# A Strategy Game Involving Conquering of Regions

This is a game that I saw in a book long ago. Unfortunately I cannot find the book now, and even if I can as I recall it does not provide a strategy. I will be very grateful if you can provide me a link to the authoritative rules of the game (and verify that I didn't remember any part wrong. Edit: @DqwertyC provided a link to the rules of the game "Aggression" which I believe is the authorative version ), but of course I am more interested in the best strategy, which I don't know, either. You may consider the rules below as an adapted version of the game but please clarify in your answer which set of rules you are using.

The game only requires a pencil and a piece of paper, and optionally a calculator (as stated in the book) if you are not good at maths or are simply lazy.

The game is played between two people (Alice and Bob). Before the game they agree on an odd number $N$, which should be significantly smaller than 100 but not too small. See the rules to get a rough feeling on how large $N$ needs to be to make sense.

The game has three stages.

Stage $\rm I$. Generating the Map  It begins with Alice drawing a closed curve on the paper, dividing the paper into two parts; the inner part is the current map (with a single region). Then Bob draws another curve to add another region to the map. The new region must be adjacent to an existing region and cannot fully encompass an existing region. Then it is Alice's turn, and it goes on. After the $i$th move, there should be exactly $i$ regions in the map, and each region should be adjacent to at least one other region, and no region can fully encompass another region. This stage ends after exactly $N$ moves, and there will be $N$ regions in the final map.

The rule that no region can be fully encompassed by another region is a rule that only exists in my adapted version.

Stage $\rm II$. Occupying the regions  This round starts with Alice. Alice can choose any region from the map, mark it with her favorite pattern to occupy it and attach an integer between 0 and 100(inclusive) to it. Then Bob chooses a region, marks it and attaches a number to it. This goes on until all regions are occupied. There is, however, one restriction on the attached numbers. The sum of the numbers attached by any individual player cannot exceed 100.

Stage $\rm III$. Conquering  This stage starts with Bob. Bob and Alice take turns to make a move. At each move, the player can choose to skip (do nothing), or he can choose one of his opponent's regions that is adjacent to one of his regions, and conquer it following the rules. The rules are:

• the attacking region's number must be strictly greater than the number of the attacked region
• the attacked region must not have changed hands in an earlier move (i.e., each region can be conquered at most once).

Attacking does not change the number in the attacker's region, nor the attacked region. In fact the numbers attached to all regions are fixed once Stage $\rm II.$ is complete.

This is different from the authorative rule of a "Aggression". In my adapted version, the "troops" in the conquered region can be used in a later stage.

This stage ends when both players choose to skip.

Goal of The Game After stage $\rm III$, the player with the most regions wins. The attached numbers are irrelevant.

If there's anything unclear about the rules, feel free to ask. I will be happy to follow up.

Now try to find out the optimal or the suboptimal strategy, smart brains!

Side note: It is not clear in the authorative version who is the first to make a move at each stage. I believe that can make a really big difference.

• So if Alice conquers Bob's region... Then what's the value of the conquered region? Also, on stage II: is the sum of ALL the regions less than or equal to 100, or is it simply the sum per player. – Jakob Lovern Jan 19 '18 at 18:00
• The value remains unchanged (same as before conquering). And for the second question, it's the sum per player. – Weijun Zhou Jan 19 '18 at 18:05
• If N=5 will there be 5 regions or 10? In other words, does one turn in Stage I consist of both players creating a region? – Aaron Dietz Jan 19 '18 at 20:37
• If $N=5$ there will be 5 regions. – Weijun Zhou Jan 19 '18 at 20:52
• This looks like the game "Aggression", by Eric Solomon. There are slight differences in that version of the game (i.e. regions can be entirely surrounded, always 20 regions total), but the game play described is almost identical. papg.com/show?2XNG – DqwertyC Jan 19 '18 at 21:30

It is not noted what happens to the attackers armies in the post; however, the comments linked a game called Aggression, which does seem similar, so I'll be using that as a reference (and as the authoritative ruleset).

Since the attacking region does not lose troops in Aggression, the best strategy (assuming I'm Alice) is to make sure I draw the biggest damned closed curve I can fit on the piece of paper, leaving a sliver of room to draw the rest along the side, such that every piece has to touch my big piece.

Once the game begins, I must immediately claim that largest piece with the number 100. This makes my country invincible.

At this point, you've already tied or won the game, depending on what Bob does.

If Bob claims more than one country, you win - since every country is touching yours, and troops don't move around, as soon as Bob takes one other country, you can instantly take Bob's main country, forcing him to only pass, while you can take every other country around your large curve.

The optimal strategy for Bob at this point is to claim any country using the number 100 as well (otherwise he will instantly lose his army when the game starts). Depending on the size of N, he may or may not Tie the game.

N   | Win/Loss for Bob
-----------------------
2   |  Tie
3   | Loss
4   | Loss (Possible Tie if Bob takes 100 in a country that has 2 neighbors)
5+  | Loss


Note that since regions can only change hands once, as soon as N is 5+ Bob will lose. If regions are left empty (so that Bob can take them permanently, assuming that you can attack blank regions), and Alice always targets Bob's stationed troops with each attack, Bob will only be able to "maintain" his current number of countries, since Bob will lose all his original countries to Alice eventually. That would leave Alice with however many countries Bob started with + 1 (her own original), ensuring a win.

Note: this relies on one key point as well; as long as there isn't enough room on the piece of paper to draw a second layer of small areas, you win. If Bob goes and gets another piece of paper and tapes it to the side to increase the game area, this strategy doesn't work.

• This feels like a very cheap strategy, certainly not in the spirit of the game. Really, you get to do this once, and if you attempt it a second time the other player will quit. – DqwertyC Jan 19 '18 at 22:02
• @DqwertyC You're right, but it is optimal - guaranteed not gonna lose. Nobody said I had to guarantee fun. – Aify Jan 19 '18 at 22:04
• While it is indeed cheap it is a working strategy. Thank you for your interest and effort. – Weijun Zhou Jan 20 '18 at 3:36
• Win is not always guaranteed, you should note that. Example: For N=4, assuming under your assumption that your 100 land touches everything, then look at the remaining 3 lands. Wherever you put the 2 lands, bob places last and can connect those two in with the third remaining land (fourth in total). One of those 3 will be your 0 and Bob can force a draw by having 100 in the middle of the three lands (forces your other 0 land to be next to bobs 100 land); he claims your 0, while you claim his other 0. Draw forced for N=4 by Bob; where you wrote a loss for Bob in your post for that case. – Vepir Jan 22 '18 at 13:48
• @Sneftel As I read the question; Each starting land must be occupied as players take turns until all land is occupied, at least with value $0$ if you have no army left. $II.$ does not state you can skip occupying land, in fact, it goes on until all initial land is occupied. The $III.$ states you can skip attacking, though. – Vepir Jan 22 '18 at 15:35