The Game of Risk - Numberphile

Mathematician and game enthusiast Marcus du Sautoy discusses the iconic game of Risk. More links & stuff in full description below ↓↓↓
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[@numberphile]

Around the World in Eighty Games (Amazon affiliate link): amzn.to/3snW2bD More videos with Marcus: bit.ly/Marcus_Numberphile

[@SirJimmySavileOBEKCSG]

Luvlay, luvlay!

[@TRUMAN_THE_TRUE_MAN]

Never asked + I’m 100% better than Numberphile ☠️💀

[@Bibibosh]

I wish they made a "Conways Game of Life" game. Like a game like life but in the game you score points and eat stuff.

[@14Penfold88]

I only watch James Grime videos

[@cavejohnson982]

Oh could you please do a Video on the Train game?

[@jansenart0]

Risk: the only reason any of us know what Kamchatka is.

[@bobrong9645]

With the exception of people who live there, I suppose.

[@asahearts1]

Same with Ukraine until recently. Most people who knew where it was probably knew about it from this or maybe a documentary about human trafficking. 😂

[@Evilyoo]

and Yaktkusk

[@Leonhardt_Nukryst]

Russia Mains in Hearts Of Iron IV:

[@hive_indicator318]

Not me. I know it because of Axis & Allies. But it's almost the same thing

[@qawi272]

His kids: „Maybe today we can beat dad!“ Him: „So the mathematics of topology…“

[@Shvetsario]

Nerds make games unfun

[@iamamish]

@@Shvetsario yeah, using their brains to create strategies, those damn nerds

[@behemoth9543]

@@Shvetsario Humans are designed to optimise the fun out of everything. Which is why we love games of chance and gambling so much - random chance is the only way to prevent optimal strategies from winning every time and removing the whole point of playing in the process.

[@Shvetsario]

@@behemoth9543 I actually hate games like that, like World of tanks for example, I shoot 5+ accurate shots at point blank and somehow they all "miss"

[@75338]

@@iamamishYeah, when my buddy calculated a no-risk win strategy for a Space Hulk boardgame mission it just wasn't fun anymore.

[@smartereveryday]

This video is beautifully shot. I also really enjoyed the stop motion!

[@ExtremeUnction1988]

You’re gettin smarter everyday!

[@Inuyasha10121]

Its funny that Marcus points out "Australasia is a weak position to hold because of its isolation." When I played with friends back in high school, we joked about "The Great Wall of Indonesia" strategy where someone would take Australasia and build a massive army in Indonesia that was super hard to get past, eventually Zerg rushing Asia if they were unchecked for long enough. Granted, if someone else took Asia you kinda get steamrolled, but it was such a nightmare to build up enough to finally breach the wall while also trying to fend of whoever had Europe. It almost always came down to someone taking Asia, trying to stay neutral in material with Australiasia, then once they took Europe they point to Indonesia and go "YOU IN PARTICULAR"

[@garrettgarcia2592]

Yeah he's just flat out wrong.

[@Inuyasha10121]

@@garrettgarcia2592 I wouldn't say he's "wrong," its more that there are social elements to games that certain purely mathematical models can't necessarily capture. Markov state models are phenomenal tools for teasing out interesting features of complex systems, but there isn't a Markov model for "games that me and my group of friends play." For instance, early when we were kinda learning the game, no one gave Australia a second thought because of the exact things he said in the video. Now, within my friend circle, taking Indonesia early is a terrible idea because everyone else turns to you and goes "NOT TODAY" specifically because we have history in the game of that event blowing up. I think for the claims made in the video, Marcus was fine.

[@GameBrigade]

​@@garrettgarcia2592he isn't. Australia turtle is very difficult to win in a real game of competent players.

[@xapheneon]

The big factor imo is that only one territory borders Australia, so that's the natural defensive position and holding it denies asia

[@GameBrigade]

@@xapheneon and it limits your options. Australia is bad

[@johndoe-rq1pu]

We've missed two huge strategic ideas here, both related to the number of players. The more players there are, the more difficult it will to conquer and hold a large continent. You're more likely to oppose multiple players. If you take Australia, you're likely to be the first one with a continent, because it's small and remote, and there's only one player bordering you so you can devote 100% of your army to that front. No one ever has reason to go through your territory to get anywhere else. Secondly, attacking does give you an advantage in that battle, but attacking is always a negative sum proposition, the best case is you lose no troops, but you are still dividing the same number of troops between more territories. So the real result is that the attacker will be better than the defender on average, but the 3rd-xth players will all be better than either the attacker of defender in the short term. Even if you attack and conquer a new territory, the bystander players will then have an army advantage to contest your new continent before you receive any benefits. This is all without thinking about the politics of conquering territories and making yourself appear strong. There is another unfortunate factor which is cards, which depending on your ruleset, possibly/probably render all other strategies irrelevant.

[@InsaneZeroG]

When factoring in cards, the attacking player (assuming they're) successful) isn't just better off long term (nominally) due to the extra territory, but also they get a card for doing so. The card economy, so to speak, is basically what decides games. Sets of cards (matching trio or one of each) can be redeemed by a player on their turn for extra troops for deployment. This redemption amount, in base rules, is progressive with the first redemption yielding 4, then 6, then 8, 10, 12, 15, and then 15+n*5 for each set after that (where n is sets after the 6th set). This not only accelerates the game, but because of another rule where a defeated player gives their cards to the player that conquered them, means that a player who, after redeeming for 40 troops defeats a player and takes their cards, could drop another 45 troops from a possible set and then take out another player and make 50 more and so on.

[@willemm9356]

@@InsaneZeroG I assume this video is talking about the European version, where this progressive redemption does not exist. 4, 6 or 8 cards for a set of 3 equal, depending on type, and 10 cards for one of each. (And a number of other differences as well)

[@Lotschi]

absolutely right about this! I love to take South-America if I can. It gives me a bit of extra troops and with a bit of luck I can later easily expand to north america. What I think is a shame is that online there are a lot of players being very passive, stacking troops in one continent for half am hour. It becomes more a game of patience than of strategy and that annoys me.

[@VoicelessRabbit]

Yes, I think they narrowed the scope of the math to basic 4 player games with no cards... Wish they would state those assumptions more upfront.

[@ewef9871]

there is also a meta. If everyone wants America, than it maybe much more worthwhile to go for africa

[@davidanderson2357]

One small correction (unless the rule has changed): the attacking player is not committed to seeing an attack all the way to a decisive conclusion; he can break off any time he wishes, which comes in handy if he sees that his attack is failing.

[@Elandil5]

Only applies to bigger army battles and I have seen plenty of times in my games where a friend with just one soldier would destroy my 15.

[@dannygjk]

@@Elandil5 not the point he made.

[@TornadoGames1]

yes and another one: the defender can choose whether to throw one die or two, which makes it nearly impossible to calculate the chance of anyone winning since it's strategy based

[@Cjnw]

Also, at around 4 minutes, the garrison army cannot attack; it would have been two dice each, not three for the attacker and two for the defender.

[@1dgram]

​@@TornadoGames1no, it just reduces the defender's chances further by making a sub-optimal choice

[@connormoriarty3410]

Great video, but I can't leave without giving praise to your editor who put together those animations/stop motion. They were FANTASTIC. Clearly a ton of time went into them. And they helped visualize and understand the topic. Keep it up!

[@numberphile]

👍🏻 to Pete

[@abydosianchulac2]

Not just helpful, but incredibly fun. I loved the little Australasia disco party!

[@lachychops2]

​@@numberphileGreat job Pete!

[@Varangian_af_Scaniae]

"Clearly a ton of time went into them" Sure telling an AI that you want a short clip of a toy soldier spinning around on a RISK board is hard work🙄

[@connormoriarty3410]

@@Varangian_af_Scaniae that’s not what they did but ok 🙄

[@jonsmith1956]

In my experience, turtling is usually the best strategy. Hold a continent and keep conquering one country per turn to keep gaining cards. Doesn't matter if you hold that country or not, in fact, it could be better to lose it so you can re-conquer it the next turn. Build up armies from continent points, build up armies from trade-ins, and expand when the time is right. When you can take out an opponent in one turn, do it, gain their cards, and trade in for more armies

[@georgegreene8744]

That makes it about conquering players Around the board rather than countries On the board -- kind of a different game -- also one that's harder on real-life friendships. Players can get in their feelings about WHO gets selected as a target.

[@blah2blah65]

@@georgegreene8744 Ideally the friends you game with will understand non-cooperative games are best played by using optimal strategies, which sometimes means ruthlessly attacking a single player despite other options. Now if every game results in attacking the same player and it is clear it isn't about optimization, then its time for tackling the obvious relational challenge.

[@georgegreene8744]

@@blah2blah65 That's an ideal That's not reality. Reality is that your friends are not the expert game theorists they think they are, and their claim that *several* opponents *all* "playing "optimal strategies" just *happened* to result in their all attacking you first, well -- that's just gaslighting.

[@TheJoemm]

@@georgegreene8744 You all have oversensitive friends. It’s a game …

[@reaganharder1480]

My observation is that full-turtling is a very bad strategy. My oldest brother typically full-turtles in australasia and I don't think I've ever seen him win. Exactly what is optimal strategy depends on the play of those around you, but establishing strong borders on a continent is almost always required for success, at least in my circle of friends. The other thing that you must be aware of is who borders you. Having strong borders with a strong enemy can be just as futile as weak borders with a weak enemy, and as you establish yourself as a world power it is highly likely that other players will conspire together to minimize the threat you pose.

[@SparkyRoosta]

3:58 Greenland can only throw two dice because they have to keep one army back to occupy Greenland

[@albank.5386]

Was looking for this comment. Also you have to invade (initially) with the number of armies corresponding to the number of dice thrown. Odd that the game guru wouldn't know these things

[@FurlogTheGiant]

Defensive 6 defeats Offensive 6

[@brandonfrancey5592]

@@albank.5386 I'm sure he does but like everything you need to simplify. If he had 50 armies there, you still can only roll a max of 3 die so the most common roll in Risk is 3 attacks vs 2 defence.

[@osbourn5772]

I think this was corrected in the video

[@hamiljohn]

Correct

[@LetsGetIntoItMedia]

Math(s) is cool and all, but that stop motion cotton explosion was 🔥

[@DaTux91]

Stop Motion Cotton Explosion is a great band name.

[@LetsGetIntoItMedia]

@@DaTux91 hahaha🤘🧑‍🎤👨‍🎤👩‍🎤🧶💥

[@cuseyeti_one8three]

The cards are quite important as well. If you do not attack, you cannot win a new territory and your turn ends without you receiving a card. If you take a territory every turn, you are guaranteed a set of matching cards after at most 5 turns. The extra armies received for completing a set increase each time and absolutely will influence the end game.

[@angarizna9337]

especially later in the game as the number of armies per matching set gets ridiculously high.

[@Queso2469]

My dad and I wrote a Monte Carlo simulator for Risk when I was a kid, with entire attack chain calculators and expected remaining units. It because pretty obvious pretty quickly that attacking was better at large unit counts, but the last couple rounds where defense wins dice ties had a pretty huge influence in non late game battles. Extended attack chains also quickly lose some of the advantage since you're losing a unit every time you move out of a country. The other super interesting thing about Risk is that you can stop attacking whenever you want, so your odds of winning a battle go up and down during the battle and you can change strategy mid turn.

[@styxrakash4639]

Video was weak but this was nice

[@1dgram]

I went a step further 20 years ago and wrote a program to calculate the markov chains to produce a table of probabilities.

[@1dgram]

I'm sure many people have come to the same conclusions over the years.

[@benneem]

An important factor is the multiplayer dynamics. You grow an advantage over an opponent by attacking them a lot in your turn, but you'd be making *both* of you weaker against a 3rd player. Ideally you want everyone else to destroy their armies fighting each-other and allow you to have large forces remaining to sweep the board.

[@davidbjacobs3598]

This is true. When it comes down to three players, whoever can hang back while the other two destroy each other typically winds up on top. On the flip side, if you have the ability to fully eliminate one player, that can often win the game as the cards you get will let you continue the attack, or reinforce the defense of your lost armies. But if you fail to eliminate them, the third player will now have an easy time doing so. (Sometimes, it's possible to "guard" certain opponent spaces to ensure you'll get the elimination later -- although this can be risky as well, if that opponent proves aggressive.)

[@AnttiBrax]

Multiplayer dynamics as in trying to affect your friends' actions with over the board politics, banter and dealing. 😅

[@bossman983]

This is why Australia being connected to Asia is so incredibly important to its strength. When the other three are fighting over the most contested continent (Asia), no one will want to waste their troops on a wall of armies (since you only need to defend one territory). The number of times myself or another player swept the board after spending half the game doing almost nothing Down Under is staggering.

[@FilipCordas]

@@bossman983 Problem is when you get continent players are more likely to start attacking you unless they hold a continent as well like with and since they attack on the strong point you lose a lot of troops more then you get by holding the continent. Cards are more important then the continent I won several games by holding siberia getting a card every turn and creating a big army. No one wants to attack since they will lose a bunch of troops and then after some of the continent holders attack each other you just take the single troop territory.

[@georgegreene8744]

Of course -- with more than 3 players RIsk is first and foremost a diplomatic game. That's why everybody I knew who played it in high school decided to switch to Diplomacy when they went to college.

[@manmanman2000]

Another important factor in attacking is that you get cards for a sucessul attack which you then can exchange for additional armies

[@robcannon5]

You always want to take at least one Territory on your turn. I usually like to only take one or per turn and then wait till i can get the reinforcements before launching a big attack

[@Lightning_Lance]

"defending" usually means exchanging a territory back and forth with a player who has a neutral/favorable disposition (usually in Asia) but otherwise hanging back, and "attacking" would mean doing more than that and actually trying to ruin someone's continent bonus or conquer it for yourself

[@robertewalt7789]

Going for opponent’s cards is the strategy that worked best for me. Wait until your neighbor has four or five. If you can wipe him out, these four or five added to your cards means you can get more troops, which you can place and maybe attack another opponent.

[@AndreSomers]

@@robertewalt7789hmmm? Is that a house rule? I don’t think that’s an official game rule.

[@robertewalt7789]

@andre No, getting the opponent’s cards when you eliminate him is an official rule. The strategy I learned is to attack an opponent with 4 or 5 cards. The official rule says you have to turn in for armies if you start have more than five. Adding the the cards from the conquered opponent to you may give you two sets to turn in - maybe a huge number of armies.

[@staticwombat]

This is fascinating. I few years ago I became curious about whether defenders REALLY had an advantage, when it felt attackers did better. I'm awful at stats, so I set up a Monte Carlo simulation that ran 3 attackers vs 2 defenders a million times and found that attackers would win 7.8% more than defenders (ie, for 1000 defender wins, there would be 1078 attacker wins). It was a really unexpected outcome. One of the statements I made was that a monte carlo simulation was unnecessary, as you could figure this out using stats, so I'm glad to find the challenge wasn't quite as simple as I'd originally thought! EDIT: Well, I never thought I'd revisit that project, but turns out my code from last time was wrong. For every 1000 defender wins, there will be around 1268 attacker wins. That's a significant advantage).

[@clausewitzianwar]

The "stop motion" animation in this video is really fun

[@starrmayhem]

there's another game called diplomacy which i think its topology is an interesting discussion. that game doesn't have any dice or any random elements, on top of that everyone makes a move at once, so the only advantage one has is the positional advantage of the topology. it would be interesting to know the chances of winning for each starting country and how would it affect decisions.

[@FedericoLatini]

More game theory than statistics

[@rogerkearns8094]

The most difficult win seems to be Italy's.

[@GeekRedux]

So much of Diplomacy depends on personal interactions that I think calculating these chances would be impossible.

[@QuantumHistorian]

@@GeekRedux Yeah, the entire point of diplomacy is the social dynamics of it. The topology is secondary to the game-theoretical aspect of who to ally with whom when, and when to betray.

[@florianlipp5452]

The map of Diplomacy is such that it has an in-built mechanism to create draws. (The famous "great stalemate line" running from St.Petersburg to North Africa). A player sitting firmly on one side of this line can always defend it but can always be squeezed out of centers on the other side of the line. And as there are exactly 17 centers on both sides of the line (and you need 18 centers to win) the game is pretty drawish.

[@Intrafacial86]

I remember a version of that railway game where it was based in German territories (possibly the original version?). I was against 5 other players (iirc) and it was my first time playing the game, but I actually ended up blowing them all out of the water because, while they went head to head against eachother competing for a few long connections, I stacked up on multiple short railways that never interfered with any of their plans. Nobody (not even me really lol) had any idea how well I was doing until the game was almost over, and even then it wasn’t until the scores were being tallied that we were like “. . . Oh.” I ended up having around twice as many points as the second highest scoring player.

[@BedrockBlocker]

One very important factor of this game are cards, which you get for attacking countries, and if you defeat a player you get their cards. You can turn in cards to deploy a number of troops. There are two widely used risk variants, one where the card rewards are constant, and one where they increase linearly, reaching 50 after the 6th-or-so turn-in. In the first variant, the game is very defensive and it's advantageous to hold a continent with a large troop bonus, like NA or Europe and the game not seldomly fizzles out to some kind of game-theoretic stalemate. In the second variant, the game is super aggressive as eliminating one player can start a domino effect of turning in cards for a big army to defeat more players. It's not necessary to hold large continents at all since later in the game the bonuses are inconsequential compared to the massive card rewards. It's much more important to amass lots of troops early and get into the position to eliminate players who hold many cards.

[@kotori87gaming89]

Thank you! Card strategy has a huge impact on the game and it isn't even mentioned in this video. Proper use of cards is vital. I play the second variant, and continents are only significant in the beginning of the game. By the middle of the game, it isn't the continents themselves that matter, but each player's access to other players and their cards that matters most.

[@BedrockBlocker]

@@kotori87gaming89 well-put!

[@Kylted]

The numbers matter, but so does understanding the psychological part of the game. In classic risk with fixed cards, the psychological element is of the most importance, where technical skill and numbers matter more in other maps and modes. Excellent video!

[@acid_ibis4214]

I was scrolling down going which risk GM is gonna be the first name I recognize in that chat. Yahhh you win.

[@UnBalancedBlitz]

not first but yeah 😂 i also hit GM

[@HudsonHornet250]

Imagine seeing you here!

[@mallorierisk]

👀

[@vampirechicken]

The World Champion of Risk is in the comment section!

[@jpe1]

The “correct” Risk strategy depends very heavily on what strategy the other players will be using. In a tournament setting where most players don’t already know each other, but can be trusted to have a sophisticated understanding of game theory and the current “state of the art” of how Risk is “best” played, then an analysis like what is presented in this video makes sense. When playing with friends/family where much is already known about expected strategy used by the others, then any number of strategies that would appear sub-optimal might be actually be best. Also, there was no discussion of how many rounds have been played (thus how many cards are in other players hands) and if someone turned in cards early, the increasing value of turning in sets of cards definitely impacts decisions on relative levels of aggression for any particular round.

[@elenna_alexia]

pretty sure the most common way to play risk is with fixed valued for trades. progressive trades really do change the flow of the game a lot. you're getting at something important here though which is that fundamentally the game is as much or more about psychology than knowing optimal strategy as long as there are more than two players (and it's really not a great game to play with only two people). in a multiplayer setting even the best player's game can often end very quickly if they provoke someone into slamming all their troops into them, allowing others to capitalize, or if they get teamed on.

[@Laezar1]

the best strategy is to never attack little timmy because he will then decide to attack you relentlessly until he loses

[@michaelbauers8800]

Sometimes this is called the metagame.

[@88porpoise]

​@@elenna_alexiaAs far as I understand, progressive is the default in North America and fixed elsewhere.

[@oasntet]

I wasn't even aware there was a variant with fixed turn-in values for cards. That would really change the late game and maybe even overall strategy quite a bit. It may, in fact, make it occasionally worth it to not attack every turn just to get a card.

[@chaoslord8918]

I've played quite a bit of Risk in my day, mostly in the form of Risk II on PC, which is a great way to crank out a lot of games quickly (and with as many players as you want). Risk II also has additional optional territories and connections that can change up the game, as well as Same Time mode, which only allows for attacking one space into enemy territory at each border, which dramatically changes how the game is played.

[@swrpggm]

Turtling in Australia early in the game has always been the best strategy for players. You build your army up and let the other players take each other out. You then go on the offensive when someone is about to take over Asia and use the stockpile of troops you gathered while turtling. It's important to keep a significant force on the border of Australia as an anchor in case things don't go well on the attack.

[@PrometheanConsulting]

Turtling... has always been the best strategy [for beginners] - fixed your sentence. The problems are legion. 1) You cannot protect your exterior positions because you are not splitting your forces so you lose them. 2) You control fewer territories so you gain fewer reinforcements each turn. 3) You must attack successfully to gain cards (which is where the big swings come in). You have limited targets which are also built up. 4) You need to avoid offending your neighbor(s) or one will wreck you and the other will sweep up your cards.

[@drzerogi]

@@PrometheanConsulting Point 3 is key here. The progressive card system is the biggest reason why the Austrailia turtle strat is so weak. But there are people who play the game without it, and that - and lack of skill in the game - is why the Australia turtle still sees success and why so many people swear by it. I've had to show many pro-AT players the error of their thinking in game when the progressive card system is used.

[@Procrastinus]

Works best with three players too. I had a long standing Risk game where one person did that every time. Made the game boring and predictable, which isn't the purpose of a game. It basically breaks the game as if the two other people attack each other, then Australia comes in after and wins (besides cards eventually determine who has the most armies, not so much territory). You get a stalemate where the first one who dies of boredom loses. I warned him that if continues to ruin the game by making it unfun, we simply won't play it anymore. He kept doing it. We stopped playing.

[@Matthew-bu7fg]

the best continents to go for can also depend on the amount of players playing the game. Playing with a larger amount of players can make going for Australia/Oceania a much better strategy as it can be difficult to establish continent control early on and the extra two troops for taking Australia early can be beneficial in asserting control over the game

[@christopherbelanger6612]

As long as you immediately push and don't sit in Australia

[@kieronparr3403]

I always start in oceania

[@evanbasnaw]

I like to hold Australia to do 2 things: 1. bolster my troop count a bit so I can add them elsewhere. 2. Poke into Asia just to prevent the top player from having it at the end of the round.

[@kieronparr3403]

@@evanbasnaw you sound like my kinda player, if I can't go aus I go s. America

[@caleblatreille8224]

I was in a Risk club in high school, we met three times a week, and it's amazing how the strategy outlined here (and a lot of variations mentioned in the comments) became apparent just through repetitions and an ever-growing sample size. I feel really lucky that the members were inclined to discuss exactly these sorts of analyses and were more interested in the theory than the competition lolol.

[@chemicalfrankie1030]

The fact the lower the nodes the easier to defend, the easier to win the game? I think that’s the first deduction everyone does when playing the first time… Risk is like poker. Since heavily influenced by luck, you don’t play the game but you play the players…

[@DemPilafian]

Did the club members discuss trash talking strategies to nudge or trick other players to attack each other and not attack you? The real winner of a battle is usually all the players except the attacker and the defender.

[@caleblatreille8224]

@@DemPilafian Yeah, we 100% did actually. That's the real advantage of Australasia and South America: their low opportunity cost in the early game when there's more than 3 players.

[@nickandres7829]

My friends and I played a lot of Axis and Allies, and most games came down to "gang up on Russia or the UK?" and "went for a Pearl Harbour (usually a German all out attack on UK's Home Fleet or Karelia in Russia) and lost and so it's basically already game over."

[@floodo1]

Same, I didnt need match to sus out these strategies as a kid, most importantly that two player isn't fun and in multiplayer the social game trumps all (-8

[@davidbjacobs3598]

As a strictly mathematical analysis, this is pretty solid, but it's worth noting the player dynamics are pretty key here. The game can become very political and interaction-heavy. Truces can effectively change the topology of the board (assuming they are honored). If one player looks particularly strong, others may gang up on them. Basically, the simply fact that there are always more than two players is a major factor in determining the outcome of the game. That said, I like to play in South America. Fast to take over, and still relatively easy to defend (two borders instead of one). Unlike Australia, it also has better mobility -- Africa and North America are both solid defendable countries, and you can move into whichever is weaker, using the bonus you've gotten faster than anyone else. Sometimes you can even win by just staking down and raising armies: players attack each other instead of you, weakening them for an endgame where you cash some cards for massive payout and obliterate. It's a very flexible little continent.

[@xapheneon]

In the version I played (might be homerule) the defender can choose to defend with one or two dice. This made the game hugely defender favoured because you can choose how many units do you risk, depending on the chance of beating the attackers roll.

[@joelinsb8650]

I am very sure that experience (and probability) say you really should defend with two dice rather than one as long as you can. When the attacker attacks with three dice (also best for attacker) the two-dice defender will very likely inflict greater casualties on the attacker than suffered by defender, maybe 50% or more extra casualties. I would love to see charts of expected casualties in big battles (say 40 attacking 30) you could roll one die/dice once and remove a large number of armies, say remove 30/20 attackers/defenders if roll a 6, 25/25 attackers/defenders if roll a 5, 25/30 attackers/defenders if roll a 4, etc. Why attack when the dice always favor the defender? Attack when your attacking group is so much larger than the defending group that you can wipe out defenders and live with the casualties because of the harm done to the defender in defender losing hold a continent and a lot of territory, especially if a third player does not have a lot of armies in the particular area of attack to turn against you after you weakened yourself wiping out the second player.

[@TheRealScooterGuy]

@@joelinsb8650 -- I made such charts at one point in the past. As you expected, rolling the highest number of dice that you are legally allowed to roll always gives you the best odds you can get as either attacker or defender. Rarely would I ever attack with less than three armies (and thus three dice) because of this. (Sometimes, you need to though.) (For those with reading comprehension issues, I am not saying that rolling the maximum number of dice gives you better odds than your opponent has, it merely gives you the best odds you can have in the roll, even if they are still against you.)

[@owenbegowin9335]

The editing/stop motion in this video is so fun! I like it

[@jimmyzhao2673]

Indeed. The cotton ball explosion was awesome. 🔥

[@6B26asyGKDo]

You'll love "Western Spaghetti by PES' on KZbin.

[@Slavir_Nabru]

I always go for South America first. It takes two stacks to defend but expanding to either North America or Africa only add one stack each. It also helps to not get drawn into the Australia build up.

[@DaTimmeh]

Africa results in a 4 point border, 2 more.

[@Slavir_Nabru]

@@DaTimmeh Apparently it depends on the version. Originally there is no connection between East Africa and Middle East, so you're only defending North Africa and Egypt, they seem to have altered it in newer versions.

[@stegwise]

The North and South America combination is 8 per turn at 3 choke points. The Monroe Doctrine is my go to strategy.

[@RitchieDiamond]

Both Africa and SA are essentially a 1 point hold, guarding relatively peacefully in Egypt and Peru, with the threat of retaliation to whoever breaks you.

[@peri5552]

my man explains risk so beautifully that I was actually gonna think that it’s a good board game (it isn’t). It’s a classic for sure, it’s very light complexitywise, it’s easy to teach but it’s so dated, so dated that there are many other and better board games with the same mechanics(territory control is a mechanic).

[@peteyoung3124]

Perfect timing on this video. I had just suggested that when we get sick of Wingspan at our game nights, we could play Risk next. We hadn't had it at a game night since before COVID. 😂

[@xtieburn]

One thing about Europe though is that most of the spaces threatening it are Asia, and Asia has a real hard time stabilising, so Europe can be surprisingly solid despite its many connections. (There is also the fact that until later on, often people just want to break bonuses, and that just means counting how many people you are connected to, not all the individual spaces. Typically each person will be gunning for a complete territory and both Europe and N. America have 3 of those.) Of course, this also means relying on Asia being a bit of a mess while building up elsewhere and thatll wreck you if Asia does successfully stabilise... Whole thing can get real complicated even before you add in the social game.

[@lifetake3103]

The fact that people will look to break your bonuses lends itself why you want as little of your countries connecting to other players countries as possible. Because even if only 1 person is bordering you in say South America from both NA and Africa you still need to defend both of those connections to defend your bonus.

[@Puschit1]

Europe is still way too hard to hold. You just can't against experienced players unless you have an alliance with someone, usually whoever controls NA, agreeing to not break each other's 5-army bonus and concentrate your efforts on the remaining fronts. And even in that scenario, the NA player has a huge advantage over the EU player. Never go for Europe as your first continent unless you play with first timers.

[@PhilbertDeZwart]

Maybe we were doing it wrong but we always tried to dash for Australia in the beginning because once you had it you could just park a big army in the one country and leave the others empty. Conquering the other continents is always a longer term campaign in which the 2 extra armies every term come in very handy

[@ColonelSandersLite]

One of the things about risk is that the viability of various strats changes a lot depending on the number of players. Australia and S. America look a whole lot better the more players there are. Fewer players make them less attractive.

[@GodwynDi]

@@ColonelSandersLite I'm not sure thats true. Even with fewer players controlling them is better than not. Its not like a player is limited to only controlling them. Australia pays for itself in terms of troops in 2 turns, and every turn afterwards is a net profit. S. America as well.

[@Puschit1]

You did it "right", whoever gets Australia first usually wins unless the others seriously gang up on that player.

[@mrsunshine63755]

The 1960 and 1970 boards and pieces were the Best!

[@eas2252]

This game and Axis and Allies were my favorites. Unfortunately, nobody in my life was or is willing to endure a game.

[@Flamewarden_Honoushugoshin]

In my experience starting in Australia was always the best tempo for launching a conquest of Asia, since whoever was trying to get Asia without a foothold for extra production was at Risk (heh) of losing their poorly defended flanks and thus their production. North America and South America are good starts too in my experience, but yeah it's usually pretty quickly resolves into a 3-player game (whoever came out on top in the Australia-Asia skirmishes and whoever claimed the Americas, and whoever got squeezed in the middle as the wild card, usually in Africa. That third player would rarely win, but they held a lot of power in forcing the other two to respect them or else end up getting flanked).

[@Sgt_SealCluber]

Yup, getting 2 extra armies from an easy to defend location is great. On a different note I like the special win condition cards that were introduced. Makes the game not take 3 days, lol.

[@asahearts1]

It breaks down when you're playing against people familiar with the game, though.

[@fredsanford5954]

The key to making Australia work is to also take SE Asia/Siam in Asia. It keeps you from getting pinned, and breaks any potential Asia bonus from other players, and once that happens, Asia can be a bit of a free-fire zone where you take one territory (and a card), and just build up until you're strong enough to blitz Asia to take and hold Ukraine/Middle East/Alaska and everything inside.

[@asahearts1]

@@fredsanford5954 Do you usually play with people who let you start with all of Australia? Do you ever spend the game fighting for control of Australia and get steamrolled by someone who got South America earlier on?

[@Tecnoc22]

@@asahearts1 The group I play with has players willing to go all in on Australia, regardless of outcome. So if you start with territory in Australia with them you can also go all in on Australia and assure mutual destruction, or you can cede the territory and try to win starting somewhere else.

[@hamiljohn]

With 3, you can only roll 2 dice on attack

[@JamesSmith-np1hs]

Risk is a game of diplomacy with two major variations in progressive card trading and fixed. Taking one territory per turn for a card is key and not pissing of your neighbors is key. The people who understand this are experienced players. The more experienced players you have the worse south america and austrailia become. If everyone is smart then everyone holds at least one continent. And lets their neighbor hold theirs. The greedy, agressive and weak players are targeted first, which is the australia player. Not to mention people fight for australia and north america often, and the more you fight the weaker you are to the third player, who is watching everyone else delete their troops. No-one should ever let anyone hold asia, as thats where players recieve their cards by trading territories and seven troops per turn is too much production.

[@hamiljohn]

@@JamesSmith-np1hs yet, in the video the example was 3 armies attacking 2 armies, and with 3 armies you can ONLY attack with 2 die.

[@Mavocad0]

I feel like that's a huge oversight on his part. That's exactly what I thought when I saw that, had me questioning if I was wrong lol If you roll three dice you're committing three armies to move. Meaning if you only have 3 armies on a territory and you commit all of them to move if you win then you'll be leaving your territory with no defending army. That's when you can only attach with 2, and one dice if you have 2 armies and you can't attach with 1 army

[@Valchrist1313]

@@JamesSmith-np1hs Australia has only one path to attack, thus only one place that is needed to defend. That's a strong-point, not a weak-point, and anyone choosing to attack it a) must sacrifice a massive portion of their army and b) becomes the weak player who is holding Australia

[@Bluhbear]

The important thing was that he was attacking with three, for the purpose of that discussion. He could have armies he left back, which he didn't mention, and the visuals people neglected to depict, for the sake of simplicity. His wording was a bit ambiguous, but it really doesn't effect the math, which was the actual point.

[@420Fanatic]

Thank you for doing risk! One of my favorite if not favorite games of all time.

[@ludvighoelstad326]

Afghanistan in game migrated about 1000km north-west. this mountainous land suddenly gained a coastline almost looks like they mixed it up with Kasakhstan.

[@darthrainbows]

Regardless of where the dice advantage is, attacking always weakens you, even if you lose no armies in the attack, because it forces you to split your armies across more territory. In a 3+ player game, the only player(s) advantaged by an attack is/are the one(s) not involved at all. I always found that in order to win, you have to appear strong enough for the other players to be wary of attacking you, but not dominant enough that other players must attack you anyway - at least until it's too late for them to do anything about it.

[@mikesmith6838]

True, but you MUST take another country on your turn to draw a card and that requires attacking. As the "book" value goes up with each book cashed in, the card is more important than the cost of the attack; generally.

[@darthrainbows]

@@mikesmith6838 that presumes your cost in the 3-5 turns of attacks is less than the gains for turning in the set. That is not at all guaranteed, especially in the early game for the American rules or in the Euro rules (where there is no escalating reward for sets).

[@danielbremner4142]

@@darthrainbows Correct. If you are playing fixed cards you will get a trade every three turns with each card being worth about 3 troops, counting the territory plus two’s. It is still beneficial to take a territory if you are only trading one troop per turn. In progressive you live and die with your cards. The easiest way to put a player in the dumpster is with a card block or a reverse card block. Check out Kylted or Thekillpetestrategy for more tips.

[@WreckItRolfe]

It always weakens you unless you are taking a border country (Iceland, Alaska etc)

[@Puschit1]

That's why games among experienced players take forever.

[@kzisnbkosplay3346]

I was taught to start in Australia, but more recently I have been leaning towards South America. It's almost as easy to defend, so gives you an early game boost, but it's a lot easier to move on to a new continent from there.

[@glenjamindle]

Both terrible continents. North America and Europe are best. Africa, then South America, then Australia. You'll never hold Asia.

[@mauriciofrieri4834]

Also South America doesn't get as mucha attention as Australia. Whenever I play I feel like most players try to get Australia or prevent some else form getting it but SA goes under the radar

[@RitchieDiamond]

To play SA while staying friendly with your neighbours in NA and Africa requires you to keep a stack alive in Asia to take cards, which isn't always possible. SA is a tricky position to play, but can be done.

[@usageunit]

My dad is a stubborn Australia guy, I used to be a South America guy, but these days I feel like North America is the best reinforcements to entry points ratio, and combined with diplomacy and cards has always worked out well for me.

[@paulsimpkins1540]

I always use S. America as the means to overwhelm N. America, because you have the same number of entry points to defend if you hold both vs. if you hold only N. America.

[@ChrisDreher]

Back in the mid/late 1980s, I wrote a program to brute-force compute whether 3-dice attacking was better than 2-dice defending. It ran every permutation and then sorted the dice from low to high and compared. In AppleBasic, this took minutes to run. Eventually, it showed attacking was better, though I had a hidden bug for a few weeks that claimed defending was better. Doh!

[@natescode]

Nice! It is always fun to code little programs to find answers like that.

[@disgruntledtoons]

To win the game you *must* attack on as many turns as possible. You get a card at the end of each turn during which you take a territory, and turning in sets of cards earns the player doing so progressively higher numbers of armies each time this is done. After a while the armies gained for doing this are ridiculously high, and in a long-running game this provides the majority of the armies raised during the game.

[@oasntet]

Evidently, the standard game mode outside the US doesn't have progressively higher rewards for cards. It's just fixed (at 5?) with, I think, the bonuses for owning the territories on the cards.

[@Zwiezwerg92]

I would love to see Numberphile videos covering some more recent board games!

[@caterpillermiller]

Risk is the reason my family doesn’t have family game night

[@DemPilafian]

The correct strategy is to play so that *you win every roll of the dice.* This is accomplish by convincing the other players that their best move is to not attack you but to attack each other.

[@aviasegel]

Love the stop-motion animations! They're really fun :)

[@mikemooney83]

The number of opponents greatly affects the maths. With the max number of opponents, Australia is definitely the way to go. With the minimum number of opponents, it's North America followed quickly by South America. (7 bonus armies and only 3 places to defend) You could easily do another video on the variables he never mentioned. (Bonus cards, more detail on the maths behind the dice rolls, etc.) Maybe a series of videos. This one only scratched the surface!

[@dbrfour]

I feel as if I've been waiting for this video since I subscribed five years ago.... Love it

[@FutureAIDev2015]

I *knew* there was a reason I loved that one scene from Katawa Shoujo where you play Risk with the student council president!

[@Austin-fc5gs]

Love the stop motion animations! I also love Ticket to Ride

[@AbiGail-ok7fc]

I think the video underestimates the value of Australia. Sure, if the choice is between Australia and Asia, the latter would be a better pick. But timing is important. Australia now vs "6 more regions before I conquer Asia" is a quite a different choice. Australia is also a lot easier to defend, as you can only be attacked on one front.

[@shadeblackwolf1508]

I assume all this is for risk international ruleset. Some european countries actually play by slightly altered rules: Each turn you may either attack, or reinforce 3 per owned territory. Not both. Continent bonus still applies in both cases. This has the effect of drastically increasing the value of continent bonuses to the point players are unlikely to let you start a turn owning a major continent under any circumstances. Moving troups in the end phase is capped at maximally 7 unit-countries. Meaning you may move a unit to a connecting country up to 7 times. This makes the Where of your troops more critical. On the whole this makes risk a much slower paced game. Risk is also mych more commonly played with fixed prize values instead of escalating ones, such that the max value of 3 cards will always be 10.

[@macedindu829]

Risk is so fun. My roommate back in the day had it on the PS2, which obviated the need to set up boards and roll dice and whatnot. Damn we played a ton of games of that. Usually 4 or 5 players. There's a lot of dynamics to it, especially when you have random placing.

[@chubs2312]

The stop shoot animation is 🔥

[@bzqp2]

There is a great, more advanced online version called "Warzone" (previously known as "Warlight"). Has a very nice community that is very active even though the game is over a decade old.

[@bzqp2]

It has many custom maps available and there are always some games you can join! (both real time and multi-day)

[@austinnguyen9107]

Really cool video, played this game before and didn't expect a video on all the maths

[@boomerhippie]

There was a variation that we played back when I was in college. This involved two changes. First, we did not use cards at all other than in the initial starting set up. Second, we allow the defender to roll three dice, if the defender had at least three units in the defending area. The effect of these two changes was first much longer games. There were games that lasted over 24 hours. Second, a lot of time was spent building up huge armies. For example I've seen 600 units in Indochina defending Australia.

[@jako7286]

There is a standard variation called “Capitals” where you choose one of your starting territories to be your Capital, and it defends with up to 3 dice instead of up to 2. This makes it nearly twice as difficult to capture (I think you need on average 1.7x as many troops to capture a defending capital). In this game mode, the objective is capturing all of the capitals, not the whole world.

[@benchbenchbench]

A future video on Carcassonne would be great

[@astrocoastalprocessor]

❤ agreed!

[@culwin]

The problem with Risk is that you always have somebody (or multiple people) who do things against their own best interest, and go to war with someone in a way which only hurts both players.

[@RibusPQR]

A bit like real life then.

[@kieranharwood7186]

There are a lot of situations in board games where it's best to be the third player, sitting back and not getting involved in the scuffle. And, of course, that makes it so much more frustrating when other players fail at risk assessment like this. If you don't retaliate then it turns out that what they did was the right play, but if you do retaliate you're now dragged into a conflict that the other players will take advantage of. I'm sure that there are no analogies to real world politics here or anything ;)

[@hendrikd2113]

In Risk attacking with a full stack against a full stack (3+ vs 2+) gives a small advantage to the attacker. Therefor not attacking is against your own interest.

[@kieranharwood7186]

@@hendrikd2113It depends. If you are in Europe, it may be better to not attack NW Africa, because if you win then S. America can now attack you with their stack of troops, whereas if you wait, S. America might attack Africa for you and then you can move in behind them and take it for yourself. 1-on-1, you should always try to be attacking, you get the slight edge in the fight and you also get to pick where the fight is, can call it off whenever you want and are the one standing to gain. If there's a third player though, you want to commit as little as possible to getting that extra card for taking a territory. The ideal turn in a roughly balanced 3+ player game is to take a single undefended territory and then move out of it, ceding it to it's original owner. They then do the same and you've each lost 1 or 2 troops for a card. If you could somehow alternate doing so between the other players then, after a few turns, you've spent virtually nothing, built up your forces and have a grip of cards to hand in whilst everyone else has been smashing their forces to bits getting nowhere. Furthermore, if you are playing the version where you have a hidden objective (such as eliminate a specific player), you aren't tipping your hand as to what you are trying, whereas going on the attack means either showing where/who you need to get or wasting resources on relatively meaningless land.

[@loki2240]

My brother was 5 years younger than me, and he always attacked on his turn until he couldn't attack anymore. We always played with at least 4 players. And sometimes, he would have some lucky rolls and mess up plans for world domination! 😂

[@stephencahill7821]

I wish that they had done an interview with an expert Risk player before this, because that was honestly pretty pedestrian. I figured that the attacking vs defending advantage for an individual battle was well known since the game came out. I remember doing brute force probability calculation problems of that in junior high, so I can't believe that the misconception lasted any significant amount of time.

[@PeterFendrich]

I'm sure I'm not alone when I say that as a child risk was truly my first step into the broader world of gaming... I still remember the day my grandmother bought it for me for my birthday. Objectively there's lots of reasons to dislike risk, or things to complain about it, and I recognize them all as completely true, but the game will always hold a special place in my soul.

[@Nordlys]

One big advantage of Australia is that it has no backdoors, in addition to netting you a continent bonus. You can amass an army on the one tile facing the world, likely outpacing anyone in Asia who has to defend multiple fronts. Then, you can safely attack with your entire army when it suits you, and safely back down if you get unlucky dice rolls. I do agree on the notion that North America gives the best bonus compared to territories to defend (assuming the ones behind your line of defense only requires 1 army each. It does however have entrance from 3 different continents, which makes it likely that 3 different players will want to breach your defense on all sides, and hard to hold from the start. If two players are sitting and amassing armies (turtling), the one with the best army gain will obviously be at an advantage. Australia could easily get islandlocked by whoever is in South-East Asia if that player does well.

[@japplek]

More episodes about more boardgames would be great! Did you know you can make a complete Turing Machine in Magic the Gathering?

[@carltonleboss]

Wait, what?

[@FedericoLatini]

Yep, as long as each optional effects is considered compulsory

[@irri4662]

Illuminatee next please

[@Miata822]

I loved Risk when I was young. There is also a psychological element to the game. Whether or not you seem to be aggressive on one front or another influences your opponents placement of their own resources.

[@Janovich]

2:29 you missed a connection between southern europe and middle east region

[@tiagoleite9452]

Great video! Please make another one about Ticket To Ride!

[@Jimmy-Chin]

I wish you showed the Markov Matrix and the probabilities

[@jimmyzhao2673]

The solution of this problem is trivial and is left as an exercise for the viewer.

[@nestoreleuteriopaivabendo5415]

​@@jimmyzhao2673 r/AngryUpvote

[@briancompton6412]

Exciting analysis! Thanks! I wonder how many fans of Risk know about, and have played, RISK LEGACY, the game that permanently and uniquely changes after every game.

[@sevenofzach]

now i do!

[@jex_au1102]

An important part which was missed is that most common rule sets include the use of the cards. When you gain at least one territory in a turn, you gain one card. When you have a set of cards you can trade them for more armies (the number of armies depends on the rule set, if it's a progressive or set based system). You also get one bonus army for each territory you control for which you also hold the card for. So cards can be hugely influential. The attacker might mathematically be more likely to win a fight to the death (at least in the 3 vs 2 die scenario), but the extra reward of gaining at least one territory makes attacking extra attractive.

[@SStephenMC]

I love playing risk but I always feel like the games i play go on for too long

[@mytube001]

Same! I don't think I've ever played a game of Risk that was shorter than three hours. 4-5 hours is the norm, at least with four players.

[@aaronbaker2186]

Missed a few interesting points. Taking at least one territory gets you a card. Armies from card sets can have a big effect on the game. The attacker rolls their attack, and *then* the defender rolls. In a 3 armies attack 2 armies scenario, if the attacker rolls 663, the defender can choose to roll just 1 die. If the attacker rolls 611, the defender can roll 2 dice. The defender gets to choose after the attacker rolls.

[@authenticallysuperficial9874]

I've never heard that defender rule

[@hastyscorpion]

I think that might be a house rule of yours. All the rules that I have found said that you have to announce the number of dice your are going to role and then both players roll simultaneously.

[@aaronbaker2186]

@@hastyscorpion huh, looks like you are correct (Hasbro says so). That actually has a noticeable effect on strategy.

[@ClayHales]

That rule is almost always overlooked, and it makes defending with a large army brutal on the attackers.

[@dylanwolf]

Hi Numberphile. I love all your videos and have been following you for years. A huge thank you to all your presenters and back room tech guys for all the work that must go into producing your superb, entertaining and informative content. A question - are there any interesting maths studies on starling murmurations? ----

[@GabrielHernandez-sn3el]

I think everyone else has hit on a lot of the key decisions of risk, but I wanted to summarize a few I have found to be helpful. 1- you should always conquer at least 1 territory per turn. The card bonuses are huge. 2- Australia is better than South America in the early came because there is only one entry point. South America is better for mid game, because you can attack through North America (which is relatively easy to defend) 3- North America is better than Europe because it has fewer areas to defend while giving the same bonus. 4- you should almost always reinforce your troops so that areas facing the enemy have the most troops, and rear areas have a single troop. Not 100% sure on the math, but I suspect having more troops in the attacking country(border country) is better than having (2 ) troops on every interior country. Goes back to attackers having the advantage. 5- you can defend Asia easier by placing your troops in Eastern Europe. You can defend from one country instead of 2(Afghanistan/irkurts?). It acts as a natural choke point . 6- depending on how your allocate troops at the beginning, North America and South America is where you should focus. You can control the borders of these two continents from only three territories (Brazil, Greenland and Alaska). 7 bonus troops for only 3 entry points is one of(if not the best) in the game. 7- I’ve found Africa to be the hardest to control outside of Asia. It tends to get attacked from all sides, and does not have enough bonus troops to make it worth the trouble. I only ever go for it after already securing NA and SA.

[@bubbahottep8644]

You cannot earn Risk Cards without attacking. So a strictly defensive policy will lead to inevitable defeat.

[@WilliamWallace14051]

It looks like somebody had fun with the animation. :-)

[@xapimaze]

Great video. Risk rule: "You must always have at least one more army in your territory than the number of dice you roll." You cannot attack with 3 dice if you only have 3 armies on the attacking country. Risk cards really matter: It is usually optimal if you can capture a risk card on each turns. And, try to be the one who eliminates an opponent who has risk cards. If the attacker rolls 3 dice and the defender rolls 2 dice, then theoretically: Red wins 2890 / 7776, so 37.2% of the time (meaning white loses 2 armies). White wins 2275 / 7776, so 29.3% of the time (meaning red loses 2 armies). They tie 2611 out of 7776, so 33.6% of the time (meaning both lose an army each). You want to avoid attacking if you cannot roll more dice than your opponent. It might be worth the "risk" though, if it means getting a card or not, or if you have a chance to eliminate an opponent and get his risk cards. I'd love to see the work-up of this with Markov matrices.

[@DrAvery-lc6bs]

In a game with 3 players, I just don't have the patience to win. But I always get to decide who does win. The first player to make a serious launch into my territory will find themselves on the receiving end of a death march intended to debilitate their largest armies and weaken their important territories until I can't attack further. Now, at this point, I'm spread so thin that defeat is inevitable, but the opposing player's losses are such that the third player is presented with the opportunity to punch massive holes into enemy territory.

[@NoNumbersAfterName]

I remember one game, a championship game at school, where one player had been boasting about his best strategy, and that involved taking North America and attacking from there. But, since the rest of us knew that that was his plan, we all focused first on taking out all his territories outside North America (he left them with only one army each and swarmed everything in Ontario), then ensured that he lost at least three armies each turn until he was defeated. Then the rest of us played against each other. He was so confused why his foolproof strategy was being beaten.

[@jansenart0]

I always thought that water crossings should double the effectiveness of defenders, as should certain territories have added defensive bonuses (central America and Ural, for example).

[@theadamabrams]

There are several RISK variants (often themed) that add bonuses to defenders or attackers in different situations. RISK 2210 is particularly interesting imo.

[@VernAfterReading]

Another key topology factor is how deeply you can attack in one turn. You can only move in the number of soldiers you attacked with (once you go to fortify stage, you can't attack further that turn). Territories with only a single or a few points of entry become huge. You can set up a buffer and a reserve to make it so you can always just recapture back and still get your bonus.

[@kxjx]

Yup, and critical how many splits are required to eliminate a player in a single turn. Travelling salesman problem.

[@chrischuter1530]

You made a point, saying about Australia being a bad country to hold. In the same way there is dependence between dice, I suggest there is also dependence between the payoff and the number of points to hold. America has 3 boundary points and the payoff is 5, but the payoff for Australia is 2 and has only 1 boundary point. Australia is about building income, and its easy to defend having only 1 boundary, you can quickly place troops on to defend, more so at the beginning of the game too. Your on the right track, there are a few tricks up Risks sleeves, like border extension. e.g. Having North America, then taking and holding South America. End result is the same number of boundaries to hold, but increased income.

[@LetsGetIntoItMedia]

A video aboit Sushi Go would be great. It's a fairly simple drafting game, but the best strategy *seems* to be "the strategy nobody else is going for". So you have to quickly determine what others are collecting, and do something different. Quick adaptations are key. But I don't know how to formalize that gut feeling

[@chrishillery]

Pedantry time: Sushi Go is a drafting game, not deck-building. Dominion, Star Realms, Legendary, etc. are deck-building games.

[@LetsGetIntoItMedia]

@@chrishillery Ah, thanks for the clarification. I didn't know the proper lingo. Fixed!

[@lore.keeper]

I've had this game for years but I totally forgot about it until now. This video made me remember how much I enjoyed it! Will def start playing again

[@QuantumHistorian]

I loved it as a kid, but now I find it's very flawed. It can take far too long for an unbeatable advantage to turn into victory, rolling endless numbers of dice becomes boring after a while, and it can easily turn into a tedious gridlock. _Diplomacy_ has the same basic conquer-everything vibe, and all the same flaws, but actually delivers on the strategy that Risk promises. The main problem with it is that it *will* end friendships in a way that even Monopoly can only dream of. But if you can find 6 like-minded friends (4 for the ancient-world version) together for a whole day of drinking cocktails and playing the game, you'll have a great time.

[@TheFirstObserver]

I always loved Risk 2210 A.D. growing up, since it was (to me) Risk with a better designed map and a little more player agency. I loved underwater colonies, and hopping up to the moon. Plus, Command Cards felt like a cool way to change the tide of battle.

[@koenth2359]

Not easy to believe they first overlooked the dependencies. I just did a simple 21×21 Excel matrix with weighed probabilities, and it came out that the expected losses of the defender exceed the losses of the attacker by 205/6^4 (about 0.15818) armies per throw. In this case the attacker always throws 3 dice and the defender throws 2. An interesting variation (that I like to play) gives the defender a very slight edge: he may choose the number of defence dice after the attacker's throw. The attacker's expected losses now exceed the defender's by 6^-4 (0.00077) armies per throw Edit: I corrected some typos: matrix size, and the result for 3 against 2.

[@Tecnoc22]

This was how I was taught to play, didn't actually realize it was a variation. Always felt like the advantage was more than 0.00077 armies per throw though. How was your matrix set up? The way that first comes to my mind on how to do it would be much larger than 28x28.

[@koenth2359]

Here's the range A1:J10 copied from my sheet. The layout is horrible here, and we use decimal commas, but other than that it might paste into Excel. Have fun. Poss 1 2 2 2 2 2 Top 6 6 6 6 6 6 Poss Top 2nd 2nd 6 5 4 3 2 1 16 6 6 -0,004115226 0 0 0 0 0 27 6 5 -0,006944444 -0,013888889 0 0 0 0 21 6 4 -0,005401235 -0,010802469 -0,010802469 0 0 0 15 6 3 -0,003858025 -0,007716049 -0,007716049 -0,007716049 0 0 9 6 2 -0,002314815 -0,00462963 -0,00462963 -0,00462963 -0,00462963 0 3 6 1 -0,000771605 -0,00154321 -0,00154321 -0,00154321 -0,00154321 -0,00154321 13 5 5 -0,003343621 -0,006687243 0 0 0 0

[@Chrisz0rrr0]

My friends and I calculated the optimal amount of dice for the defender to throw for every possible attacking player's throwing result and put this in a spreadsheet on OneDrive. This assumes large armies for both players, of course. I also had no idea this was a (local?) variation of the rules.

[@koenth2359]

This ( the 'variant' rule) was how I learned Risk as well (Netherlands, back in the eighties)! However, I have looked for a version of the official rules stating that, but could not find any. If anyone could tell me where to find a copy, it would be more than welcome. [Careful with links though, I noticed the youtube algorithm tends to remove comments containing them] Edit: I found a publication however, named 'An optimal dice rolling policy for Risk'. 1994 Ger Koole, Univ. of Amsterdam. In the text he mentions a difference in this respect between the Dutch and English rules. Here I quote part of his conclusion section: "4. Conclusion In the previous section we derived an optimal defence policy for the game of Risk, assuming that there are enough armies in the territories involved in the attack. The optimal policy (according to which we have to roll two dice only if the second highest die of the attacker’s roll is 1, 2 or 3) leads to an average loss per army very close to 1 2 for both players. Thus, each player can make the decision to attack or not just based on tactical arguments, as the number of lost armies hardly depends on who is attacking or defending. It is remarkable that the British rules differ from the Dutch rules in that the defender has to roll its dice together with the attacker, basically giving two possible policies, always rolling one die or always rolling two dice..." ... and, yes, I found the (Dutch?) rule back in a pdf called 'spelregels-risk-oud': " *De verdediger* moet zijn gebied verdedigen met alle legers die op dit gebied staan. Staan er 2 legers of meer op dit gebied dan mag hij met maximaal 2 dobbelstenen gooien. Heeft hij 1 leger op het gebied staan dan mag hij slechts met 1 dobbelsteen gooien. Men is niet verplicht om met 2 dobbelstenen te verdedigen als men 2 of meer legers op een gebied bezit. Als men er de voorkeur aan geeft om de legers stuk voor stuk ter verdediging te gebruiken dan mag met 1 dobbelsteen gegooid worden. *De aanvaller* gooit altijd het eerst met de dobbelstenen, daarna de verdediger . .... "

[@Chrisz0rrr0]

@@koenth2359 That's a very neat article, thanks for referencing it! Seems like our Excel of that time did not account for the interdependency between rolls..

[@somethinggeeky]

Now do Stratego, I personally played that a lot more. I would love to know the math behind it.

[@TekomiNakama798]

And to think that I only came here to watch "301 views"

[@TRUMAN_THE_TRUE_MAN]

Didn’t ask + I’m way better than Numberphile 💀🤣

[@alexyz9430]

I asked - I'm so much worse than Numberphile 🍀😭

[@FlexxibleFree]

Like 12 years ago, right?

[@RubenSilva-tk4yb]

The +1 strategy is my tried and true method. Hold either Australia or South America and then a "buffer" country. For Australia, this means holding Siam. For SA, this means holding Central America and North Africa. Then you can (1) slowly stack armies on these "buffer" countries to provide adequate defense/prepare for assaults and/or (2) attack a border country once per turn to earn a card. After a few turns, assuming no opponent breaks through your "buffer", your "buffer" countries will have enough armies due to the +2 armies compared to opponents, plus any armies you get from cards (4/6/8/10, or the double multiplier). You do this until you can creep into claiming other continents, knock out another player (and claim their cards), and/or launch longer assaults. If/when done right, when you math out the attacks, and if/when the dice favor you, you can knock out two players during your turn, claim 2-8 cards, and claim nearly half the board.

[@n0denz]

I'm interested in knowing how the math changes when you take conquering into account. There are the initial statistics of each dice roll, but then you have to factor in the probability of multiple dice rolls for one territory and the probability that you can take a second territory with what remains of your initial army. What would that even look like? But yes, attacking is generally the way to go. You have to if only to deny opponents their continent bonuses. But if you get too offensive, you'll just make it extremely easy for someone to sweep in and take all the places where you only left one army. One failed large offensive can weaken you beyond remedy.

[@user-oj8ey3yc1m]

I am Indian in class 8 but I can understand these videos I think I am lucky that I able to understand thanks NUMBERPHILE and happy birthday to GH HARDY in advance ( 7 Feb )

[@PHI_UNBX]

Nice I am also Indian 🎉🎉

[@mathsolmpiyad]

Happy birthday gh hardy in advance lengend

[@mathsolmpiyad]

I am also indian

[@anshusingh3479]

Ya u are tooo lucky

[@anshusingh3479]

Happy birthday to gh hardy in advance

[@FedericoLatini]

Fun fact, italian risk called risiko had one litte rule change that skew the strategy towards defending and turtleing

[@jacobD643]

what's the rule?

[@rcline300]

The defender also can use 3 dice. I didn’t know until now that the English rule was different

[@kieranharwood7186]

@@rcline300 We can't be expected to print a board game that has SIX dice. Much easier to change the rules and then we can save on plastic ;)

[@lorenzosotroppofigo1641]

​​@@rcline300There is a rule that skews it towards attacking too, you draw a card for every first successful attack on your turns and you can get a combo with it to get 10 troops

[@robinbernardinis]

@@lorenzosotroppofigo1641 Cards exist in all versions of Risk. I believe that the difference between Risk and Risiko when it comes to cards is that in Risk the amount of troops you get depends on how many sets of cards you turned in until that point (4 for the first, 6 for the second, 8 for the third, then 10, 12, 15, 20, 25, etc.), whereas in Risiko it depends on the specific set of cards you turn in (4 for 3 cannons, 6 for 3 infantry, 8 for 3 cavalry, 10 for one of each and 12 for two of any card plus a wild card, IIRC), so in the long run you actually get more troops in Risk than in Risiko, but in Risiko someone could get a lot of troops earlier in the game.

[@beblitz5639]

Great Video just one thing to complicate the math: If the rules have'nt changed as defender you can decide if you want to roll 1 or 2 dice to defend after you see the attackers roll. So for example if the attacker throws two sixes as their highest dice the defender can decide to only throw one die so he would only loose one troop.

[@yorktown99]

The advantage of Australia in the game of Risk is that it can only be attacked along a single, narrow front. Resources can be concentrated there without fear of an attack from another direction. The continental bonus is worth more at the start of the game when a player has few territories and fewer concentrations of force. As the game progresses, and players consolidate or are eliminated, turning in cards (representing a series of successful battles) or the bonus from simply controlling more territory ramps up dramatically.

[@plant3341]

Risk is fun because you get to explain the law of large numbers every time your friends say you're just lucky

[@ClintonDawkins]

Every dice throw is an independent event. You're probably explaining it wrong.

[@prefeitobear9209]

I always lose a friendship playing Risk. Totally worth it.

[@mandolinic]

I played a lot of Risk. My favourite strategy (which often won the game) was to take Australia, which gave a fixed bonus every turn, then burst out through Indonesia to take the Middle East and Russia, which protected the Western flank of Asia, then move Eastwards, entering America via Alaska. Move East to take Greenland, and South to Central America to take over South America. By this time, you're getting so many continent bonuses each turn that it's (almost) impossible to lose! From there, it's either Brazil to North Africa, or (for a change) Greenland to Iceland and the rest of Europe. Either way, it's just a mopping up operation. And yes - playing aggressively is the best policy. However, the traditional Risk board requires relatively few armies to take over, and so there's always a chance that an unlucky run of the dice will spoil the strategy.

[@maxheadrom3088]

In Brazil, the game is called WAR. It was 'cloned' by a local company called Grow that, to be honest, made some pretty nice board games.