Based on accounts of Ludus Latrunculorum that I have seen on the Web, although the rules are not fully known, it is believed that pieces moved and captured orthogonally, and that they jumped without capturing, and captured by custodian capture. Seega, Tablut, and Checkers have all been noted as games in which some characteristics of Ludus Latrunculorum survive.
Some reconstructions suggest that the first player puts one piece on the board, then players alternate in placing two pieces, and finally the first player puts his last piece on the board. It's not clear to me that such a technique of reducing the advantage of the first move would have been attempted in an ancient game; such measures have only started to appear in games very recently. The placement of two pieces at a time in Seega could be hypothesized as an imperfect memory of such a technique, but it seems to me that a simpler explanation for it would be that the reason is to speed up the placement phase.
The book Games Ancient and Oriental by Falkener gives a reconstruction of the game which differs considerably from the description above, but which seems to me to be a playable and interesting game with simple rules.
The arrangement of the board and pieces is the same as that of Canadian Checkers above.
Pieces move and jump both orthogonally and diagonally, despite being set up only on squares of one color. A piece can only jump an opponent's piece, although jumps are not captures. Capture is by the normal custodian capture method, without an additional delay, and along diagonal as well as orthogonal lines. Eliminating or immobilizing the opponent is the goal.
Other reconstructions have the pieces moving and capturing along orthogonal lines only.
Some of these provide each player with sixteen or seventeen men, and one Dux. The men are placed on the board alternately by the players, two at a time, and then the Dux. As with the white King in Tablut, the Dux can only be captured when surrounded on all four sides. In R. C. Bell's reconstruction, the objective is to capture all the opponent's men, and in some others, merely to capture the opponent's Dux, as in Chess or Tablut. In at least one reconstruction from this group, only the Dux can jump.
A computer version of Ludus Latrunculorum for the Macintosh, and an activity kit about Rome for British schoolchildren, give a reconstruction with twelve pieces for each player on the back ranks of a 12 by 8 board.
The main accounts of Ludus Latrunculorum that are known are a brief mention in De Lingua Latina by Varro, and a somewhat longer reference in the Laus Pisonis, formerly attributed to Saleuis Bassus, which still gives little information about how it is played. There are, however, also brief mentions in Ovid and Martial; and two statements by Ovid would seem to confirm that custodian capture was used. A paper by Ulrich Schädler notes a particularly informative remark by Ovid in the Tristitia which leads me to the conclusion that pieces could move forwards and sideways, but jump backwards as well. Restricted backwards movement usually implies promotion, but this is not always true, particularly when some limited capability of backwards movement is present. Another paper by the same author notes that the 12 by 8 boards in some archeological finds in Britain were not likely to have been used for Ludus Latrunculorum, as boards known to be used for that game found elsewhere typically had sizes such as 8 by 8, 7 by 8, 8 by 9, or 10 by 10.
The Laus Pisonis indicates that each side had a fairly large number of men, and that is a good reason to omit the piece placement phase found in Seega as part of the game, in my opinion.
Also, since there are references to captures being relieved, some reconstructions have a delayed custodian capture, where a piece is merely immobilized when it is flanked, and then is removed on a later turn as the move for that turn.
So, here is my reconstruction of Ludus Latrunculorum, based on the available information, and the various other reconstructions that have been attempted:
m m m m m m m m m m m m m m m m m m m d m m m m . . . . . . . . . . . . . . . . M M M D M M M M M M M M M M M M M M M M M M M M
Each player has 23 men and one Dux, placed on the board as shown.
A Dux can move one space in any orthogonal direction, and can also jump over the player's own or the opponent's pieces without capturing, making as many jumps as possible, as in Halma or Chinese Checkers.
A Man can move one space forwards or sideways, but not backwards, but may still jump in all directions.
Capture is by the custodian method. One captures one of the opponent's men if that man is next to one of your own men, in an orthogonal direction, by moving a second man so that the opponent's man is flanked on two opposite sides by your men. A man can move between two of the opponent's men without being captured.
The opponent's Dux can only be captured by being surrounded on all available sides; in the corner, this requires only two men, and on the edge of the board, only three are required. When one of the opponent's men is in the corner, it can be captured by being surrounded on the two available sides as well; on the edge of the board, since two opposing sides are available, it is not possible to capture a man using only one man on one side with the edge of the board on the opposite side.
One additional rule is that a man can also be captured as a Dux can be captured: thus, if a man moves between two men on the edge of the board, when a third man of the opponent moves to cause it to become surrounded on a third side, it can be captured as well.
The objective is to capture the opponent's Dux or all the opponent's other men.
One additional rule that might have been present is that no piece, not even the opponent's Dux, can jump over a Dux.
While I think that a delayed custodian capture, where removing pieces that were immobilized previously by being flanked, is a sufficiently unusual type of move as to have been unlikely to have been the rule, a more plausible type of delayed capture might have been this: flanking a piece only immobilizes it, but only immobilized pieces are subject to being captured by displacement (as in Chess). Thus, the capturing move also advances one of one's own pieces, and the game could then be thought of as attempting to reflect the principle of concentration of forces in warfare. In the case of the Dux, obviously it would have to be one of the flanking pieces that makes the capture; I would be inclined to go with the rule that either a flanking piece, or a third piece, can carry out the capture by displacement of an immobilized enemy man, if this type of capture is used.
It is interesting that in this case, capture by displacement, immobilization by the custodian method, and jumping would all exist as moves in the game.