Should I make a Tips Thread/Guide?

[Ceaseless]

Connect 4 has been a fun game where it's solved enough to know the answers but complex enough to still enjoy working out the rationale for them. I'm wondering if there are enough people who would even read the Connect 4 forums to make it worth posting some ideas that help follow the general concepts in the game.

[play4row]

yeah i'd read it. the forum is quiet right now but these threads get found by people searching months later, so a guide has a long tail even if only a few of us reply now. odd/even threats is the part i'd want most, that was the slowest thing for me to actually start seeing on the board.

[Ceaseless]

yeah i'd read it. the forum is quiet right now but these threads get found by people searching months later, so a guide has a long tail even if only a few of us reply now. odd/even threats is the part i'd want most, that was the slowest thing for me to actually start seeing on the board.

Yeah, parity is definitely on the top of the list of things to nail down.

[play4row]

the rule clicked fast for me, seeing which threat squares actually get reached took way longer. counting a threat on a row the column never fills up to, that kind of thing. a couple of worked examples would do more than the definition.

7x6 or 9x9 though? you covered the odd height difference in the other thread, and i still don't know which one people here actually play.

[Ceaseless]

the rule clicked fast for me, seeing which threat squares actually get reached took way longer. counting a threat on a row the column never fills up to, that kind of thing. a couple of worked examples would do more than the definition.

7x6 or 9x9 though? you covered the odd height difference in the other thread, and i still don't know which one people here actually play.

Actually gets reached? You're going to have to give examples of what you were confused by there. If they're not reached, what happened, did they get undercut by a different threat? Did the opponent execute a tactical finish that ended the game first? I'm unsure if what you're describing is even a parity issue or if something else was intervening.

As for which one is played, you can check the game page and see recently played games. You can check player profiles like Arena players to see how often they play, etc. You can even host yourself and see where you're getting games.

[play4row]

not parity i don't think, it's the step before. early on i'd count every threat i could make, including ones sitting high in a column, then the game ends with that column three deep and half of them were never real squares. nothing intervened, i was just miscounting.

and fair, the game page would answer the 7x6 question quicker than asking here.

[Ceaseless]

not parity i don't think, it's the step before. early on i'd count every threat i could make, including ones sitting high in a column, then the game ends with that column three deep and half of them were never real squares. nothing intervened, i was just miscounting.

You're going to have to post an example of what you mean here. What you're describing is way too vague.

[Ceaseless]

Even numbers are integers divisible by 2. So 0, 2, 4, 6, 8, 10… all even. Odd numbers are integers not divisible by 2. 1, 3, 5, 7, 9… all odd. Adding or subtracting an even number from an integer will not change if it is even or odd. On the other hand, adding or subtracting an odd number from an integer will change its parity from odd to even, or even to odd. In connect 4, a horizontal 4 in a row has 4 disks of the same parity, they’re either all odd or all even. A diagonal 4 in a row has 2 odds and 2 evens.

When I refer to a column’s parity, I will reference the column by its empty slots. In the base game where a column has 6 slots, the column is of even parity. When a player plays a disk into it, it now only has 5 empty slots, and so will have odd parity. So anytime a player plays a disk, they are changing the parity of the column they play in. Since the base game begins with an even number of columns, after the first player's turn, there will be an odd number of columns with the opposite of their starting parity, and after the second player’s turn it will be even again. I will call the first player red and the second player yellow from here on. So on the base game’s 7 columns 6 rows deep, the base state of a column is even. After the red’s turn, there will be an odd number of odd columns, and after the yellow’s turn, there will be an even number of odd columns again. For example, after red’s first turn there would be 1 odd column, and after the yellow’s turn they can either change another even column odd to make there be 2 odd columns, or they can change an odd column back to even so there will be 0. I will also refer to the parity of a threat by the row their empty slot occupies. So for example, if I have a 3 in a row, and the intended fourth slot that’s currently an empty disk in row 3, that threat is an odd row threat.

What is the significance of the parity of a column? If both players only played their disks one after another in one entire column, one player we’ll call Player A would start in the column, and their opponent we’ll call player B would finish it. So when they moved to the next column, the player order remained the same, Player A leading and player B following. However, under the same circumstances in an odd column, Player A would start the column, and Player A would finish it, meaning if they moved on to the next column it would be Player B’s turn to start the column. This makes for a major difference between boards that start with odd columns vs boards that start with evens.

When a threat exists, no one wants to play in the disk directly below the threat, as if the threat holder does it their opponent blocks the threat, and if the opponent does it the player fills in their threat. As a result, that disk, and every disk above it, is functionally removed from the game until players are ready to push through that slot. So for a column with 6 disks, if a player has a threat in row 3, no one is playing in disks 2-6 of that column, eliminating 5 slots. So while the disk is technically an even column with 6 slots, it’s functionally an odd column with 1 slot.

Note that how many disks are removed is not only based on what row the threat was on, but how many disks were in the column to begin with. If a column has an even number of disks, an odd row threat functionally removes an odd number of disks, changing the functional parity of the column, while an even row threat removes an even number of disks, which will not change the parity. If the column has an odd number of disks, an odd row threat functionally removes an even number of disks, which will not change the parity, while an even row threat functionally removes an odd number of disks, changing the functional parity of the column.

In the base game, each of the 7 columns are 6 disks tall, meaning all of the 7 columns are even columns. As a result, if both players just kept playing up columns, the first player would get the odd rows and the second player would get the evens. The natural state of the game means they would need to do things like block 4 in a row, which would involve some criss-crossing between columns, but it would always be able to sort itself back out as long as no threats reshape the board too badly. As a result, if the game is played where all of the disks except an empty final column are played, red wins if they have any threats on odd rows 1, 3, or 5 and yellow wins if they have any threats on even rows 2, 4, or 6 (the lowest one wins the day if they both had threats.) If the game had been played to all but 2 empty columns, the result is the same for the second to last column too, as there is no parity change for even columns.

To easier illustrate some parity statement, consider the following variant of Connect 4 where instead of aiming to get 4 disks in a row, each player has specific disks they must get. I will label the columns on the board from left to right A, B, C, D, E, F, G… And the rows from bottom to top 1, 2, 3, 4, 5, 6… So D3 would be column 4, row 3. I might say red wins if they get D3, but yellow wins if red doesn’t get D3. Or I might say red wins if they get D3, but yellow wins if they get D2. These variants of Connect 4 are easy for a player to play, as most moves will just be waiting moves until the key disks emerge. So why bother? Because they simulate certain endgames of Connect 4, where each player might have key slots like the fourth disk in a row, and they’re just filling disks until someone finally has to give in.

So we noted that red would win with odd row threats and yellow wins with evens on the standard 7 column 6 row board. So the “red wins if they get D3” is an easy red win. “Red wins if they get D3 and yellow wins if they get D2” is an easy yellow win, since yellow wins on row 2, and there is no way for red to get to D3 unless yellow fills D2 first anyways. On the other hand, if I had flipped “red wins if they get d2 and yellow wins if they get d3”, yellow would be able to claim d2, and red would claim d3, so neither player would win, in other words, a draw. So what happens if these ideas compete in multiple columns?

When it comes to even row threats in the base game, yellow dominates. To use my hypothetical Connect 4 game again, say yellow wins if they get all of the following slots, a2, b2, c2, d2, e2, f2, and g2, but red wins if they get any one of them. It turns out to be an easy yellow win. All they have to do is take the disk above red every turn. Red gets every odd slot, and yellow gets every even. So even if yellow had to get all of row 4 and row 6 as well, they would still win. This means if yellow had a diagonal where they already achieved both of the odd rows, they could handle getting both evens without issue. https://boardgamearena.com/archive/repl ... 5;&goto=11 Look at this example of yellow inevitably filling in 2 minor even row threats without incident. Any time red played in an odd column yellow would just take the disk above it, and any time red played in an even column yellow just took an even column, which will always exist since after red’s turn there must be an odd number of the initial column types. For a horizontal, the only thing stopping yellow from taking over the entire row to themselves is a connect 4 threat from red below. https://boardgamearena.com/archive/repl ... 55;&goto=7 Here yellow only has 1 disk in row 4, yet can pretty much guarantee the entire row for themselves taking the disk above red for nearly the entire game. Thanks to the row 3 disk below in the center, red can’t undercut them in row 3, and row 4 undercuts row 5, so the only thing red has left is a row 1 check. Yellow can easily block it, and criss cross back in row 2, as soon as red takes one of those row 2 slots yellow just takes the other.

Odd row threats aren’t quite as cut and dry in the base board. It turns out threats that create parity changes like odd row threats reshape the values of the board. For example, red wins if they get D3, but yellow wins if they get a2, b2, c2, e2, f2, or g6… is actually an easy red win, despite yellow greatly outnumbering their threats. Red can play in D1, and from there just take the disk on top of yellow until yellow is forced to play D2. Not only will red have taken every other row 2 disk outside of the d column, they can even take all of the row 4 and 6 disks outside of the d column. In a less extreme example game where red’s odd row threat faces yellow’s even row threat, red’s row 3 threat trumps yellow’s row 2 threat and red just takes the disk on top until the game is over. https://boardgamearena.com/archive/repl ... 5;&goto=22

However, that parity shift keeps flipping back and forth. Red lacks the ability to dominate all of row 3 like yellow dominated all of row 2 in the previous example. Say red wins if they get D3, but yellow wins if they get E3. It will be a draw, forget about a full row, red could not even guarantee 2 odd row disks. There are an even number of disks in the A, B, C, F, and G columns, so all of their moves between red and yellow can be cancelled out. While D and E can only fill 1 disk before a threat is exposed, so D1 and E1 as a pair are also an even number of disks. Therefore, it is inevitably red’s turn and they must take D2, giving up D3 to yellow, taking D4, yellow takes D5, and red takes D6. Now it’s yellow’s turn, and only E2 remains, which they must give up, red takes E3, and no more win conditions remain. https://boardgamearena.com/archive/repl ... 5;&goto=23 In this game red’s F3 odd row threat is cancelled out by yellow’s E5 threat, leaving yellow to win on B4. (Red’s B5 threat is a moot point since yellow’s b4 is directly below it, undercutting it.)

This means yellow can use an odd row threat to cancel out red’s odd row threat, even though yellow normally could not win with an odd row threat by itself. This also means if red had a diagonal that filled out 2 evens, and they only needed 2 odds to go, yellow could block it by forcing red to give up one of those odd row slots to them. It also means if yellow had two odd row threats that each won the game, such as a horizontal 3 in a row on row 3 open on both sides, yellow could win with that. These numbers keep expanding accordingly, if there were 3 odd row threats, red gets 2 of them and yellow gets 1, each given up by the other player. If there were 4, it’s paired 2 and 2, and so on.

Keep in mind the difference between same columns and different columns. A red row 3 threat can defeat any number of yellow row 2 threats in different columns, but would be undercut by one in the same. A yellow row 3 threat and a red’s row 3 threat cancel out in different columns, but if they’re in the same column, not so much. This also applies to racking up more threats. Yellow can win with 2 odd row threats in different columns, but if they’re in the same column they all succeed or fail together.

So red can make use of odd row threats to win, and while yellow normally wins with evens, odd row threats can be used for defense, or even victory in higher numbers. Red’s even row threats are unable to make a win on their own though. While an odd row threat can change them into seemingly lethal items, that disappears as soon as the odd row threat does, making them redundant or meaningless for such purposes. However, they still hold value as support. They can undercut opposing threats or even support your own. A particularly notable tool in the base game is how row 2 actually meaningfully undercuts row 5. Separate parity threats can undercut each other, though usually this is obvious, since many of them are vertically adjacent. So the main divides are 3/6 and 2/5. 3 and 6 have clearly defined roles that make undercutting largely irrelevant, as the 3 already beats the 6 in most cases anyways. But 2 and 5 are less clear. https://boardgamearena.com/archive/repl ... 5;&goto=18 Look at how red is using their threat on E2 to keep E5 out of the game, leaving the competition between yellow’s B2 and red’s G3. Normally, without F2, red would have to decide which to give up between E5 and G3, but the E2 threat, normally eliminating disks E1-E6, is actually only removing disks E1-E3, as E5 would already have gotten rid of disks 4-6. So E2 has actual parity significance when handling an odd row threat in a higher row within the same column. As soon as yellow inevitably allows red to fill G3, red can sacrifice E2 for the E5 to win the day over yellow’s B2.

[Ceaseless]

Even numbers are integers divisible by 2. So 0, 2, 4, 6, 8, 10… all even. Odd numbers are integers not divisible by 2. 1, 3, 5, 7, 9… all odd. Adding or subtracting an even number from an integer will not change if it is even or odd. On the other hand, adding or subtracting an odd number from an integer will change its parity from odd to even, or even to odd. In connect 4, a horizontal 4 in a row has 4 disks of the same parity, they’re either all odd or all even. A diagonal 4 in a row has 2 odds and 2 evens.

When I refer to a column’s parity, I will reference the column by its empty slots. In the base game where a column has 6 slots, the column is of even parity. When a player plays a disk into it, it now only has 5 empty slots, and so will have odd parity. So anytime a player plays a disk, they are changing the parity of the column they play in. Since the base game begins with an even number of columns, after the first player's turn, there will be an odd number of columns with the opposite of their starting parity, and after the second player’s turn it will be even again. I will call the first player red and the second player yellow from here on. So on the base game’s 7 columns 6 rows deep, the base state of a column is even. After the red’s turn, there will be an odd number of odd columns, and after the yellow’s turn, there will be an even number of odd columns again. For example, after red’s first turn there would be 1 odd column, and after the yellow’s turn they can either change another even column odd to make there be 2 odd columns, or they can change an odd column back to even so there will be 0. I will also refer to the parity of a threat by the row their empty slot occupies. So for example, if I have a 3 in a row, and the intended fourth slot that’s currently an empty disk in row 3, that threat is an odd row threat.

What is the significance of the parity of a column? If both players only played their disks one after another in one entire column, one player we’ll call Player A would start in the column, and their opponent we’ll call player B would finish it. So when they moved to the next column, the player order remained the same, Player A leading and player B following. However, under the same circumstances in an odd column, Player A would start the column, and Player A would finish it, meaning if they moved on to the next column it would be Player B’s turn to start the column. This makes for a major difference between boards that start with odd columns vs boards that start with evens.

When a threat exists, no one wants to play in the disk directly below the threat, as if the threat holder does it their opponent blocks the threat, and if the opponent does it the player fills in their threat. As a result, that disk, and every disk above it, is functionally removed from the game until players are ready to push through that slot. So for a column with 6 disks, if a player has a threat in row 3, no one is playing in disks 2-6 of that column, eliminating 5 slots. So while the disk is technically an even column with 6 slots, it’s functionally an odd column with 1 slot.

Note that how many disks are removed is not only based on what row the threat was on, but how many disks were in the column to begin with. If a column has an even number of disks, an odd row threat functionally removes an odd number of disks, changing the functional parity of the column, while an even row threat removes an even number of disks, which will not change the parity. If the column has an odd number of disks, an odd row threat functionally removes an even number of disks, which will not change the parity, while an even row threat functionally removes an odd number of disks, changing the functional parity of the column.

In the base game, each of the 7 columns are 6 disks tall, meaning all of the 7 columns are even columns. As a result, if both players just kept playing up columns, the first player would get the odd rows and the second player would get the evens. The natural state of the game means they would need to do things like block 4 in a row, which would involve some criss-crossing between columns, but it would always be able to sort itself back out as long as no threats reshape the board too badly. As a result, if the game is played where all of the disks except an empty final column are played, red wins if they have any threats on odd rows 1, 3, or 5 and yellow wins if they have any threats on even rows 2, 4, or 6 (the lowest one wins the day if they both had threats.) If the game had been played to all but 2 empty columns, the result is the same for the second to last column too, as there is no parity change for even columns.

To easier illustrate some parity statement, consider the following variant of Connect 4 where instead of aiming to get 4 disks in a row, each player has specific disks they must get. I will label the columns on the board from left to right A, B, C, D, E, F, G… And the rows from bottom to top 1, 2, 3, 4, 5, 6… So D3 would be column 4, row 3. I might say red wins if they get D3, but yellow wins if red doesn’t get D3. Or I might say red wins if they get D3, but yellow wins if they get D2. These variants of Connect 4 are easy for a player to play, as most moves will just be waiting moves until the key disks emerge. So why bother? Because they simulate certain endgames of Connect 4, where each player might have key slots like the fourth disk in a row, and they’re just filling disks until someone finally has to give in.

So we noted that red would win with odd row threats and yellow wins with evens on the standard 7 column 6 row board. So the “red wins if they get D3” is an easy red win. “Red wins if they get D3 and yellow wins if they get D2” is an easy yellow win, since yellow wins on row 2, and there is no way for red to get to D3 unless yellow fills D2 first anyways. On the other hand, if I had flipped “red wins if they get d2 and yellow wins if they get d3”, yellow would be able to claim d2, and red would claim d3, so neither player would win, in other words, a draw. So what happens if these ideas compete in multiple columns?

When it comes to even row threats in the base game, yellow dominates. To use my hypothetical Connect 4 game again, say yellow wins if they get all of the following slots, a2, b2, c2, d2, e2, f2, and g2, but red wins if they get any one of them. It turns out to be an easy yellow win. All they have to do is take the disk above red every turn. Red gets every odd slot, and yellow gets every even. So even if yellow had to get all of row 4 and row 6 as well, they would still win. This means if yellow had a diagonal where they already achieved both of the odd rows, they could handle getting both evens without issue. https://boardgamearena.com/archive/repl ... 5;&goto=11 Look at this example of yellow inevitably filling in 2 minor even row threats without incident. Any time red played in an odd column yellow would just take the disk above it, and any time red played in an even column yellow just took an even column, which will always exist since after red’s turn there must be an odd number of the initial column types. For a horizontal, the only thing stopping yellow from taking over the entire row to themselves is a connect 4 threat from red below. https://boardgamearena.com/archive/repl ... 55;&goto=7 Here yellow only has 1 disk in row 4, yet can pretty much guarantee the entire row for themselves taking the disk above red for nearly the entire game. Thanks to the row 3 disk below in the center, red can’t undercut them in row 3, and row 4 undercuts row 5, so the only thing red has left is a row 1 check. Yellow can easily block it, and criss cross back in row 2, as soon as red takes one of those row 2 slots yellow just takes the other.

Odd row threats aren’t quite as cut and dry in the base board. It turns out threats that create parity changes like odd row threats reshape the values of the board. For example, red wins if they get D3, but yellow wins if they get a2, b2, c2, e2, f2, or g6… is actually an easy red win, despite yellow greatly outnumbering their threats. Red can play in D1, and from there just take the disk on top of yellow until yellow is forced to play D2. Not only will red have taken every other row 2 disk outside of the d column, they can even take all of the row 4 and 6 disks outside of the d column. In a less extreme example game where red’s odd row threat faces yellow’s even row threat, red’s row 3 threat trumps yellow’s row 2 threat and red just takes the disk on top until the game is over. https://boardgamearena.com/archive/repl ... 5;&goto=22

However, that parity shift keeps flipping back and forth. Red lacks the ability to dominate all of row 3 like yellow dominated all of row 2 in the previous example. Say red wins if they get D3, but yellow wins if they get E3. It will be a draw, forget about a full row, red could not even guarantee 2 odd row disks. There are an even number of disks in the A, B, C, F, and G columns, so all of their moves between red and yellow can be cancelled out. While D and E can only fill 1 disk before a threat is exposed, so D1 and E1 as a pair are also an even number of disks. Therefore, it is inevitably red’s turn and they must take D2, giving up D3 to yellow, taking D4, yellow takes D5, and red takes D6. Now it’s yellow’s turn, and only E2 remains, which they must give up, red takes E3, and no more win conditions remain. https://boardgamearena.com/archive/repl ... 5;&goto=23 In this game red’s F3 odd row threat is cancelled out by yellow’s E5 threat, leaving yellow to win on B4. (Red’s B5 threat is a moot point since yellow’s b4 is directly below it, undercutting it.)

This means yellow can use an odd row threat to cancel out red’s odd row threat, even though yellow normally could not win with an odd row threat by itself. This also means if red had a diagonal that filled out 2 evens, and they only needed 2 odds to go, yellow could block it by forcing red to give up one of those odd row slots to them. It also means if yellow had two odd row threats that each won the game, such as a horizontal 3 in a row on row 3 open on both sides, yellow could win with that. These numbers keep expanding accordingly, if there were 3 odd row threats, red gets 2 of them and yellow gets 1, each given up by the other player. If there were 4, it’s paired 2 and 2, and so on.

Keep in mind the difference between same columns and different columns. A red row 3 threat can defeat any number of yellow row 2 threats in different columns, but would be undercut by one in the same. A yellow row 3 threat and a red’s row 3 threat cancel out in different columns, but if they’re in the same column, not so much. This also applies to racking up more threats. Yellow can win with 2 odd row threats in different columns, but if they’re in the same column they all succeed or fail together.

So red can make use of odd row threats to win, and while yellow normally wins with evens, odd row threats can be used for defense, or even victory in higher numbers. Red’s even row threats are unable to make a win on their own though. While an odd row threat can change them into seemingly lethal items, that disappears as soon as the odd row threat does, making them redundant or meaningless for such purposes. However, they still hold value as support. They can undercut opposing threats or even support your own. A particularly notable tool in the base game is how row 2 actually meaningfully undercuts row 5. Separate parity threats can undercut each other, though usually this is obvious, since many of them are vertically adjacent. So the main divides are 3/6 and 2/5. 3 and 6 have clearly defined roles that make undercutting largely irrelevant, as the 3 already beats the 6 in most cases anyways. But 2 and 5 are less clear. https://boardgamearena.com/archive/repl ... 5;&goto=18 Look at how red is using their threat on E2 to keep E5 out of the game, leaving the competition between yellow’s B2 and red’s G3. Normally, without F2, red would have to decide which to give up between E5 and G3, but the E2 threat, normally eliminating disks E1-E6, is actually only removing disks E1-E3, as E5 would already have gotten rid of disks 4-6. So E2 has actual parity significance when handling an odd row threat in a higher row within the same column. As soon as yellow inevitably allows red to fill G3, red can sacrifice E2 for the E5 to win the day over yellow’s B2.

The following is an initial draft that covers some of the parity concerns of the base game, specifying as such. Key details will shift when you change the numbers, so I may make revisions to add that in or include it in a separate post. I'd also need to think about if this text is easy enough to read and how good the examples are.