Certain hands require a bit more flexibility in how you treat them. That is, they don’t always adhere to the fundamental rules that have been set down so far. By learning how each hand can and cannot work for you, the better you will be at the game of Yahtzee.

A decision that comes up frequently is how to play two pair. For example, if it is early on in the game and the roll is 2-6-2-6-5 you may wonder if you should keep both pairs and try for the Full House or if you should keep just the higher pair and re-roll three dice. Since Yahtzee obviously has no category for two pair, keeping both means that you are going all out for the Full House, which is a Lower Section category. In keeping with one of the fundamental themes of getting ahead in the Upper Section, it is better to keep only the higher pair in this situation early on in the game. If the score box for the higher pair is already filled, it is still better to keep the lower pair than to keep both. Keeping the single pair early allows much greater opportunity at the time that it is most beneficial to be opportunistic.

When confronted with a roll that contains both a pair and a Small Straight, assuming that the Small Straight, Large Straight, and the Upper Section entry corresponding to the pair are all unfilled, several factors have to be considered. Among them, the type of Small Straight, how many re-rolls remain, and the value of the pair all need to be looked at. Based on the fact that a Small Straight will occur relatively frequently and a Large Straight will not, the following rules will summarize the best strategy for this situation. After the first roll, you should keep the pair only if it is high such as 4’s or better and the Small Straight is 1-2-3-4 or 3-4-5-6. Otherwise you should keep the Small Straight. Also, you should keep all Small Straights of 2-3-4-5 regardless of the value of the pair. After the second roll, it is automatic that you want to keep all of the Small Straights.

The next dilemma you may face is if you encounter the Small Straight and a pair but your Small Straight is already filled. Do you then keep the pair or take an early shot at the Large Straight? Well, if the Upper Section category corresponding to the pair is open you should keep the pair unless you have 2-3-4-5. This is true after either the first or the second roll. If the Upper Section category corresponding to the pair is filled you should keep the highest single die after the first roll, and keep all straights after the second roll.

What if the first roll produces a natural Full House? This might surprise you but on the first roll of a hand early in the game when the corresponding Upper Section and Full House boxes are unfilled, it is best to break the guaranteed Full House and roll for the 3 of the kind. The reason is the powerful potential of this hand to make the 4 of a kind or Yahtzee on one of the two re-rolls. Faced with the same decision after the second roll, you should break the Full House again if the 3 of a kind is 4’s or higher, even though there’s only one roll left to improve the hand. Again, the intent is to improve the 4 of a kind to get ahead in the Upper Section or a Yahtzee. When a Full House is made after the final roll, you should always score it as such on your score card. Finally, if you have made a Full House and the Upper Section category corresponding to the 3 of a kind is filled, you should still break it to roll the 3 of a kind after the first roll, but hold the guaranteed Full House after the second roll. In this situation where the Lower Section 4 of a kind is also filled, always keep the Full House.

At the beginning of a Yahtzee game, the scorecard is empty which means you have tons of possibilities. Given what you now know about how many directions a game can take, it stands to reason that your early choices will have a great significance on your prospects for the entire game. Now, it is important to learn how you should approach the beginning of a game of Yahtzee.

The proper strategy at a game’s outset can be summarized by three major themes:

• Follow the dice

• Get ahead in the Upper Section (or at least stay on average)

• Be opportunistic with “difficult” hands such as Yahtzee, Large Straight, and the 4 of a kind.

Following the dice simply means that you should take what the dice give to you. You need to remember that throughout the game you’ll be trying to make hands that are either X of a kinds or straights. Early in the game, it’s not important to pick out specific categories and try to complete them. Rather, you should take the good scores that present themselves. The idea is to go with the flow. If a roll contains three or more of any particular value, it’s generally best to hold these dice. For example, if your first roll of the game is 3-6-6-6-4, the best play here is the obvious one. Hold the three 6’s. If the first roll is 1-3-4-5-6 then go with the small straight and hope to get a large one. There is no reason to force the issue when the game is new. Just enjoy the luxury of being able to get what you are given.

When it comes to getting ahead in the Upper Section, it should be done as quickly as possible. This improves your chances of netting the 35-poitn bonus for making a total of 63 points or more. Remember that 63 points represents an average of three numbers in each category. Therefore, if you can get ahead by filling in one or more of the Upper Section categories with four of a particular number you will be well on your way to securing the bonus points at the conclusion of the game. Final hand combinations that contain 6’s cause a lot of dilemmas if you don’t get them up top as soon as possible. A lot of 6-6-6-6-x should be scored in the Upper Section if possible. In fact, any 4 of a kind early in the game is best applied to fill a box in the Upper Section, even four 5’s or 6’s, which most people automatically use for their 4 of a kind. When it comes to 3 of a kinds, this had too is best entered in the Upper Section instead of a 3 of a kind. Remember that the 3 of a kind is fairly common so you will likely have other opportunities in the game to fill this spot. Early in the game 3 of a kinds are best applied to fill the Upper Section, which keeps you on pace to earn the 35 point bonus.

Being opportunistic is the last theme of strategy that you need to recognize. Aside from Yahtzee, the two toughest boxes to fill in the Lower Section are the 4 of a kind and the Large Straight. With respect to the 4 of a kind, remember that the score for this hand is the total of all five dice. Therefore, a hand of 1-1-1-1-4, which is worth 8 as a 4 of a kind, is far inferior to a hand of 6-6-6-6-4, which is worth 28 as a 4 of a kind. In the early stages, consider filling the 4 of a kind score box only after the corresponding Upper Section box is filled.

You need to remember that there are only two straight hands, the Small and the Large. The Small Straight is relatively easy to make and there are many opportunities to fill this box during the course of most games. Therefore, you should never keep three dice of a Small Straight early on in the game. For example, it is not wise to keep a 3-4-5 hoping to get a 6 or a 2.

Making the Large Straight is considerably more difficult. It stands to reason then if a hand with a strong Large Straight potential arises, you should actually go for it. For example, if you roll a 2-3-4-5-5 you have a far better chance to go for the Large Straight rather than keep the two 5’s. You may notice that this possible straight is “open ended” which means that it can be completed on either side with a 1 or a 6. In a “closed end” situation such as a 3-4-5-5-6, it would be to your benefit to not go for the Large Straight and instead keep the 5-5.

To better understand how a perfect Yahtzee player performs, there was actually a time when a computer was encoded to play 100,000 games using the optimal strategy. Basically the following information was determined:

The high values for both the small and large straights are expected and worth about 30 points per game on average. This has to do with their high frequency of occurrence and their high scoring values of 30 and 40 points respectively. In fact the small straight only has about a 2% chance of being filled on your score card as a 0. The Yahtzee category, despite being relatively rare, has a value of about 25 points per game basically due to its high scoring value for points for the first Yahtzee and the additional 100 bonus points you get for every Yahtzee after the first one.

The computer also noted that the first six entries, Aces through Sixes all have the same averages, even though there is a slight chance of getting more Fives and Sixes, each with an average score in excess of three times the number, compared to the Aces and Twos. The Aces and Twos each have an average score of less than three times the number.

Besides Yahtzee, the category most likely to get a zero marked down for it is the four of a kind. This is due primarily to its low frequency of occurrence. In fact, in approximately one in every three games you will most likely have a zero in the score box. Of course if the dice always did what you wanted you would never have to fill any score box with zero. However, that is highly unlikely as well. Even with flawless strategy it is sometimes unavoidable as well as inevitable that you’ll eventually end up playing with dice that simply do not want to cooperate.

It should be no surprise to anyone that Yahtzee is indeed the hardest hand to make. Also not surprising, especially to those who have played the game before, is that there is a high frequency of assigning a zero to the Large Straight box as well. Also interesting to note that in roughly 10% of the games you play, the Aces box is going to be filled with a zero as well.

As a game progresses from the first hand to the 13th hand, the expected value of all the score boxes decreases. The more hands you roll the more that number decreases. What this means is that the longer a particular box remains unfilled, the lower its average score is likely to be. The rate at which the expected scores of the entries fall is closely related to the difficulty in achieving them. Therefore, the Large Straight and Yahtzee entries, with their consistent drop offs in averages, are somewhat difficult to hit in any point in the game.

The highest possible score in the game of Yahtzee is 1,575 which would require making 13 straight Yahtzees. The lowest possible score is 5 which can only be achieved by purposely filling in every box with zero except the Chance box where you are required to put in the total of your dice which has a lowest possible value of 5.

Basically, if you are like most people who are competitive, you are always looking to improve your score in any game. In this particular instance, you are looking to improve your Yahtzee score. The way to do so is to play as close as possible to the optimal strategy as you can get.

In mathematical terms, “optimal strategy” is the method of play that maximizes the expected score for a game. In a player’s terms, it is simply the best way to play that tells you the correct play for every possible position in the game. If properly instructed, a computer can already do this, and would eventually play a perfect game of Yahtzee. However, there are two problems with this. The first is time. Even the most sophisticated computers can’t play fast enough to crunch the numbers in anything approaching a reasonable time period. The second problem is that even if a computer did play a perfect game, would you even realize it? Since in Yahtzee we do not know in advance what we are actually trying to accomplish, it is hard to know the outcome we are going to get.

In order to get the optimal strategy for Yahtzee, each possible position in every single path of the game that it could take would have to be considered. Yet the greatest counting tool, the computer, wouldn’t even be effective enough to do this. That means that there is only one way to do this. The number of necessary calculations has to be reduced to a more reasonable figure. Basically you have to analyze the game in reverse. By actually starting at the point where all the score boxes have been filled and working back in time towards the first roll it is possible for perfect choices to be made when faced with any decision encountered during a Yahtzee game. This is called “dynamic programming” which thankfully has already been done by the experts in the game of Yahtzee. Otherwise, it would take you about 50,000 hours of non-stop work to figure it out.

The end result of all this mathematics and computations is that with optimal play your expected score in order to achieve an optimal game would be 254.6 points. There are no approximations in this number. No matter where you are in the game or what the current state of your score card, the perfect play can always be determined. It is truly the precise recipe for playing each and every hand in every imaginable situation.

The value of 254.6 points is also important. Knowing that it comes about by playing optimally means that there is no possible way that your average score in the long run can ever exceed 254.6. If someone claims to have an average score of 300, either they haven’t played many games or they are a little prone to exaggeration.

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.

The best thing about Yahtzee is that every game is different. No matter what you do you will never play the same game twice. A lot of that has to do with the statistics of the game. It just means that there are numbers and some odds involved in this game, although you certainly don’t have to memorize them. However, they do give you a better understanding of the underlying complexity of the game that seems so simply when you play it.

Let’s start with the roll of the dice. When you roll five dice, you will see many different outcomes. Mathematicians call each of these outcomes a “combination.” A combination is a grouping of elements in which order does not matter. If we wanted to, we could list all of the possible combinations of the dice, but since there are 252 distinct combinations that may be rolled with five dice, let’s just say that it would talk a while! Basically the combinations would range from 1-1-1-1-1 to 6-6-6-6-6.

Not all of these combinations are equally likely though. For example, if you look at the last two combinations 5-6-6-6-6 and 6-6-6-6-6, you may think that the Yahtzee is harder to get and therefore less likely. If you think that then you would be correct. The fact is to get a Yahtzee you would have to roll each of the same number, and in this example it would be a 6. On the other hand, 5-6-6-6-6 has five possible ways of occurring. One way is for one die to be a 5 and the rest to be the 6’s. Another way is for the second die to be a 5 and the rest 6’s, and so on until you have five ways of rolling that combination. So, for a roll of five dice, the hand of 5-6-6-6-6 is in fact five times more likely than the hand of 6-6-6-6-6.

Essentially you are saying that the hand of 5-6-6-6-6 has five times the weight of 6-6-6-6-6. A weight can be similarly be assigned to each of the other 252 combinations. For example, a hand of 1-2-3-4-5 is 120 times more likely as a hand of 6-6-6-6-6 and so forth.

Again, this may sound confusing to you, but it is actually just common sense. If you have more numbers you can roll then you have more chances to roll them. That is why it seems so easy to fill up the Upper Section of the score card as compared to the Lower Section.

A Yahtzee is obviously the best hand in the game of Yahtzee. Provided that the score box for the Yahtzee is still empty your first Yahtzee is scored in that box and you earn 50 points. Subsequent Yahtzees made in the same game earn a 100-point bonus. They are usually tallied up by placing a check mark in the appropriate spot in the Yahtzee score box. The scorecard only has enough room for three checkmarks, but you are allowed to score as many Yahtzees as you can make during the course of the game.

If you have any subsequent Yahtzees you must also score them in an open score box, which often leads to confusion regarding the rules governing this requirement. Since Yahtzee always qualifies for many of the other score boxes such as 3-of-a-kind, 4-of-a-kind, Full House, or Chance in the Lower Section, as well the corresponding numbers in the Upper Section. For example, if you roll a Yahtzee made up of 5’s and your Fives box is empty on the Upper Section you can then score five 5’s in the Fives Box for a total of 25. In addition to that box score you will also earn the 100 point bonus on top of the points specified by that box’s scoring rules.

There are different rules in which a Yahtzee can be played. This can vary by game version or release, even though they come from the same manufacturer. The explanations are shown in the game rules, but will also be explained here. Of course, each group of players may also make house rules such as allowing a Yahtzee to be used as a Joker only if it was scored as 50, or never allowing a Yahtzee to function as a Full House. Regardless of which variation you play, none of these subtle rule differences alters the strategy advice that you should follow. That brings us back to where the Yahtzee can be scored.

Yahtzee may also be scored in either of the remaining two Lower Section categories, which are the Small and Large Straight; according to the games “Joker Rules” provided that the Yahtzee box is filled with either 50 or 0, and the corresponding Upper Section box is filled. If only non-corresponding Upper Section boxes remain, you have no choice but to fill one of them with a 0.

If you make a Yahtzee, but have already filled the Yahtzee box with -, you can score the hand in an open box in the manner described above. In this case, you earn only the score associated with the category that you fill. You may not score 50 points for a Yahtzee or the 100 point bonuses for subsequent Yahtzees.

One last note about scoring additional Yahtzees; though the rules may appear complicated the best move tends to be very straightforward if you score a second or third Yahtzee. If the Upper Section’s corresponding number is open it is almost always best to total up the 5 numbers and place it in the number that it corresponds with. For example, if you get a 3-3-3-3-3 and it is your second Yahtzee, and your Three box is open then take the 15 points. If it is not open the Joker rules will usually be in effect, so you should choose the open Lower Section score box that earns the most points between the ones that are left such as the 3 of a kind, 4 of a kind, Chance, Full House, Small Straight, or Large Straight.

Since there are only two parts of Yahtzee – the rolling of the dice and the scorecard – it is important that you fully understand how the scorecard works so that you can complete the game fairly.

There are two sections of the scorecard; the upper and the lower sections. The top of the scorecard, or the upper section, consists of six categories. They are the Aces, Twos, Threes, Fours, Fives, and Sixes. The scores in each of these categories are calculated by adding all of the dice that display the chose number in a finished hand. For example, if you finish a hand with a 2-2-2-4-6 and you choose to score the twos, you would then input a 6 into the scorecard in the Twos box of the upper section of the scorecard. Alternatively, if the player already has a score in the Twos box or only needs a score for the Fours and Sixes box they would score it as 4 in the Fours box and 6 in the Sixes box. If absolutely necessary, the hand could also be used to record a total of 0 in the Aces, Threes, or Fives box. A score must be recorded after each and every hand regardless if you made anything that you need. That is when the 0 may be used in boxes that you do not have a score for.

At the bottom of the upper section there is another section called the “bonus.” This bonus is very important to the game since a bonus of 35 points is awarded when the total in the upper section is at least 63. The value 63 represents an average of a three-die tally for each entry. That means that you have 3 Aces + 3 Twos + 3 Threes + 3 Fours + 3 Fives + 3 Sixes for a total of 63 points. It then stands to reason that if you are below average on one of the entries, you must achieve better than average results on another box in order to receive the 35 point bonus.

The bottom of the scorecard consists of seven categories. They include the 3-of-a-kind, 4-of-a-kind, Full House, Small Straight, Large Straight, Yahtzee, and Chance. The scoring for each of these categories varies so here is an example for each:

• 3-of-a-kind – A hand in which at least three dice show the same number. The score for this hand is the total of all five dice. For example if your final roll is 5-5-5-6-2 the total would be 23.

• 4-of-a-kind – A hand in which at least four or more dice show the same number. The score would again be the total of all five dice. For example, if you have a 4-4-4-4-3 the total would be a score of 19.

• Full House – A hand containing three dice showing one number and two dice showing another number. For example, a 2-2-2-5-5 is a Full House. The score for this hand is fixed at 25 points.

• Small Straight – A hand containing four dice in sequence such as 3-4-5-6. The score for this hand is fixed at 30 points.

• Large Straight – A hand containing five dice in sequence such as 2-3-4-5-6. The score for this hand is fixed at 40 points.

• Yahtzee – A hand containing 5-of-a-kind such as 6-6-6-6-6. The fixed score for a Yahtzee is 50. If you make more than one Yahtzee in the same game it must be used to fix another score box but it gives you an added bonus of 100 points.

• Chance – This is an “any hand goes” entry. The score for this hand is the total of all five dice. Any hand can be used as a chance and is usually used when you can’t find another box to make an entry in. For example, if your final hand shows a 2-2-4-5-6 it would be scored as a 19 in Chance if unable to be used anywhere else.

At the end of each hand, one score box must be filled. Once a box is filled, it can no longer be used. In order to get the highest final score, it is preferable to fill in each box with as high as a value as possible. But on some occasions, you will have one or possibly more score boxes filled with a value of zero.

For example, consider a position near the end of the game in which only two score boxes are open – Aces and Full House. Then you roll this hand – 4-4-4-5-6. That is a dilemma in the Yahtzee game because a score must be entered into either the Aces or Full House category, since the rules state that one score box must be filled after every hand. However, the combination does not equal a Full House, nor are there any Aces in the hand. Therefore, in this case, it is necessary to fill in a zero to either of the open score boxes. Granted you are going to lose some points, but you will have to do it as per the rules.

This may sound rather confusing but once you look at the scorecard you will clearly be able to figure out which score goes where. It is well mapped out and in no time at all you will be scoring like a professional.

While it is important to understand that people have come up with the optimal strategy and have actually found a solution to this game, most likely it won’t serve any practical purpose to your game. It is almost to complex to read, much less memorize. Even more so, it is too vast to even comprehend enough to actually apply it to the game of Yahtzee. Therefore, you actually have to learn the basic strategy in which to play the game so that you can reach your optimal strategy in real life.

The basic strategy of Yahtzee is an approximation of the optimal strategy. Its purpose is to mimic the optimal strategy while retaining sufficient simplicity so that it can be applied by regular people who are playing a regular game. Essentially it is a set of simple guidelines that grasp the essence of the optimal strategy but allow it to be put into practical use.

With the exception of Chance, each and every score card category can be considered either an “X of a kind” or a “straight.” All of the upper categories can be considered X of a kind hands since the aim is to get as many of each value, such as Aces, as possible. The 3 of a kind, 4 of a kind, and Yahtzee entries are clearly X of a kinds while the Small Straight and the Large Straight are clearly straights. A Full House is a 3 of a kind plus a pair. Understanding that the game of Yahtzee consists of only these two types of entries has important strategic ramifications. In particular, there are 10 different X of a kind categories, but just two categories of straights. Given this information, it seems logical that a winning strategy would embody a bias toward trying for an X of a kind hand.

Another fact you need to consider about the game is that higher valued dice are worth more than those with lower values. For example, three 6’s are more valuable than three 3’s. There are several reasons for this and not just the obvious either. First, hands such as 3 of a kinds, four of a kinds and Chance are scored by adding all of the dice together. So it’s best to make X of a kind hands with dice of a high value. Furthermore, in the Upper Section you are trying to score at least 63 points in order to get the 35 point bonus. The added importance of higher valued dice is obvious here. Getting ahead in the Sixes category puts you ahead of the average whereas getting an extra Ace only puts you one point ahead. In addition if you score less points in the Fives category for example, it is more difficult to overcome then if you score less than average in the Twos box.

The Chance entry is unique in that it is neither an X of a kind nor a straight. It’s simply the sum of all five dice. As such, it may be used any time with the knowledge that it will never have a zero score. When playing competitively it is never proper to purposely try for Chance unless you are at the end of the game. Instead, this entry serves as a catch-all in case things go wrong for you in the midst of a hand. In other words, winning strategy dictates that Chance should be reserved as a safety mechanism only for an unwanted outcome.