5.4 1st July 2003: Various minor tweaks, main sections updated are Q1.3 (new addresses for some WWW links, including updated links to Hasbro's website and for XFrisk and removal of some dead links) and Q3.3 (new risk calculators and other web sites)
5.33 27th February 2002: Updated Q1.3 (WWW links) including updated links to Hasbro's website which now includes PDF's of lots of editions of Risk's rules, and removal of some dead links
5.32 4th May 2001: Updated location of US Mirror of FAQ, non-HTML versions of the FAQ and my email address
5.31 10th Oct 2000: Updated Q1.3 (WWW links), updated Q3.3 (new address for RiskOdds calculator)
5.3 18th Jan 2000: Cleaned up HTML so validates as HTML 4.0 Transitional, updated Q1.3 (WWW sites, downloadable versions of risk), link to RiskOdds calculator Q3.3
5.23 6th Feb 1998: Updated Q1.3 (WWW sites, downloadable versions of risk)
5.22 3rd Nov 1997: Updated email and WWW addresses throughout
5.21 5th Feb 1997 : Updated email addresses throughout
5.2 7th Aug 1996 : Added a new section Q4.12 for the variant Multiple Earth Risk, updated sections Q4.4,Q4.5 (corrected attributions from Schmittberger book), added Dragon article number for Nuclear Risk to Q4.3, Q5.1, added more comments on Castle Risk Q2.2.3, added mirror site link for this FAQ Q1.2, updated Q1.3 (other sites on the WWW).
5.1 19th Jan 1996 : Minor corrections, additions to Q1.3 (computer games) and rewrite of Q2.5.3 (risk cards).
5.0 5th Jan 1996 : Added other WWW links, some new variants, information on computer versions of Risk, minor corrections.
This FAQ answers questions about the board game RISK by Parker Brothers games (now part of Hasbro). The FAQ assumes you own a copy of the game (including the rule book). There are three main parts: A discussion of the many variations in the "official" rules, some basic probability analysis, and a list of rule variants people have come up with.
This file is far from complete. Contributions would be appreciated.
Well, you've got it once clearly! But maybe you've saved it and can't remember where you found it, so, here goes!
The FAQ is on the WWW at http://www.maths.nottingham.ac.uk/personal/odl/riskfaq.html
It is also mirrored at http://www.geocities.com/daleallenbaker/riskfaq.html (this may download more rapidly so, if the Nottingham link is slow, give it a try!)
The FAQ is available by email by dropping a note to Owen Lyne: email@example.com
Pointers to the FAQ are occasionally posted to rec.games.board, and the rec.games.board FAQ also contains pointers to where this FAQ is held.
There are a lots of sites on the WWW with Risk information. For example try Yahoo. Risk appears in their directory under Recreation:Games:Board Games:Risk .
A great compilation of Risk links and information is available from http://risk.start4all.com.
Another favourite of mine is (you can find these and all the
others I am aware of in my bookmark
Risk, As Defined by Andrew (moved since FAQ 5.33)
Hasbro's WWW site includes PDF copies of many editions of the rules which is very nice. They can be found from Hasbro: Game and Toy Instructions (search for Risk in the dialogue box, new address since FAQ 5.33).
There are also some computer implementations of Risk kicking around, though I can't vouch for any of them, not having tried them myself. Again Yahoo proves useful for Hasbro's computer version of the game released in 1997 (and sequel RISK II, in 2000). That series of games comes in the Yahoo directory under Recreation:Games:Computer Games:Titles:Recreation and Sports:Board Games:Risk Series.
Nessysoft produce RiskIt for Windows which is shareware.
Virgin Publishing released a DOS game called Computer Risk in 1989 (supports a mouse, up to 6 players, including computer players), which my information source told me is better than RiskIt or the freeware WinRisk. Unfortunately this isn't shareware - anyone know its current availability?
XFrisk is a computer version which uses the X11 window system for graphics and runs on most UNIX and UNIX-like platforms. It is a multiplayer game played on a TCP/IP network, see the project homepage for more details (new since FAQ 5.33).
There also some Macintosh Risk games too, available at http://hyperarchive.lcs.mit.edu/HyperArchive/Archive/_Game/_Board/, they are net-risk-11.hqx and risk-ii.hqx. I don't have access to a Mac, so I have no personal knowledge of these programs, but I believe thay are both freeware or shareware.
I'd like to expand this section of the FAQ in future revisions. Let me know what should be here! Mail me at: firstname.lastname@example.org
You'll probably pick up quite a bit just from the discussion of the implications of some of the rules, but here are a few very basic tips (thanks to Niels Ull Jacobsen for suggesting these). More suggestions can be found on the WWW pages mentioned in Q1.3. Also note that different sets of rules make for quite different games, so some of this advice may be less relevant to the way you are playing.
Any suggestions for tips for this section gratefully received, but please, only those for beginners. I don't want the FAQ to go into really detailed strategies!
Yes, several. There have been a large number of different versions published in the U.K. alone. In the U.S. the rules have been copyrighted six times: 1959, 1963, 1975, 1980, 1990, and 1993. The 1963 edition refers to the 1959 version as "the original French rules". The U.S. rules have subtle -- and some not-so-subtle -- differences from edition to edition, and many of the rules that have been "standard" in the U.K. editions for some time have only recently made it over to the States.
The answers in this FAQ are drawn from old and new versions of the U.K. and U.S. rules. (The Australian rules are fundamentally the same as the more recent U.K. rules.) Rules that appear to be the same across all versions are identified, as are rules that vary. In addition to the official rules there are also the inevitable numerous "house rules" that people invent according to what they think the game should be like; some of these will be discussed as well.
The maintainers of the FAQ (see Sources, Q5.1) would be particularly interested in hearing of any official rules that differ from the ones discussed here, but are also interested in additional "house rules".
In all versions of the game, one way to win is to be the last surviving player. In a two-player game there could be neutral armies left at the end, but usually it means taking all 42 territories on the map. In some versions this is the only way to win. The U.K. rules (old and new) allow an alternative victory conditions: secret missions (Q2.2.1). The new U.S. edition offers a third scenario: capitals (Q2.2.2). Both alternatives tend to make for a shorter game.
Missions are offered in some versions of the rules as an alternative way to win the game (see Q2.2). Missions have been available in European editions for a long time, but have only recently appeared as a variant in the U.S. Missions are used only if there are more than two players.
When playing with missions, each player is dealt a "mission card" at the start of the game. Each player keeps his mission a secret from the others. Missions are dealt after colours are chosen. Some sets of rules (including the new U.S. rules) clearly specify that missions are dealt before territories are allocated, while others leave this ambiguous.
There are 14 (only 12 in the 1998 US edition) different missions, as follows:
i) Kill a certain colour. There are 6 of these, one for each colour. If fewer than 6 colours are going to be used in the game then the cards corresponding to the unused colours are removed from the deck before missions are given out. (This is quite clearly stated in at least some versions of the rules, and failing to do so doesn't make much sense.) If you draw your own colour, then your mission changes to mission (iv).
ii) Conquer some combination of continents. There are 6 of these:
Conquer Asia and South America;
Conquer Asia and Africa;
Conquer North America and Africa;
Conquer North America and Australasia;
Conquer Europe and South America and a 3rd continent of your choice [this mission is omitted from 1998 US edition];
Conquer Europe and Australasia and a 3rd continent of your choice [this mission is omitted from 1998 US edition].
iii) Occupy 18 territories with at least 2 armies in each territory.
iv) Occupy 24 territories (no restriction to 2 or more armies in each).
The old U.K. rules are unclear as to whether the unused missions should be kept secret, so no one knows what other people might have. Later editions clearly state this to be the case. Similarly, it is not clear in the old U.K. rules whether an eliminated player's mission is revealed. Certainly it is useful to know the 14 missions well so you can see if someone is on the verge of completing one of them, even if you don't know whether it is their mission.
In the old and new U.K. rules, to succeed in a mission of type (i) you must kill the last remaining unit of that colour yourself (it doesn't matter who kills the rest of them; it's all or nothing on their last unit). If someone else kills the last one, or you draw your own colour, then your mission changes to mission (iv) (so several people could be trying to get 24 at once).
In the new U.S. version of the rules this is not true. In the case of the mission of destroying a colour, anyone may complete your mission. "For example, if your mission is to destroy all the yellow troops and another player actually removes the final yellow armies from the board, that player has helped you complete your Secret Mission."
To win the game you have to complete your mission. At that point you show your card and the game is over. The old U.K. rules are unclear as to whether this could be during another player's go, in the case that you have 24 or more territories and someone else kills the last of your target colour. The new U.K. rules say clearly you can only win during your own go. This prevents the above leading to a simultaneous victory by two players.
Some people play that you have to hold your victory condition for a full turn (that is, one go for each player), which makes all missions except type (i) harder. This appears to be a house rule, not in any official version. Using this house rule you would have to agree at what point you reveal your card -- after completing your mission but before the full turn, or just at the end of the full turn. (The latter could lead to arguments if people can't remember when you first met the conditions; e.g., if a retreat (see Q4.2) might have added an army to one of your territories to achieve mission (iii).) It is clear that the only thing that could go wrong for someone completing a type (i) mission would be to be wiped out in that following turn.
To create a clear rule for mission completion you could state any of the following:
i) You can only win at the beginning of your go (will tend to make continent missions harder).
ii) You can only win at the end of your go (this is basically equivalent to "during your go" because you don't tend to worsen your position vis a vis completing a mission during your go).
iii) You can only win a full turn after revealing your mission and maintaining its conditions in the interim.
One could no doubt come up with still more rulings here.
The missions make for a much faster game with more uncertainty -- the strongest looking player may be nowhere near winning. Of course, if you take over the whole board you have completed most missions (and the 18 with 2 in each would just be a matter of time) and you certainly win.
Slowing the missions down with the "hold for a turn" rule makes the game rather more like world domination Risk -- if someone can hold those quantities of territory for a turn they are well on the way to a victory.
As with Secret Missions, this is an alternative victory condition meant to shorten the game. Play is similar to World Domination, but before play begins, each player chooses one of his territories to be his capital. The choices are revealed simultaneously. The objective is to capture all (or some number of) capitals. If you lose your capital, you are not out of the game. (But you aren't doing very well!)
Note that Capital Risk has nothing at all to do with Castle Risk; see Q2.2.3.
This is a very different game, also put out by Parker Brothers, that uses some of the mechanics of Risk while changing and adding a lot of things. Most seasoned Risk players seem to feel that Castle Risk is an abomination. Since this comment has been in the FAQ I've had some defenders of Castle Risk, though they mostly use it to come up with better variants, rather than playing it straight. And now, since adding that comment, I have had messages defending straight Castle Risk -- so maybe you will like it?! The 1992 U.S. edition of Risk has Castle Risk on the other side of the board.
Some of the differences include: The map is different; it is not a game of world conquest. There are castles (of course), which are the goal of conquest; in this regard (only) it has some similarity to Capital Risk (see Q2.2.2). Instead of Risk cards that are cashed in for extra armies, one gets cards that can be used to gain special abilities, such as bonuses on attacks or defense, ability to attack across water, etc. The attacker cannot roll more than 2 dice against a castle space.
In the U.K. rules and the 1959 U.S. rules, the Risk cards are dealt out as evenly as possible (omitting the jokers), and each player then places one army from their initial allocation of armies in each territory thus assigned to them. In later U.S. rules, the start is not determined using the cards; players take turns putting armies in unoccupied territories until each territory has one unit.
In the 1959 U.S. version, players start with only the one army per territory. In all other versions, after each territory has a single army, players take turns distributing the remainder of their armies, one a time, on their territories. In the old U.K. rules, you can place at most 4 armies in a single territory at setup. The new U.K. rules and the U.S. sets have no such limit.
The total number of armies per player is the same in most versions except the 1959 U.S. edition, and depends on the number of players:
6 players - 20 armies each
5 players - 25 armies each
4 players - 30 armies each
3 players - 35 armies each
2 players - 40 armies each (this varies between editions)
This figure can be altered to give a different style of game. If it is lowered, the game will start more slowly and turn order and luck will play a greater part. If the number is increased, the game will be bloodier and the action will begin more quickly.
The old U.K. rules state clearly that on each of your turns you may do one of two things, either reinforce or attack. If you choose not to attack then you get additional armies that you may allocate among your territories.
In the new U.K. rules, and all versions of the U.S. rules, you always get your armies for territories owned regardless of whether you then attack. This makes for a quite different game but is clearly equally "correct".
An interesting house rule is to allow reinforcements to be placed only after all attacks are finished. This makes sneak attacks impossible.
One thing that appears to be the same in all versions of the rules is the number of armies you get: 1 army per 3 territories you control, rounded down, minimum of 3, plus extra for complete continents or for sets of Risk cards cashed in.
In the rules where you forfeit reinforcements if you attack, you still get armies for complete continents or Risk cards even if you attack, which would seem to make owning continents even more important.
The "1 per 3" reinforcement armies can be placed anywhere, subject to limits on number of armies per territory (see Q2.8). The same is true for armies gained by turning in sets of Risk cards. If, however, a territory you occupy appears on a Risk card you are turning in, the two bonus armies (if you're using that rule) must be placed on that territory.
Some players claim that the extra armies gained for controlling a complete continent must be placed in that continent. Nobody has yet cited this in any official version of the rules, so it might be just a house rule.
There is a lot of variation here. All the rules seem to agree that you get a Risk card only if you conquered a territory during your go, and that you can cash them in in sets of 3 of the same symbol or 3 all different symbols. (The pack includes 14 of each symbol plus 2 jokers which can be used as any symbol. A joker plus any other two cards can thus form a set, and any 5 cards must contain a set.)
The U.K. rules offer two methods. The "standard" scheme is:
A set of 3 Artillery is worth 4 armies
A set of 3 Infantry is worth 6
A set of 3 Cavalry is worth 8
One of each is worth 10
Jokers are of course almost always used to make sets of one of each.
This scheme means it is impossible to gain huge numbers of armies in one go unless you control most of the board. It also tends to give little incentive to hold your cards once you have a set, unless you have a chance of improving the value of the set. (Once you have 5 cards you must have a set, and must play a set at the start of your next go.)
We invented a house rule which allowed you to cash in Risk cards at any stage in your go (provided it was before you picked up a new card) and that you could put the armies on gradually as you attacked and saw where they'd be needed. This made the cards a bit more powerful. With other sets of rules for armies-per-set this is probably rather over-the-top.
The U.K. rules also list the possibility that the number of armies does not depend on the type of set, but rather the number of sets cashed in by all the players so far, e.g., 4, 6, 8, 10, 12, 15, 20, 25, 30, etc. This sequence (with later sets continuing to increase the payoff by 5) is also used in all editions of the U.S. rules.
Personally I think this makes Risk cards too important; with the other scheme it is not worth forfeiting territory reinforcements to gain Risk cards usually - to make a set takes 3 cards, for each of which you have forfeited at least 3 armies. However sometimes there are other reasons to attack and the cards are a nice bonus. Players who use the increasing payoff for sets seem to agree that it is paramount to conquer at least one territory each turn so as not to fall behind on acquiring cards. Two players will sometimes agree to leave one territory weakly defended so they can take turns conquering it! Thus it makes for a very different game, but equally valid.
The 1980 U.S. rules and the new U.S. rules suggest making card sets worth only one more than the last. Thus the first set is worth 4, then five, six, etc.
Some players have house rules giving different sequences of payoffs. For instance, you can play that the sets pay 4, 6, 8, 10, 12, 15, 20, but then the payoffs decrease again: 15, 12, 10, 8, 6, 4, 6, 8, etc. Or that after 20 the next payoff is 4, then 6, 8, etc. This can make it more important to be able to cash in sets of cards at the right time.
In some rules the actual territories on the cards are completely meaningless apart from at the start of the game when they are used for deciding initial territories (see Q2.3). In other rules, the territories are worth a bonus when cashing in a set of cards: If the set includes one or more territories you control at that time, you get a bonus of two armies for each such card. Each pair of bonus armies must be placed in that card's territory. The 1980 U.S. rules state that you can claim no more than 2 armies through such use of Risk cards, leaving you the choice of where you want them out of the territories whose cards you have.
In some rules the territories on the cards are used for both purposes.
The old U.K. rules make no mention of any possibility of gaining Risk cards by wiping out other players. The new U.K. rules, and all U.S. rules, state that if you destroy a player's last unit you get any Risk cards they had at that moment.
In general, the rules always allow you to turn in sets of cards at the start of your go. If you have 5 or more cards, you MUST turn in a set.
If you don't get another player's cards when you wipe him out, there is no way to get above 5 cards. When you get your 5th card, you will be required to turn in a set at the start of your next go.
If you do get cards by wiping someone out, it is possible to go way over 5 Risk cards. In the new U.K. rules, you simply hold the extra cards until your next go, at which time you are required to trade in enough sets to get you below 5. In most U.S. editions, when you gain cards by wiping out a player, you are required to trade in sets immediately to reduce the number of cards which you continue to hold to 4 or fewer. You can also trade in cards if you do have sets, however many cards you hold. This could mean that you trade in a set that you had already completed, having hoped to improve it with the new cards but not done so.
However, the U.S. rules since the 70s state that if by capturing Risk cards a player holds "6 or more" cards, he must turn sets in until he has "4 or fewer." This (presumably intended as a clarification) opens up a couple of loopholes. Firstly, where you may have exactly five cards, and the card at the end of your turn then puts you at six, allowing you to cash in two sets of cards at your next go. Secondly it is now unclear whether you may continue to turn sets in once you have reduced your holding to four or fewer cards. This rule tends to reduce the desperation plays sometimes seen under the older U.S. rules, where a player holding (say) 3 non-matching cards would go all-out to wipe out a player holding 1, in the hope that the 1 extra card would give him a set that he could cash in in mid-turn to gain the armies to wipe out everyone else, since under these rules it is clear that you may not turn in sets mid-turn if you don't get up to 6 cards mid-turn.
One implication of the new U.K. rule cited above, as opposed to the U.S. rule, is a disincentive to wipe out other players. In holding their Risk cards as well as your own, you become an inviting target for other players to destroy you (and a threat due to the cards you will have to cash in next go), especially if you have weakened yourself significantly in destroying the first player.
A variation suggested in some sets of rules (such as 1980 U.S.) is to allow special powers if the attacker holds the card with either the territory being attacked or attacking from. By revealing the Risk card the player may re-roll any one die on each battle involving that territory. The card is not used up and goes back into the player's hand. You cannot use more than one card per battle, but may use more than one card during a single turn. Again, this is only for the attacker, not the defender. See Q2.6.3 for definitions of attack and battle.
While not explicitly outlawed by the rules it is generally considered against the spirit of the rules to donate cards to another player, trade cards with other players or simply throw cards away. (You might want to do this if you have five cards but only a set of three cannons and are using the rules where a set's value depends on the type of cards in the set, see Q2.5).
Again there is variation between sets of rules. This much seems to be agreed: The attacker and defender each roll some number of dice, and the highest die of the attacker is compared to the highest of the defender; the loser (defender wins ties) loses one army. If both sides rolled at least two dice, the second highest dice are compared and the loser again loses one army.
The rules also agree that the attacker can roll up to three dice, and must have more armies in the attacking territory than the number of dice rolled. If he wins, he must move into the conquered territory at least as many armies as the number of dice rolled on the last attack. (He can of course move more armies if he so wishes, but must always leave at least one army behind.) Thus one occasionally rolls fewer dice so as to avoid having units sidetracked. One can also start out rolling three dice and then switch to rolling fewer dice once the defender is down to 1-2 armies. There is no requirement that each attack be done "to the death" before switching to another attack or changing the number of dice being rolled.
There is less agreement about the defender's dice, and the sequence of rolling. See the next two questions.
The number of dice the defender can roll is explicitly different in different sets of rules. In most editions the defender rolls 1 die if he has 1 unit in the territory, and can choose to roll 1 or 2 if he has 2 or more units. Some sets of rules (New Zealand and U.K. circa 1990) state that the defender can only roll 1 die unless he has at least 3 units, which weakens the defender. It also removes the interesting decision an attacker has when he doesn't actually want to take a territory but is happy just to reduce it to 1 army (when you get to the redeployment rules you'll see why that might be) since he no longer needs to worry about reducing the defender from 2 to 0 in one throw. I have on occasion rolled just one die as the attacker because the defender only had 2 units left.
The old U.K. rules are quite clear that the attacker rolls the dice first, then the defender rolls. Though it isn't explicit, this suggests that the defender can choose how many to roll based on the attacker's dice, since this would seem to be the only point of rolling later.
This sometimes causes the defender to use only 1 die when he was entitled to use 2. If faced by 2 sixes you only roll 1, whereas faced with the second highest being a one you roll 2. (In both cases this assumes you are not trying to let the attacker through with minimal losses to get to a mutual enemy.)
The U.S. and new U.K. rules specify that both attacker and defender decide beforehand and roll simultaneously. This leads to a faster game and it is rare that the defender rolls 1 when entitled to 2.
The 1980 version of the U.S. rules states, "An attack is actually one or more battles which are fought with dice. The object of an attack is to capture a territory..." Elsewhere in the same rules, it says an attack is "fought to capture one or more territories on a turn." So it's unclear whether an attack is a string of battles for the same territory, or all the battles of a player's turn. But a battle is definitely a single cast of the dice. The 1993 version of the rules, unfortunately, omits this wording.
Once a player has finished with all his attacks (if any) for the turn, he can "redeploy" (known as "fortification" in some sets of rules). This means moving any number of armies from one of a pair of neighbouring territories to the other (as usual you cannot leave a territory unoccupied). This is all the movement of armies that is ever allowed in the game except upon winning a battle.
The old U.K. rules impose a maximum of 7 to be moved. This set of rules also denies reinforcement armies if you attack. Together, these rules mean that a buffer zone of 1-unit territories controlled by your opponent can be quite a useful defence. (Hence there is some reason to drive an opponent's territory down to a single unit without actually taking it.)
Some people remove the limit of 7 (as in the U.S. and new U.K. rules). What is probably a house rule, however, is to allow the movement of the group (perhaps limited to 7) through an unlimited number of connected, controlled territories. Another variant (in the 1980 U.S. rules) that makes for quite a different game is to allow you to move all your units entirely freely within any connected set of controlled territories (subject to leaving 1 in each). With random starting positions this would make it even more important to connect your troops up. Yet another house rule is to allow each single army to move to an adjacent controlled territory, provided that after all such movement there is at least one army per territory.
A territory may never be left empty and all the standard rules impose no upper limit. A variation from the 1980 U.S. rules is to allow only 12 armies per territory. If during placement you are unable to place some armies, then you lose them.
The 1980 U.S. rules offer a variation that simulates the influence of commanders, or great generals being at a particular battle. Once per turn, each player may change one of the dice he rolled to a six. Thus a roll of 1, 2, and 3 may become a roll of 6, 2, and 3. The rules are clear in stating that this may only be performed by an attacking army.
If the defender must choose how many dice to roll before seeing the attacker's roll (see Q2.6.1), then he should always roll as many dice as permitted, assuming he is trying to kill as many attacking armies as possible.
If the defender gets to see the attacker's roll before deciding, then he should generally roll 2 dice if the attacker's second die is 3 or less and 1 die if the attacker's second die is 4 or more. If the defender is down to exactly 2 armies, then he should roll 2 dice (assuming the rules permit him to) if the attacker's second die is 4 or less. This is because rolling one die and losing a single army forces the defender to roll only one die the next time, so he's better off trying to get lucky while he can still roll two dice. If the defender has exactly 3 armies, however, he should still use the "second die is 3 or less" rule, rather than playing conservatively to avoid losing two armies. If the defender has exactly 4 armies, he should roll 2 dice if the attacker's second die is 3 or less, or if the attacker's roll is 4-4. With 5 or more armies, always use the "3 or less" rule.
It depends on whether the defender gets to see the attacker's roll before choosing how many dice to roll (see Q3.1 and Q2.6.1). If the defender decides before seeing the roll, then here are the odds of the various possible outcomes given the number of dice being rolled:
Attacker Defender rolls: 2 dice 1 die rolls: +-----------------------------------------------------------------+ | Att lose 2: 29.26% (2275/7776) | Att lose 1: 34.03% (441/1296) | 3 | Def lose 2: 37.17% (2890/7776) | Def lose 1: 65.97% (855/1296) | dice | Each lose 1: 33.58% (2611/7776) | | +-----------------------------------------------------------------+ | Att lose 2: 44.83% (581/1296) | Att lose 1: 42.13% (91/216) | 2 | Def lose 2: 22.76% (295/1296) | Def lose 1: 57.87% (125/216) | dice | Each lose 1: 32.41% (420/1296) | | +-----------------------------------------------------------------+ 1 | Att lose 1: 74.54% (161/216) | Att lose 1: 58.33% (21/36) | die | Def lose 1: 25.46% (55/216) | Def lose 1: 41.67% (15/36) | +-----------------------------------------------------------------+
Thus, in the most common case (3 dice vs 2 dice), the attacker loses an average of 0.921 armies to the defender's 1.079, which is a ratio of about 5 to 6.
If the defender gets to see the attacker's roll before choosing how many dice to roll, and applies the strategy given in Q3.1, the ratio when the attacker rolls 3 dice is about 1.001, i.e., the attacker loses very slightly MORE armies than the defender.
Again, it depends what version of the rules you're using. The following
analysis assumes that
(a) the defender must choose how many dice to roll without first seeing the attacker's roll, and
(b) the defender is allowed to (and therefore should) roll 2 dice if he has exactly 2 armies remaining.
If the attacker has enough armies that he will always be able to roll 3 dice (and is willing to do so), then the number of armies he expects to lose before conquering a single territory is, on average:
0.8534144 N - 0.2213413 (1 - (-0.525359)^N)
where N is the number of armies initially in the target territory. This is only an average; any particular battle of course depends on how the dice may fall! Still, this formula has a few interesting corollaries.
The increase in defensive value obtained by adding a second army to a territory that had only one army is far greater than that obtained by adding that army anywhere else. In eliminating a one-army territory, an attacker expects to lose about 1/2 of an army; in eliminating a two-army territory, he expects to lose about 1 1/2 armies. Adding a third army to a two-army country costs the attacker only about 3/4, and additional armies cost amounts that converge rapidly toward 0.8534 (about 5/6). So a good rule of thumb for setting up a defence against possible total annihilation (if that is your only concern) is to put two armies in each territory, with all leftover armies piled into a single place.
Because of the alternating sign on the exponential term, it is generally slightly more cost-effective for defence (and makes no difference to offence) to have an even number of armies. For instance, two countries with four armies each will cost an attacker 6.42 armies to get through (not counting those left behind), while dividing those same armies into three and five will cost him only 6.34 (see Q3.4).
If the defender gets to see the attacker's dice before choosing how many dice to roll, and follows the strategy given in Q3.1, then the number of armies the attacker loses in conquering a territory is, on average:
1.0010293 N - 0.3891440 + 0.6147449 * (-0.1563182)^N
when N is greater than zero. Here the exponential term diminishes much more rapidly as N increases, so the advantage of even numbers over odd numbers of defending armies is less pronounced. The advantage of adding the second army to a territory still holds.
If you want to find out more about the expected losses from a risk battle there is a really neat web calculator available from Scott Bartell, here is some information About RiskOdds.
Other pages that discuss and/or calculate odds and losses
Daniel Drake's Complete Odds for Risk Attacks, Ismo Puustinen's The Risk Probability Calculator and Martin Oldfield's Some probabilities for Risk (all these sites are new additions to FAQ 5.4).
Based on the formula in Q3.3, you can estimate how many armies you'll need to conquer all of a player's territories, including the armies you have to leave behind as you go. This again assumes that you're always attacking with three dice, and are never forced to "split" your armies to pursue two different paths of conquest.
As a fairly good approximation, you can take 5/6 the total number of defending armies, plus 7/9 the number of territories, minus 1/9 the number of territories that contain exactly one army. You might also add 1/18 the number of territories that have exactly two armies, but this is unlikely to be significant. This is the number of armies you expect to lose, including those left behind as you move. You should have at least 3 more than this so you can expect to be able to attack with three dice all the way to the end, plus as many more as you feel makes for a safe margin against bad luck.
The above approximation is for rules where the defender and attacker roll their dice simultaneously. If the defender gets to see the attacker's roll before deciding how many dice to roll in defence, the approximation becomes: take the total number of defending armies, plus 3/5 the number of territories, minus 1/10 the number that contain exactly one army.
The odds that your first three cards will form a set would seem to be about about 1 in 3, since given any two first cards there is one symbol that will turn them into a set. But the odds are shifted slightly by the fact that, once you have one or two cards, you know those symbols are less likely to be drawn. The possibility of a joker shifts the odds far more, such that the chance is actually 42.28% that three randomly selected cards will form a set. (12.37% of the time it will be a set of three the same; 29.60% of the time it will be a set of one of each; and 0.32% of the time it will include both jokers and thus could be either type. See Q2.5 for why it might matter.)
The odds of four cards including a set are 81.70%. However, if someone has three cards that did not form a set (for instance, you're pretty sure they'd have cashed in a set if they could've), and they draw a fourth card, the odds that the fourth card gives them a set are only 68.29%. Five cards, of course, always include a set.
If you're wondering whether someone else's cards might include a set, you can sometimes get more precise odds by considering your own cards, particularly whether you are holding any jokers. If you're looking at one joker (or remember that one joker has been cashed in since the last time the deck was shuffled), the odds that someone else with 3 cards has a set are 38.06%. Someone with 4 cards is 79.87% likely to have a set. If you know that the other person has neither joker, the odds he has a set with 3 cards are 33.41%; the odds he has a set if he has 4 cards are 77.80%.
Risk is the elemental war game and its rules are very resilient to changes and additions. Below are a few variants.
Reserve one army for the Martians (Green, of course). After everyone's taken a turn, draw a card from the deck. Put three Martian armies there. Reshuffle the deck.
They will fight until either they are destroyed or take the territory. If they do win, give them a card.
After that, each time it becomes the Martian's turn, draw a card and place all the armies they have coming to them (three minimum) evenly among the territories (those they already hold and the newly drawn one). Odd armies are placed in the new territory. Then the Martians will attack as long as they are able; until they are down to one army in each territory (and zero in the newly drawn one). Martian attacks are chosen with this objective: To gather all their armies together into one mass. They attempt this by, firstly, all groups except the largest will head toward the largest. Then the largest will head toward the second largest. They will always take the shortest path. You can roll a die if there is a tie.
Finally, the Martians only exchange risk cards when they have 5 or more (possible under some rules). They exchange until they have 4 or less.
When a territory is attacked, the defending army has the option to retreat into an adjacent territory that is held by the same player. The attacking army then loses one unit and must move at least one unit into the territory. This means that an attacker must have three armies to attack an army with a route to retreat. The defender declares whether or not he wishes to retreat once the attacker first begins to attack the territory (before any decisions about number of dice, etc.).
This variant was devised by several navy chaps and published in issue 34 of Dragon Magazine, February 1980. The article was titled 'Feel Like RISKing Everything?' (thanks to Jason Tuchelt for finding this information in Dragon magazine's index in issue 112.) The full rules run to nearly three pages, but the basic ideas were as follows.
At most five players play -- red tokens are reserved to represent nukes.
Two new overseas borders (a la Brazil to North Africa) are added, to connect Madagascar to Western Australia and Eastern Australia to Peru. The continental bonus for South America and Australia are raised to 3, and for Africa to 4, to reflect the increase in borders.
Starting on a player's third turn, he may take up to half of each turn's reinforcements (rounded down) as nukes instead of regular armies. Bonus armies for continents or sets of Risk cards can be taken only as regular armies. Nukes are placed just as regular armies, but can NEVER be moved.
Starting with a player's fourth turn, he may use nukes for attacks. Any number of attacks per turn can be performed with nukes. The player states where the nuke is launching from, and the target; the target can be ANY territory on the board. The defender either removes one army from the target, or defends with a nuke of his own. The defending nuke must be in the target territory or one adjacent BY LAND. If the target has any nukes and is down to only one army, the target's nuke MUST be used for defense. Other players with adjacent nukes may also choose to provide a nuke for defense. The attacking nuke, and the defending nuke or army, are removed.
If a territory becomes devoid of armies because of a nuclear attack, it becomes a nuclear waste land, meaning no armies may occupy it for at least two turns. Place two coins in it, topped by a piece from the attacking player's box. At the beginning of each player's turn, he removes one coin from each wasteland marked by his pieces. When all coins are gone, the territory is left EMPTY. A nuclear waste land may be nuked, but the number of coins on it never exceeds two.
When a territory containing nukes is taken over by conventional armies, the attacker first moves armies into it according to the standard rules. Half of the nukes (round up) are removed. The remainder are immediately used by the defender to attack ANY territory on the board (including the one just taken over) in accordance with the rules given above. If this results in the creation of a nuclear waste land, a piece from the original attacking player is placed on the coins (as it occurred during his turn).
A player may nuke himself. (This might be done, for instance, to create a waste land as a barrier to a conventional attack.) Players with adjacent nukes may use them to defend the player from nuking himself!
In order to get continental bonuses, a player must have armies on ALL of the territories in that continent.
This really speeds up play, as well as make the game comical and unpredictable. Every time double fours, fives, or sixes are thrown a nuclear melt down has occurred (a double is counted whenever two dice of the same colour match). The top card is drawn and all the armies in the territory shown on the card are removed. A penny (or similar marker) is placed there to show that the area is a wasteland.
For the rest of the game any army moved into the territory is halved, fractions dropped. However, the territory no longer counts towards holding a continent. If a continent is completely wasteland then no one can get the armies for completely controlling it. Units don't die off as they spend time in the territory or when they leave. One army must be left as with normal territories.
Optionally, if a territory has a second melt-down, clean-up efforts have succeeded in removing the radiation. The territory is restored.
As another option, armies may not be required to stay in an irradiated territory. If played this way, an attrition rule should be added so that at the beginning of that player's turn, after he places all armies, the army is halved, rounding down (three armies become one, one army becomes none). A single unit is allowed to attack from a wasteland space.
Instead of random, frequent melt-downs, this rule provides more strategy. After placing armies, a player may choose to forego any attacks that turn and make a tactical nuclear strike. The player uses one of the cards in his hand and the country on the card is nuked in the same way as described in Three Mile Island Risk. The player may then fortify to end his turn.
The idea here is that each player is of a certain continental background (European, Asian, North American) and has the ability to convince the people living there to revolt sometimes.
At the beginning of the game each player puts an army next to the list of continents on the board to denote his background. Two players may choose the same continent.
At the start of any turn, a player may choose to use a card to have a revolt instead of attacking that turn. He places his armies as usual then reveals a card. The card must have the name of a territory on his continent. All the armies there are changed into his color army and are his. So, if the blue player has ten armies in Central America, the red player may take them as his own with the proper card.
Under this variant, any territory can attack any other territory even if they are not adjacent. The attacking territory immediately loses half its armies (rounding the loss up), then the attack proceeds as usual. If the attacker breaks off the attack to conduct a different attack, including attacking the same target from another location, the airlift is over and resuming the airlift-based attack requires again losing half the remaining armies.
Variants on the variant include limiting each player to a single airlift attack per go, and/or requiring that the player be holding the card for the target territory.
This is really a different game that uses the Risk game pieces. This is in Schmittberger's book, if you wish to check it out.
Each player is allocated 20 armies (25 sometimes if there are only 3 players). Then each person is dealt 5 territories and armies are placed (one at a time in cycles). All other territories are neutral and have 3 armies. These defend optimally but do not attack.
Each person has a set of goals (public). After setup each person is again dealt 5 goal territories (ensuring these do not match ones they start with). To win the game a player must take (and hold for a turn) all their 5 goals.
This rule makes attacking along a sea-lane more difficult. It gives the board a bit more character geographically, makes continents easier to hold and makes it more worthwhile putting significant garrisons on islands.
For the first 4 rounds of combat across any sea-lane the defender is allowed to roll an extra die (even if this means rolling 3 dice). This can mean (if the attacker is also rolling 3) that 3 armies are at risk per round of combat. If the defender loses on all 3 dice (rather unlikely!) but has only 2 armies he simply loses those 2.
This variant notices that the difficulty of "kill red" type missions depends greatly on the number of players. Low number of players, they are harder than continent missions, high number easier. So, either you drop them to balance things, or, if you realise what fun they are, you play like this.
Only the "kill" missions are used. They are dealt out to the players. Each checks if they have their own colour. If any does they reveal it and cards are shuffled and dealt again, until a successful deal is made, i.e. all players have to kill other players. You do not need to finish off your victim yourself, so, as soon as one player is eliminated, somebody has won! This means there is no sitting around for eliminated players while the rest play on, and the game should be very tense. As well as weakening your target you need to ensure nobody else gets too weak!
This version of the game should only be played if you consider standard world domination Risk to be far too short a game! You can play double-Earth Risk, triple-Earth Risk or even-higher-ple Earth Risk.
For n-ple Earth Risk, you need (funnily enough) n Risk sets. The boards are positioned side by side, and Kamchatka on Earth 1 links to Alaska on Earth 2, etc., and Kamchatka on board n links back to Alaska on board 1. You use the playing pieces from all n sets, and start with n times as many armies as usual. (This can lead to some huge heaps of armies initially, but if you do pile them up, that means you're very weak elsewhere ... things soon even out a bit.) Each pack of cards is shuffled and dealt independently for each board, so you have (roughly) the same number on countries on each board to start with, but once the game has begun all n packs (including 2n jokers) are shuffled together. When it comes to trading in cards, it's entirely up to the individual player which Earth each card corresponds to, so (for example) if you hold Iceland on two of the boards, and trade in the Iceland card, you can put your two bonus armies on whichever of those two Icelands you prefer. To win, a player must conquer all n Earths. Bonus reinforcements for holding a continent work as usual (ie. you need hold only one North America (not all n) to get those reinforcements), and you can claim continental reinforcement bonuses for the same continent from more than one board (eg. if you hold two South Americas and an Autralasia you can claim 6 bonus armies).
The original FAQ was drawn from Owen Lyne's copy of the rules. Other people have contributed other versions over time. Some of the contributors can be reached as:
Mathematics Department, University of Nottingham, UK
(was email@example.com - but this address no longer seems to work and I don't have a new one for him)
Some of the sources for the variants given in section 4 are (others were received as submissions for the FAQ by their inventors):
Games Magazine, August 1994
New Rules For Classic Games, by R. Wayne Schmittberger
Published by John Wiley & Sons, Inc.
(c) Copyright 1992
Dragon Magazine, issue 34, February 1980
(Posted to the net in January 1987 by Graham Wilson)