5-6 or the 1-2-4-5 and try for the Large Straight. Early in the game, you would never do this. But with time running out at the end of the game, it is now or never to fill the remaining open score boxes. Needing only one more number to make the Large Straight forces you to try for it at this point. Another example is only the Fours and Aces boxes are unfilled with no prior Yahtzee, and you roll 2-4-5-5-6. At this point you should keep only the 4. This is a horrible play early on in the game, but in the end game, you are desperately in need of 4’s and therefore the proper play.

As you can see, you may do a lot of things different in the end game then you would do in the beginning. Although it may seem like there is not a lot options involved, you would be surprised at how different strategies lead players on different paths. The point of this strategy is to learn the right path and then be able to take it so that you will most likely come out the victor of the games in which you play.

If you are behind in the Upper Section, then the strategy is to try to make up the deficit as quickly as possible before the later part of the game. Thanks to the opening strategy, you will almost never be far behind in the Upper Section at this stage. In fact, you will usually be no more than 1 to 3 points in the hole. Remember, the opening strategy dictates that you don’t fill in the Fours, Fives, or Sixes score boxes unless three or more of the value are present. Hence you’ll never be in a situation where you have a score of 12 in the Sixes box, because that means there were only two Sixes thrown, and the hand would have been recorded in another manner.

That being said, sometimes you are just going to have bad luck and you will be in a situation in which, for example, the Aces or Tows categories have subpar scores. But these minimal deficits are easier to overcome. In these cases, you’ll be trying to get a high 4 of a kind to use in the Upper Section to get back to average at least. Still, overcoming an Upper Section deficit is usually not easy. There are times when the dice simply do not want to cooperate. However, even in these cases, your plan will be to stay as close for as long as possible. Provided you can overcome the deficit with a single high valued 4 of a kind, you will generally try to hand on to that chance.

There are three other major cases to consider in the middle game. Each of these corresponds to having opportunistically achieved a difficult hand early in the game. For instance, you have the Large Straight. If you successfully roll a Large Straight early in the game, you can concentrate almost exclusively on obtaining X of a kind hands. This is because the Small Straight is very common and can be picked up somewhere in the end of the game.

If you fill the 4 of a kind box early, as you should already know that the 4 of a kind should be placed in the Upper Section early in the game, but there will be cases in which you roll another 4 of a kind of the same type, you can concentrate more on the Upper Section and on the Large Straight.

If you make a Yahtzee early, you should play the remainder of the game slightly more aggressively in pursuit of the X of a kind hands. This is because each subsequent Yahtzee gets you a 100 point bonus. As such, you will generally hold 4 of a kinds, whether or not their corresponding upper entries are filled, in an effort to get another Yahtzee. All 3 of a kinds will also receive preferential treatment, especially those that show up on the first roll. Pairs are unlikely to improve to a Yahtzee, and usually are not given special treatment.

Generally, this is the time to stop going with the flow and begin to look at certain cards as a way of filling open categories. You should at the very least stay on par with the Upper Section and if you are behind you need to play for a high 4 of a kind to score in the Upper Section to help you catch up.

If you are ahead in the Upper Section, then you can afford to continue being aggressive in the middle game. For example, if you the Fours, Fives, and Sixes filled and you are six points ahead toward the Upper Section bonus. You can play more opportunistically to try to make the Lower Section combinations, knowing that even if you fail, you can afford to fill in the less than perfect tallies for the remaining upper categories and still have an excellent chance of making 63 points and earning the 35 point bonus.

Assuming the conditions above, consider a roll of 1-1-2-2-6 with your Full House unfilled. Early in the game you would have kept the pair of 2’s, in keeping with the rule of holding the higher pair. However, now it is better to keep the 1-1-2-2 because even if you don’t succeed in making the Full House, you can fill in the Aces entry without falling behind in the quest for the Upper Section bonus. In the same manner, you can sometimes try for a hand that you know will be difficult to make such as the 4 of a kind or the Large Straight, knowing that you can record the score in the Aces or Twos categories as a fallback.

Though you are playing more aggressively, it is important to never fall behind the pace to secure the Upper Section bonus. Therefore, you must maintain vigilance while being opportunistic at the same time. For example, assume you are ahead by 4 points you can be more aggressive with the Aces and Twos unfilled then with the Fours and Fives unfilled.

When you are on par in the Upper Section and you are set to make the bonus, it is best to play similar to the opening game with some strategic differences. First, it is necessary to consider what you have already filled. You will have to proceed a little more cautiously so as not to put the upper bonus at risk. As the game proceeds, falling behind in the Upper Section becomes more and more problematic, as opportunities to make up the shortfall begin to diminish. As such, when on par, continuing to record an average of three of each value in the Upper Section should be your main concern.

Remember that earlier in the game, with a Full House or a Small Straight plus a pair, preferential treatment was given to high values of dice. When in the middle game, and only on par for the upper bonus, preferential treatment should be given to any pair that corresponds to an unfilled upper category. As such, even lower pairs of Aces or Twos, if not already filled, should now be kept in the hopes of achieving three or more of that value.

As in the case of being ahead, if the opportunity presents itself to make a 4 of a kind of Fives or Sixes when the corresponding upper category is already filled, it is generally correct to take the risk to do this. This is especially true if Chance is still unfilled and can be used.

The middle game in Yahtzee is extremely complex to say the least. At the beginning of the game, the score card is empty so you know the situation you face, and at the end of the game the score card is full so you know exactly what you need to get. In between though, the score card can be partially filled in a myriad of possible ways. In fact, after about seven hands have been played, there are more than 1,700 different ways in which the score card can be filled, considering only the categories themselves and not the scores that they contain.

Due to the level of complexity, it is simpler to discuss the middle part of a Yahtzee game in general themes than it is to cite concrete examples. The variety in the game at this point makes the study of specific examples rather contrived, since it is incredibly unlikely that you will ever find two middle games that are identical to each other.

With this in mind, you still need to consider some of the strategic themes involved in playing the middle portion of a Yahtzee game. Rather than be bound by rigid rules as in the operating game, you will try to get a “feel” for how to best approach this most challenging part of Yahtzee.

One of the most important themes of the middle game strategy is transition and planning. AS you already know, it is important to “go with the flow” in the beginning of the game. But whereas the dice dictate how you play and score your hands early in the game, in the middle it is necessary to start planning for the rest of the game. You still allow the dice to dictate your play but only up to a certain point. With several boxes already filled, you need to carefully assess the open categories as you play your hands, allowing a bias in the strategy toward filling the open spots.

The tack you take in the middle game depends to a large degree on your status in the Upper Section. Specifically, you need to look at whether you are ahead on points, on par, or behind in achieving the gal of 63 points. To figure out your status in the Upper Section, you should add the values in the boxes that are already filled. Then calculate a sum for the values in the unfilled categories by assuming that you will get three of each number that is left. Add the two totals together to see where you stand. If you land at 63 points, you are on par. At the same time, if you have more than 63 points you are ahead, and if you have less than 63 points then you are behind.

For example, if you have 48 points in the Aces, Fours, Fives and Sixes categories, and that the Twos and Threes are unfilled, you will need three 3’s, and three 3’s to get the 15 points you need to reach 63. This is planning ahead and it is something that is essential to the game of Yahtzee.

In all games, even Yahtzee, everyone will have bad luck sometime. It could be that you end your hands with 1-2-3-4-6 or 1-1-2-2-3. How should you enter these tough luck hands into the score card? This is a source of serious error among players. Never fear though, here are some answers for you.

When a tough luck hand doesn’t qualify for anything, you always have the option to use the Chance box. However, even with Chance there are certain rules to follow. It is most favorable to use Chance when you have a score of 22 or more. Anything fewer than 19 is not recommended at all, which is because most likely you will have a few 1’s or 2’s you can score instead. For example, if you have the hand 1-1-3-3-4 the best way to score it is as 2 points for Aces. The hand doesn’t equal a chance total of 22 or more, but it does contain two 1’s, so that is the proper way to score it.

How about 2-2-4-6-6? The Chance total is only 20 and there are no 1’s. Next on the list is two 2’s, so it should be scored as 4 points in the 2’s score box. A hand of 1-3-4-4-5 should be scored as a mere 1 point in the Aces category. The hands of 2-3-5-6-6, 1-4-4-6-6 or 2-3-4-4-6 should each be scored as chance and so forth.

It is important to know why a hand such as 1-1-2-2-3 is better scored as two Aces rather than two 2’s. The reason is that later in the game, it is easier to make up a smaller deficit toward the Upper Section bonus. Recalling that you need to average three times the number for each value, filling in the Aces entry with only 2 points (for two Aces) has a deficit of 1, while filling in the 2’s entry with 4 points (for two 2’s) has a deficit of 2.

An open Chance box provides a lot of valuable flexibility. You get a feeling for this when you consider that Chance should never be filled early unless the total is 19 or more. Even filling in the Aces box with a value of 1 is preferable to exhausting Chance early. Late in the game, when you are trying to make your remaining open combinations, having the Chance option open will be a big asset. It is a common mistake for players to fill in Chance too early with a low total. Finally, you need to note that you should never fill in the Upper Section categories of 4’s, 5’s, or 6’s with less than three of the number. Once again, the goal of 63 points and the Upper Section bonus is extremely important to your overall game.

To summarize hands one through four, it is imperative that you be flexible early on, be opportunistic with regard to taking shots at getting the high scoring tough hands, and save your Chance for as late in the game as possible. It is the only category in which the value rises as the game progresses.

At the same time that we’re trying to get ahead in the Upper Section, you will be filling in the Lower Section on an opportunistic basis. If you stumble into a Large Straight or a Yahtzee, you should always use it as such in the score card. It is the same for final rolls that result in a Small Straight or Full House.

When it comes to the 3 of a kind and 4 of a kind, there is a special relationship between them as there is between the Small Straight and the Large Straight. The primary reason for not filling in the 3 of a kind and Small Straight boxes too early is their frequency of occurrence. Since they are common hands, there is no great rush to complete them. However, there is a more subtle reason for keeping them open. Remember that their counterparts, the 4 of a kind and the Large Straight are among the most difficult hands to obtain. By taking advantage of the relationship between these hands, you can increase your chances of completing the more difficult combinations.

When you look at the straights, you need to remember that as long as the Small Straight is unfilled, you have great flexibility in going for the Large Straight. Usually, when trying for the Large Straight, you’ll hold a Small Straight during the play of a hand. Now, if the Large Straight doesn’t come in, the option to fill the Small Straight works as a good fall back. However, once you’ve filled the Small Straight box, the quest for the Large Straight becomes more problematic. With the Small Straight filled, you’re in a bind if you go for the Large Straight and fail.

The lesson to be learned is not to jump at the chance to fill in a 3 of a kind, or Small Straight. Flexibility in Yahtzee is extremely important. If you can keep these boxes open until their counterparts are filled, you will maintain maximum flexibility.