Cameron Browne (2008)
Palago is a tile placement game in which players strive to form creatures of their colour.
Palago has been officially released into the wild!
I'm the current Palago World Champion...
Over a quadrillion creatures in every bag!
Tiles: Two players, White and Blue, share a common pool of 48 hexagonal Palago tiles. Each tile contains a straight and a circular tip of each colour, and may be oriented in three ways such that edge colours always match.
Figure 1. Three rotations of a Palago tile.
Start: White places two adjacent tiles in the playing area, such that edge colours match.
Play: Players then take turns adding two adjacent tiles, such that at least one touches an existing tile and all edge colours match. Both tiles must be played unless the first tile wins the game.
Figure 2 shows a typical opening by White (left) and reply by Blue (right). This is probably the best opening pair for White, as discussed in the section on Opening Theory below.
Figure 2. A typical starting sequence.
Aim: The game is won by the player to form a creature (closed group) of their colour with at least one straight. Only one tile need be played if that tile wins the game for either player.
For example, Figure 3 shows a game won by Blue who has completed a blue creature with one straight. The closed white eye does not count as it doesn't have any straights.
Figure 3. A game won by Blue.
If a move simultaneously completes winning creatures for both players then the mover loses. If the tiles run out before either player wins then the game is a draw.
Tiebreaker: Players may optionally use the following rule to decide drawn games: If the tiles run out before either player wins, then the game is won by the player with the largest incomplete creature (counting straights) else the game is a draw if this count is equal.
Single holes enclosed on all six sides, as shown in Figure 4, represent a special case. A player cannot place the first tile of their move in such a hole (unless that tile immediately wins the game) as it would then be impossible to place the second tile of the move adjacent to it.
Figure 4. A hole.
Figure 5 (left) shows a situation in which either player may fill a hole by closing their group; however, they would lose by doing so as such a move would also close an opponent's group (right). Such unplayable positions of mutual disadvantage are called seki in Go.
Figure 5. A losing move for both players (seki).
However, White can quickly turn the position to their advantage by playing the two tiles shown in Figure 6 (left). This move closes the group except for the freedoms within the hole, which White can close on their next move to win (right).
Figure 6. White resolves the seki to win.
It's theoretically possible for Blue to achieve a similar win from the position shown in Figure 5 (left). However, this is unlikely as it would require at least nine tiles to close the surrounding Blue group and White is only three tiles away from a win.
This example shows how a group adjoining a single hole may be closed if the rest of the group is closed first. However, any group adjoining two or more separate holes is safe from closure; holes may therefore be used strategically as attacking moves (one hole) or to ensure group safety (two holes).
Strategy & Tactics
Players should look for moves that simultaneously achieve two disjoint threats, which constitutes a guaranteed win unless both threats can be nullifed by an adjacent tile pair.
Attacking players should be able to form an immediate threat almost every turn, which can be useful for limiting the opponent's replies. For example, Blue threatens to win from their very first move in Figure 2, forcing White to play at one or both of the red dots to stay alive. However, overly aggressive play can prove disastrous if the opponent's developing position is ignored; a much safer approach is: connect the opponent's most dangerous threats each turn to nullify them. This may give the opponent the largest group, but it's unlikely that the game will last long enough for that to matter.
The requirement that tile pairs be placed adjacent to each other makes it harder to spoil double threats set up by the opponent. This means that games are more likely to end in a kill rather than players being able to continually defend until the tiles run out.
Tip Threats: Figure 7 (left) shows that three tips in a row are a guaranteed win if left undefended, as their owner can form two disjoint threats next turn (right). This is true for both players in this example.
Figure 7. Guaranteed win for the next player.
In fact, even two tips in a row pose a similarly fatal if more subtle threat. Figure 8 shows how such a pair can be exploited by the next player to set up a win.
Figure 8. Two tips in a row can set up a win.
Note, however, that such positions should always be considered in the context of the surrounding pieces and never in isolation. For example, Figure 9 shows that Blue can end up fighting for their life if they blindly try the same trick in a different context!
Figure 9. The context can change the result.
Danger Patterns: Figure 10 shows a danger pattern called the triskelion; White can force a win if it's their turn to play by creating two disjoint threats as shown. Although White will never be presented with this exact pattern – there will always be four tiles placed after the second move – the principle holds for a number of similar patterns: look for disjoint groups, three or less moves from completion, that can both be reached with a single tile pair.
Figure 10. White can win.
Figure 11 shows a more subtle danger pattern; White has already lost this game even with next move. This can be proven using similar techniques to those used in the solution of Puzzle B below.
Figure 11. White has already lost.
Figure 12 shows two Blue tips joined to make a 2/3 circle or "open mouth" formation (left). It would be prudent of White to negate this formation by closing the circle as shown (middle), otherwise Blue may make an immediate threat next turn (right) or use it as a dead end into which a nearby group may be extended to close off two freedoms at once.
Figure 12. Open mouths should be closed.
Immediate Threats: Figure 13 shows two basic types of immediate threat that the opponent must answer next turn. These are called one-freedom and two-freedom threats based on the number of tips required to close the group.
Figure 13. A two-freedom threat and one-freedom threat.
The two-freedom threat (left) is less dangerous as it has four safe extensions that counter it with a single tile. The one-freedom threat, on the other hand, is more dangerous as it has only two feasible extensions (right), neither of which counter the threat but simply reduce it to a two-freedom threat that requires a second adjacent tile to fully address.
Figure 14 shows the six unique opening pairs, not counting reflections and rotations.
Figure 14. The six unique opening pairs.
White should only ever open with formation f for the reasons given below. Blue should only ever open with its inverse c.
White is assumed to start the game in the following examples, for consistency.
Opening a is a provable loss for White, as shown in Figure 15. If Blue replies with a rotation of a as shown then White is forced to close the open mouth, allowing Blue to set up a winning fork next turn (right).
Figure 15. Solution for opening a.
Similarly, opening b is a provable loss for White if Blue replies as shown in Figure 16. This creates an immediate threat which White must address by playing at points p, q or both (shown). Blue can then set up a winning fork next turn (right).
Figure 16. Solution for opening b.
Openings c and d
Openings c and d combine to make a probable loss for White; if White starts with c then Blue can play d to set up a win, and vice versa (Figure 17). This start looks more promising for White as they are not immediately threatened and can in fact set up a two-freedom threat of their own, as shown. However, Blue has a killer reply that negates the danger and sets up an even stronger one-freedom threat (right). White cannot address this threat without leaving a double tip for Blue to set up a win.
Figure 17. Solution for openings c and d.
If White makes their second move more defensive (Figure 18) then Blue may create two immediate threats as shown. White may address both of these next turn but cannot stop Blue from then setting up a winning fork (right).
Figure 18. Defensive play also leads to defeat.
White may follow other lines of play on their second move, but none have yet been found that allow survival.
Opening e, shown in Figure 19, allows Blue to immediately create a strong single-freedom threat (middle) to which White has only one viable reply (right).
Figure 19. Blue attacks to force the reply shown...
This sets up an overwhelmingly strong position for Blue with numerous winning attacks, one of which is shown in Figure 20:
Figure 20. ...which Blue can then exploit.
Note that Blue's attack leaves double White tips in two places which is theoretically dangerous, however Blue can force a win from this position without losing the initiative. Thanks to Mike McManaway for emphasising the danger of this opening.
Formation f is therefore White's only viable opening pair. Blue has a number of interesting replies to this opening, but none that have proven decisive.
Figure 21. The only safe opening for White.
Game #1: Annotated sample game #1 demonstrates the balance between aggressive and defensive play and features a surprising hole sequence.
White to play two adjacent tiles and set up a win.
Can Blue play two adjacent tiles to survive?
Who will win the following game?
Single Placement: Players take turns placing a single tile per turn. This makes moves easier to visualise and the game easier to play but harder to win; it's extremely difficult to set up a winning threat with a single tile and this will be extremely unlikely against an experienced and careful opponent.
Place Anywhere: Relax the tile placement rule so that the two tiles placed each turn need not be themselves adjacent. This was the way that the game was originally played, however it was found to make defensive play too easy, resulting in players developing the board without real purpose and leading to too many draws. The adjacent placement constraint introduces the hole threat and tightens the game up considerably.
Palagem: A deceptively simple multiplayer game in which players construct a target pattern (palagem) which they try to recreate on the board.
Please see the Palagem page for more details.
Palagonia is a world inhabited by creatures and shapes that can be formed by Palago tiles. You can visit it here.
Palagonia is also the name of a multiplayer version of Palago co-designed by myself and Mike McManaway.
How many creatures are there? Some initial analysis suggests that there are over a quadrillion (1,000,000,000,000,000) creatures in every bag of 48 tiles!
Check out some fractals and other mathematical curiosities made with Palago tiles.
Palago tiles and rules by Cameron Browne and copyright (c) Cyberite Ltd 2008.
Palago is published by Colour of Strategy, the makers of Tantrix. Please check out their very nice Palago site.
The Palago tile design was originally devised for an earlier game Mambo that used three colours (red, white and blue). The simplification to two colours ironed out a number of problems and resulted in a more elegant game that eventually become Palago.
The name Palago is reminiscent of "archipelago", specifically the Palau islands which look very much like a giant game of Palago played with islands and ocean. It could even be described as a kind of "Palau Go".
Palago can be played on Richard's PBeM server, check out the help file for more details. Many thanks to the server regulars who helped test the game. Please challenge me (camb) to a game any time.
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