Ascent

Ascent is a 1998 game by Gregory K. Van Patten.

Ascent is played with numbered tiles initially arranged in a diamond pattern (see starting configuration). There are four rows slanting down to the left, and four rows slanting down to the right. Also, there are seven levels (ie, the level with the "1" tile, the level with the "2" tiles, etc.).

On his turn, a player must switch two tiles from different levels, under the condition that the tile initially in the higher level has a larger number than the tile in the lower level.

• The goal of the first player is to arrange tiles in one of the four rows slanting down to the left so that the numbers form a strictly increasing sequence from the top of that row to the bottom.
• The goal of the second player is to arrange tiles in one of the four rows slanting down to the right so that the numbers form a strictly increasing sequence from the top of that row to the bottom.

Extra considerations: Passing is not allowed, so you could possibly have to make a move which achieves your opponent's object, so that you lose. If a move simultaneously achieves strictly increasing sequences from top to bottom, in both directions, then the game ends in a tie.

Here is a sample game by the author (gray pieces mean switches).

First player has just moved, and threatens to win
by switching the 2 and 3 on the lower right side. 

.Second player defends by switching the indicated 3 and 6, and can now win by switching 4 and 5. Note that if second player had tried defending
by switching 1 and 7, first player would have won immediately. 

First player switches 3 and 4, and can now win by switching 2 and 5. 

Second player switches 3 and 7, and can now win by switching 6 and 7.


First player switches 2 and 5, and can now win by switching 4 and 6. 

Second player defends by switching 5 and 7,
and can now win by switching 2 and 3. 

First player switches 2 and 3, and can now win by switching 4 and 5.

Note the [3 3 5 6] row slanting down to the right does is not a
win for second player, because it does not form a strictly
increasing sequence (it contains more than one "3" tile). 

However, the first player neglected to consider this 1,3 switch,
which results in a second player win.

Kensington

Kensington was designed by Peter Forbes and Brian Taylor in 1979, and published by Arxon, among others.

The game is similar to Nine-Men-Morris in a Tessellation of squares, triangles and hexagons,

Pieces are placed on line intersections. Players take turns placing their fifteen pieces on empty intersections. Then pieces are moved one at a time from one intersection to another (a player passes if he does not have a valid move). 

• If a player has pieces on all corners of a triangle, he can reposition one enemy piece. 
• If a player has pieces on all corners of a square, he can reposition two enemy pieces.
• The object of the game is to control the corners of a hexagon.

The game has a design weakness that can be patched with the following rule: 

• It is not permitted for a player to rebuild a triangle or a square until two turns have passed.

 

 

The game had above than average visibility back then, and was quite probably backed by a nice marketing budget. It appeared more than once, at least, on the american Games, the french Jeux et Strategie and the italian Per Gioco magazines. However, it became somewhat obscure after the 1980s. It can be played at Ludii.

 

The board can be (better) used to play Christian Freeling's Lotus, a Go-like game.

Chase

Chase was designed by Tom Kruszewski in 1985 and published by TSR.

Clark D. Rodeffer and me (João Neto) wrote an article about it in Abstract Games Magazine #9 (2002). It is a remarkable game that, sadly, is almost forgotten. I recommend reading the article and try yourself the game.

Some words from Steffan O'Sullivan in 1997:

Bumps and exchanges are other key elements to watch out for in the game.You can carefully set up a fork, thinking you've got a good capture going next turn, when suddenly he makes a bump move which protects both pieces at once and threatens a move to the Chamber!

One thing chess players have a hard time adjusting to is the concept of exchanges in this game. It seems like a good exchange, trading a "1" die to capture a "6" die, but after the exchange, you both have the same number of dice totalling 25! Very puzzling - you have to learn to think differently than in chess.

As your dice pool dwindles, you become less flexible. Yes, you still have 25 movement points total, but if those are concentrated on five dice as opposed to spread out over nine dice, you'll find it's harder to make all kinds of moves. Most dice will have high numbers, and you can't creep up on the enemy. Even worse, you can't afford another loss at that point, and the game gets very tense. Most of the strategy centers around the Chamber: he who has the most dice at any given moment is ahead, unless his position is really lousy.

Position is something that takes time to learn: there are key rows which radiate out from the Chamber that are very important. It's also important to keep at least some of your dice together. Not only does this allow for Exchange moves, it also makes for some good bumping. But this is something you'll learn best from playing the game, so I won't go into detail.

Here's another review from Sid Sackson in GAMES #68,

Tripples

Trippples is a 1972 William T. Powers' race game, where the board is constructed with a set of pieces marked with their possible moves.

According to BGG:

Trippples is played on a 8×8 board as opponents race from one corner to the other. On the board are placed, in a pre-movement round, tiles with three arrows which point to three of the eight surrounding spaces. Each player plays these tiles alternatively until they have all been played. Movement is made by following the arrows, and avoiding the neutral zone in the middle. The game ends when one player reaches his goal or when both players cannot make legal moves.

The key game mechanism is that your move options are determined not by the three arrows on the tile under your (transparent) game piece, but by those under your opponent's piece. Thus, each move is chosen to try and limit your opponent's next as well as to try and guarantee yourself a good selection of choices. This adds an element of bluff and second-guessing to the game.

The rulesheet:

The game does not have the property of converging to the endgame, it seems possible to walk on circles, never be able to reach your final corner. That's why the rules have two extra winning conditions for these types of situation. To slightly reduce this problem, it's better to consider that empty tiles allow for unrestricted moves.

There's a version using hidden information, where all tiles start upside down, and are only revealed when a piece stops there. It's also possible to add an element of luck, by shuffling the tiles and placing them on the board at random.

The review at Games & Puzzles #42:

This game has patent US3820791. When searching this game I came by the blog The Dreams of Gerontius, which has a post about Tripples, and mentions an much older 1894 patent (US519326) for a game with some board design similarities, 

This is a game for two or four players, with capture and scoring themes.

[addendum] There's also another similar game I wish to mention, Schada from c.1936 [1][2]. 

Here each player occupy their first two rows with sixteen pieces. The game is a variant of Halma, where captures are possible. The moves are decided by the available markings at the square the piece is standing.

[1] Fred Horn - Schada, AGPC Quarterly 18 [2] (2016)
[2] Fred Horn, Kerry Handscomb - Schada, Abstract Games Magazine 20

Diu Fang

Diu Fang is a Chinese traditional game, apparently from the Central Asia region. In China it is known as square chess, game of squares, or playing squares (fang means square).

The game is played on a board with 7x7 or 7x8 intersections (but there are other sizes depending on the region),

The rules as stated by Olli Salmi:

• Each player has enough stones to fill half the intersections.
• The board starts empty and each player, in his turn, drops one friendly piece on an empty intersection. 
• When a player has four pieces making the corners of a square, he captures an enemy piece from the board. However, any piece forming a square cannot be captured.
• When all pieces are placed, pieces start moving to any adjacent empty intersection. Moves are mandatory, even if it means unmaking a square.
  
• Wins the player that captures all enemy pieces.

A Chinese text translation (from the previous link) states:

[...] The goal is to form a "fang" (square) by occupying one square of the board, which allows the player to capture one of the opponent's pieces. [...] Once all the pieces are placed, each player removes one of the opponent's pieces and begins moving their pieces. The objective remains to form squares and capture the opponent's pieces. Players can also dismantle and reform their existing squares to capture more pieces. The game ends when one player loses all their pieces, resulting in a loss.

"Diu Fang" has two variations: "Wei Fang" (aggressive strategy) and "Tian Fang" (peaceful strategy). In Tian Fang, players agree not to aggressively encircle the opponent's pieces, making the gameplay less confrontational and more harmonious compared to Wei Fang, which is more competitive and intense.

It is not stated if it is possible to form squares with 45º or other angles.

The game is reminiscent of Nine Men Morris and Alquerque (some boards have diagonal connections). It seems to have the same «win strategy», as in Morris, that allows the alternate movement of making and unmaking a square, thus capturing a piece in every two consecutive turns.

Details about this game (or family of games) are also described by Shimizu et al. in [1].

it is possible to note the diagonal lines at the board's middle

The authors state, besides the previous rules:

• The first stone must be placed at the middle (assuming a 7x7 board)
• "The players take turns in placing stones on the points of intersection of the lines, with the aim of creating a square around a field, while at the same time preventing their opponent from doing so."
  
• "The players take turns in taking a stone of the opponent which is not part of a square off the board and then moving one stone vertically or horizontally from a neighbouring point to the point left vacant by the removed stone. When a player manages to create a square of four pieces of his own he is allowed to remove an opponent’s piece from the board. This is repeated again and again. So a player aims to create squares himself, as squares of four stones are safe and no stone can be taken from it." This paragraph seems to mix two different rules. Each piece move means unconditionally removing an enemy piece. Then, an extra capture is possible after a new square if made. This would make for a much faster game.
  

The Ludii portal has some reconstructions that can be played (search for Fang).

In the main Fang variant, Ludii ruleset mainly follows [1]:

• 7x7 board, with diagonals in the square formed by the central 3x3 lines. 24 pieces per player. 
• Black plays first. Players alternate turns placing a piece on one of the empty spots on the board. 
• The first stone must be placed in the central spot. 
• They attempt to orthogonally surround an empty point with four of their pieces, making a square. 
• When all of the pieces are placed, players alternate turns removing one of their opponent's pieces from the board. They then move one of their pieces along the lines of the board to the place left vacant by the piece they removed. 
• When a player makes a new square, they may remove another of the opponent's pieces from the board. Pieces that are in a square around an empty point may not be captured, and captures are not made in the placement phase. 
• The player who captures all of the opponent's pieces wins.

This surrounding of an empty intersection does not appear at [1], which just says creating a square around a field. Anyway, this rule implies 45º angle squares. And this ruleset does not refer to any other type of square. So perhaps Salmi's description skips this detail, which seems probable to me (using different shapes of squares feels too modern).

[1] Shimizu et al., Game boards in the Longmen Caves and the game Fang, BGS 5.

[addendum] I found this Chinese site which says (using automatic translation):

Rules for the Game "Fangqi" (方棋)

Introduction. Fangqi is a collective term for traditional two-player board games popular in northern China. The capture method resembles that of Zhizi, where forming specific patterns allows the capture of opponents' pieces. Rules, board designs, and names vary slightly across regions.

Game Phases: 

1. Placing Phase: Players alternate placing one piece at a time on an empty intersection of the board until all pieces are placed.
2. Biqi Phase (Removal Phase): If the board is filled, each player selects one opponent's piece to remove. On odd-lined boards, the second player selects first, and on even-lined boards, the first player selects first.
3. Moving Phase: Players take turns moving one piece along a line to an adjacent empty spot.

Capturing Pieces:

• Captures occur when a specific pattern is formed (e.g., a square or rectangle). The capturing player removes a specified number of opponent's pieces, prioritizing unprotected pieces.
• Common patterns:
• Square: Four pieces forming a square remove one opponent's piece.
• Rectangle: Six pieces forming a rectangle remove two opponent's pieces.
• Victory is achieved by reducing the opponent's pieces to a number insufficient for forming capture patterns.

Movement Rules:

• Generally, players can only move one piece to an adjacent empty intersection.
• In some digital implementations, a player who cannot move loses; others allow returning to the Biqi Phase to remove additional pieces.

Regional Variations and Specific Rules:

1. Daxing District (Beijing)  
   - Placing Rules: Pieces cannot be placed at the four corners.
   - Capture Patterns:
     - Dragon: Five pieces in a straight line (horizontal or vertical) capture three opponent's pieces.
     - Square: Four pieces forming a square capture one opponent's piece.

 2. Guan County (Hebei)  
   - Placing Rules: Same as Daxing.  
   - Capture Patterns:
     - Dragon: Five pieces in a straight line add two pieces in the placing phase or capture two in the moving phase.
     - Square: Four pieces forming a square add one piece in the placing phase or capture one in the moving phase.

 3. Qinghai  
   - Unique Rules: Pieces are returned to the opponent instead of being removed.
   - Capture Patterns:
     - Bar (杠): Five pieces in a line remove three opponent pieces, with the line marked as unusable for further captures.
     - Square: Four pieces forming a square remove one opponent piece and mark the square unusable.

 4. Shandong Variations (Wugunqi, etc.)  
   - Capture occurs during the placing phase, but pieces are marked and removed before the moving phase begins.
   - Unique terms like "Five Tigers" and "Three Diagonals" describe specific formations.

 5. Northeast China (Manchu)  
   - Similar to Shandong, with unique formations such as "Big Tiger Claw" and "Sky Bar".

Summary. Fangqi incorporates a rich variety of rules that adapt to regional preferences. Core elements such as strategic placement, capture through pattern formation, and movement across intersecting lines remain consistent. Each regional variant adds unique strategies, fostering a diverse gaming experience.