# King of Tokyo: Math, Mayhem, and Blown Minds

There is an regular comparison that gets batted around the gaming world surrounding the game *King of Tokyo*. It is common, in gaming circles, to hear *King of Tokyo* be described as similar to (if not the same as) a classic dice game dating back to before 1940. Both require rolling dice, holding the ones you want and re-rolling to try and get your perfect hand. Both are games of probability at their heart and both require you to play the odds to some degree. That classic game is, of course, *Yahtzee. *

I have to admit that I believe saying *King of Tokyo *is the same as *Yahtzee *is a bit unfair. It feels like saying, to use a random simile, a Kobe Steak is similar to supermarket value beef mince.

I thought, now hoping to get back into writing regularly for this blog, I would kick start the analysis articles with something that would be a quick look into the mathematics behind *King of Tokyo *as a game. There are whole papers that go into the various different types of roll in *Yahtzee *and how you should play. These go into the odds for each type of result and I believed (foolishly) the same kind of mentality could be used for *King of Tokyo*. Granted there is a difference in the number of dice (*Yahtzee *has five dice, whereas *King of Tokyo* has six to eight) but that is just a matter of scale, right?

After an evening and a half trying to wrap my head around the numbers, I can tell you that it does not work like that. Most of the time when playing *Yahtzee*, you are looking to optimise for very specific things, which makes the mathematics of working out the odds difficult, but the end goals are fixed. They remain the same. No matter how difficult the math, you know there are only 15 different ways of ultimately scoring your hand.

Now – when playing *King of Tokyo* we remove those fixed end goals as they become adaptive per roll. They are constantly changing and moving based on what you are trying to achieve as a player in that turn. This makes it a game with a ridiculous number of possible outcomes. Yes, you ultimately want to either win by victory points or by conquest (being the last monster standing) but how you get there has billions of combinations.

If you remove the restrictions around *Yahtzee *and keep in the same number of dice it becomes complicated enough, but try adding an additional die to the base five, constantly changing circumstantial goals (they are always changing based on the environment around the table) and add in the option to occasionally roll with 7 or 8 dice.

One of the countless reasons I like games is because probability is a good place for an amateur mathematics fan like myself to geek out.

That being said, *King of Tokyo *is a probability nightmare. There are too many moving parts and too many potential strategies to be able to map them out in an easy way. Instead, it is some psychoanalytical calculus bomb that makes it too difficult to even know where to begin.

Let me show you what I mean –

## The Calculus Nightmare

So if we look at the odds of rolling, as an example, six hearts in *King of Tokyo *(because we have just come out of Tokyo and want a lot of health), then we are looking at a fairly easy calculation. What are our odds of getting one heart? We have a 1/6 chance. How many dice do we have? We have 6 dice. That makes our calculation:

1/6 x 1/6 x 1/6 x 1/6 x 1/6 x 1/6 = 1/46,656

Pretty low odds, but we have two re-rolls to get what we want. Let’s say we re-roll three, then our odds are:

1/6 x 1/6 x 1/6 = 1/216

Better odds of getting what we want. If we went on to re-roll one, we would get a 1/6 chance of getting what we want.

Now let’s say we only want five hearts, and the other one can be anything? Well, we calculate five, and then waste one dice, surely? But what if we want it to be something specific? Now let’s say we want two of one thing, two of another thing, and two of a third? What are the odds on getting what we want on re-rolls? What if we were to add in additional dice, or be able to manipulate them in accordance with cards? What if you need to roll constantly changing results due to cards that are coming out each turn?

Just working out the basic combinations is difficult as, unlike in *Yahtzee*, almost every single combination of dice is a valid hand in its entirety under constantly adaptive goals, and often in multiple different ways. There are a few exceptions, especially around victory points, but a miss in *King of Tokyo *feels more redeemable than a miss in *Yahtzee *due to the flexibility of the strategies at play. You are always able to get a Chance in *Yahtzee*, but in *King of Tokyo *those dice results matter more and can help influence any number of strategies across any number of turns. I don’t know, maybe it is me oversimplifying *Yahtzee* but, for me, *King of Tokyo *offers more options per roll.

There are, in fact, 252 unique combinations of results you can get in *Yahtzee*, when you have five dice, and that is when you take the duplicate results out of the 7,776 possible rolls (so 5,5,1 and 5,1,5 and 1,5,5 are counted as one unique set as they all have 2×5 and 1×1). If we compare that to *King of Tokyo*, when you have 6 dice, that goes up to 462 unique combinations out of 46,656 possible rolls. If you unlock the additional two dice, that goes up 1,287 unique combinations of dice to out of 279,936 possible rolls!

There are 66 Power Cards in the game. If you buy three of them that gives you 45,760 different combinations in what those three cards could be. If you buy four it gives you 720,720 different combinations in those four cards. You may buy 5 (unlikely, but possible), and so end up with 8,936,928 different combinations.

Add those odds and combinations to the dice and mind blown…and that is before any re-rolls.

## So What Does This Mean?

I think what this means is that, by comparing *King of Tokyo *to *Yahtzee* (something I have been guilty of in the past) we run the risk of doing the game a massive injustice. Where there are, for all intents and purposes, 252 different unique ways a roll can go in *Yahtzee*, there are potentially billions of different ways a turn can go in *King of Tokyo *once additional dice and cards are taken into account.

What *King of Tokyo *is, is an evolution of some of the same mechanics as *Yahtzee* whist doing more towards both providing the player with options and making the game more replayable. It is this, this advanced nature, that makes it both interesting and highly fun to play. The two games share similarities, but they are also incredibly different. Plus, *King of Tokyo* has monsters, which, in my opinion, beats an abstract dice game any day of the week!

It is also this complexity, however, that makes it difficult to analyse for the board game beardy gamer. Analysis of a game like *King of Tokyo *is not simple and can easily (oh so easily!) get out of hand incredibly quickly.

So, there we have it. An article that was meant to be an analysis turning into a reason as to why the game can’t easily be analysed.

What’s your take on the subject? What’s your opinion of the *King of Tokyo/Yahtzee *debate? Let me know your thoughts in the comments below.

Thank you for your article. I must say, I never thought about comparing King of Tokyo to Yahtzee, but it is interesting to see the probabilities of rolling the various dice combinations you might need. I love the game, or rather its sibling King of New York, because there is more to than just dice rolling, unlike Yahtzee which is a roll-and-write. KoT (and KoNY) allow you to buy extra powers, you have to decide when to retreat, and in a 3 or 4 player game you can sort of team up against the person in the lead. Anyway, as I say, it’s an interesting perspective, so thank you for sharing.

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Would you recommend KoNY over KoT?

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Thanks for the analysis Luke! I always enjoy trying to keep up with you when you’re in Probability mode!

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Haha thanks, I think! It’s a nice area to geek out in to be honest, but as only an amateur mathematician it gets my thought processes in knots at times!

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Plus giant monsters. Honestly, Yahtzee doesn’t have Godzilla. I think you’re spot on that the mechanics are similar and then they change. Yahtzee is fun, but I think I’d play KoT over it now if given the choice. That said, have you played Favor of the Pharoah? Similar mechanics again, even similar unlocking of extra dice, however a fair bit of strategy in choosing what you’re taking. I call FotP “Yahtzee with superpowers” personally.

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I haven’t even heard of Favor of the Pharaoh! That sounds really cool (as soon as I’ve finished writing this comment I’m going to look it up on BGG) – I’m guessing you recommend it?

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Yes. Fun for up to four players. Multiple options for the different βpowersβ you play with. Plays relatively quickly especially once you know it. Neat components. Not too many pyramids you can βbuildβ with dice.

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I really liked reading this. Looked your insights into the game. The changing circumstances aren’t something people usually consider when they talk about KoT. Yahtzee sort of has that, based on what you’ve scored already, but not near the level as Tokyo.

Thanks!

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Ummm….I was told there would be no math.

(my head hurts :P)

Great post!!

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Cheers mate!

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