Row Impartial Terminus (RIT) Game Analysis

• Row Impartial Terminus (RIT) is an impartial combinatorial game defined on integer partitions, where moves shorten a row while preserving the partition structure.
• The game employs a unique core–remnant decomposition that splits a partition into paired rows, reducing analysis to a Nim position via odd-even row differences.
• Winning strategies for RIT rely on computing the nim-sum of row differences, aligning both normal and misère play outcomes with those of Nim.

Searching arXiv for the primary RIT paper and closely related partition-game literature. I’m going to look up the cited arXiv records for RIT and nearby partition-game work. Attempting arXiv lookup: (Gottlieb et al., 3 Sep 2025, Gottlieb et al., 2022), and related impartial-game papers. Row Impartial Terminus (RIT) is an impartial combinatorial game on integer partitions, represented as Young diagrams, in which legal moves shorten a uniquely determined row while preserving the partition condition. Its defining structural result is that every position admits a unique decomposition into a core and a remnant, and that the Conway pair of the RIT position coincides with the Conway pair of the Nim position defined by the remnant. Consequently, both normal-play and misère-play analysis reduce completely to Nim, yielding an explicit winning strategy and placing RIT in the Conway–Gurvich–Ho hierarchy as forced and miserable, but not pet (Gottlieb et al., 3 Sep 2025).

1. Formal definition and state space

RIT is played on partitions

$$\lambda=\{\lambda_1,\lambda_2,\dots,\lambda_r\},\qquad \lambda_1\ge \lambda_2\ge \cdots \ge \lambda_r>0,$$

with $\lambda\vdash n$ meaning $\lambda_1+\cdots+\lambda_r=n$. The empty partition $\{\ \}$ is the unique partition of $0$. As usual, a partition is identified with its Young diagram: a left-justified array of boxes with $\lambda_j$ boxes in row $j$, counted from the top.

A legal move is defined as follows. For a partition $\lambda=\{\lambda_1,\dots,\lambda_r\}$ and an integer $k\in[\lambda_1]=\{1,2,\dots,\lambda_1\}$, let $i$ be the largest index such that $\lambda\vdash n$0. The move replaces $\lambda\vdash n$1 by $\lambda\vdash n$2 and leaves all other row lengths unchanged: $\lambda\vdash n$3 This is called a move on the $\lambda\vdash n$4-th row. Equivalently, one chooses a column $\lambda\vdash n$5, finds the lowest row that reaches that column, and shortens exactly that row to length $\lambda\vdash n$6.

The only terminal position is the empty partition $\lambda\vdash n$7. RIT is studied under both standard conventions. Under normal play, the player making the last move wins, so the terminal position is a $\lambda\vdash n$8-position. Under misère play, the player making the last move loses, so the terminal position is an $\lambda\vdash n$9-position. The paper uses $\lambda_1+\cdots+\lambda_r=n$0 for the Sprague–Grundy value of a position $\lambda_1+\cdots+\lambda_r=n$1 in a game $\lambda_1+\cdots+\lambda_r=n$2, $\lambda_1+\cdots+\lambda_r=n$3 for the misère Grundy value, and the Conway pair

$\lambda_1+\cdots+\lambda_r=n$4

2. Core–remnant decomposition

The central structural device in RIT is the decomposition of a partition into paired rows. Given

$\lambda_1+\cdots+\lambda_r=n$5

the rows are grouped as

$\lambda_1+\cdots+\lambda_r=n$6

with the convention $\lambda_1+\cdots+\lambda_r=n$7 for $\lambda_1+\cdots+\lambda_r=n$8 when needed.

The core is defined by duplicating the even-indexed rows: $\lambda_1+\cdots+\lambda_r=n$9 Thus the core retains the lower row in each adjacent pair and repeats it twice. For example, if

$\{\ \}$0

then

$\{\ \}$1

The remnant is the tuple of odd-even row differences: $\{\ \}$2 again with $\{\ \}$3 for $\{\ \}$4. In the same example,

$\{\ \}$5

This decomposition is unique. If

$\{\ \}$6

then the core determines the even rows, and each odd row is recovered by

$\{\ \}$7

If $\{\ \}$8 is odd, the final even row is interpreted as $\{\ \}$9. The uniqueness of this reconstruction is the structural basis for the reduction of RIT to Nim (Gottlieb et al., 3 Sep 2025).

3. Odd and even rows as the mechanism of reduction

The decisive technical distinction is between moves on odd-numbered rows and moves on even-numbered rows.

If $0$0 is obtained from $0$1 by a move on an odd row, then the core is unchanged: $0$2 and the remnant changes by exactly one Nim move: $0$3 Conversely, every Nim move from $0$4 arises from an odd-row move in RIT that preserves the core. In effect, an odd-row move decreases exactly one coordinate

$0$5

which is formally identical to reducing a single heap in Nim.

If $0$6 is obtained from $0$7 by a move on an even row, then there exists an odd-row move from $0$8 to some $0$9 such that

$\lambda_j$0

An even-row move therefore changes the remnant only provisionally; the next player can restore the original remnant by reducing the odd row immediately above by the same amount. This is the copycat mechanism behind the strategic irrelevance of the core.

These two lemmas imply that odd-row moves are the genuine Nim moves of the game, while even-row moves do not alter the strategically relevant state in a stable way. A plausible implication is that the game’s entire nontrivial behavior is encoded in the pairwise row differences rather than in absolute row lengths. That implication is made exact by the Grundy theorems below (Gottlieb et al., 3 Sep 2025).

4. Equality with Nim under normal and misère play

The remnant

$\lambda_j$1

is interpreted as a Nim position whose heap sizes are precisely these coordinates. The central normal-play theorem states: $\lambda_j$2 Since the Nim Grundy value is the nim-sum of the heap sizes, this yields

$\lambda_j$3

Hence $\lambda_j$4 is a normal-play $\lambda_j$5-position exactly when this xor is $\lambda_j$6.

The paper proves the parallel misère statement: $\lambda_j$7 Combining the two equalities gives the Conway-pair identity

$\lambda_j$8

Thus the normal and misère theories of RIT coincide exactly with those of the corresponding Nim remnant (Gottlieb et al., 3 Sep 2025).

The proof is by induction on the size of the partition. Odd-row options are exactly the Nim options on the remnant, while even-row options can be answered by an odd-row move restoring the original remnant. Therefore the mex defining the RIT Grundy value matches the mex over Nim followers of the remnant. The misère proof uses the same induction, with terminal value $\lambda_j$9 instead of $j$0.

5. Outcome theory, winning strategy, and representative examples

Under normal play, let

$j$1

Then $j$2 is a $j$3-position if and only if

$j$4

If the xor is nonzero, a winning move is obtained exactly as in Nim: choose a coordinate $j$5 and reduce it to $j$6 so that the resulting xor is $j$7, then realize that reduction as an odd-row move on row $j$8.

If the opponent moves on an even row in a normal-play $j$9-position, there is an explicit response: reduce the odd row immediately above by the same amount, thereby restoring the remnant. This yields the paper’s concrete strategy statement: from a losing RIT position, every opponent move can be answered by a move on an odd-indexed row, and an even-row move is answered by reducing row $\lambda=\{\lambda_1,\dots,\lambda_r\}$0 by the same amount when the opponent has reduced row $\lambda=\{\lambda_1,\dots,\lambda_r\}$1 (Gottlieb et al., 3 Sep 2025).

Under misère play, the outcome is exactly that of misère Nim on the remnant. Consequently, if not all nonzero remnant coordinates are equal to $\lambda=\{\lambda_1,\dots,\lambda_r\}$2, optimal play follows the normal Nim rule of moving to xor $\lambda=\{\lambda_1,\dots,\lambda_r\}$3. If all nonzero coordinates are $\lambda=\{\lambda_1,\dots,\lambda_r\}$4, the position is losing precisely when the number of such coordinates is odd.

Several examples illustrate the reduction.

For

$\lambda=\{\lambda_1,\dots,\lambda_r\}$5

one has

$\lambda=\{\lambda_1,\dots,\lambda_r\}$6

Hence

$\lambda=\{\lambda_1,\dots,\lambda_r\}$7

so the position is losing under normal play. Under misère play, the remnant consists of two heaps of size $\lambda=\{\lambda_1,\dots,\lambda_r\}$8, so it is winning for the next player.

For

$\lambda=\{\lambda_1,\dots,\lambda_r\}$9

one has

$k\in[\lambda_1]=\{1,2,\dots,\lambda_1\}$0

Then

$k\in[\lambda_1]=\{1,2,\dots,\lambda_1\}$1

so the position is winning under normal play. A move to xor $k\in[\lambda_1]=\{1,2,\dots,\lambda_1\}$2 is obtained by reducing the first remnant heap from $k\in[\lambda_1]=\{1,2,\dots,\lambda_1\}$3 to $k\in[\lambda_1]=\{1,2,\dots,\lambda_1\}$4, which corresponds to reducing row $k\in[\lambda_1]=\{1,2,\dots,\lambda_1\}$5 from $k\in[\lambda_1]=\{1,2,\dots,\lambda_1\}$6 to $k\in[\lambda_1]=\{1,2,\dots,\lambda_1\}$7, giving

$k\in[\lambda_1]=\{1,2,\dots,\lambda_1\}$8

with

$k\in[\lambda_1]=\{1,2,\dots,\lambda_1\}$9

For a pure core position such as

$i$0

the remnant is

$i$1

so the position is strategically equivalent to the zero position in Nim. This makes explicit that a nonempty Young diagram can nevertheless be a normal-play zero position when all odd-even row differences vanish.

6. Conway–Gurvich–Ho classification and relation to adjacent work

RIT is classified in the Conway–Gurvich–Ho hierarchy as forced and miserable, but not pet (Gottlieb et al., 3 Sep 2025). In the terminology recalled in the paper, a game is forced if each move from a $i$2-position results in a $i$3-position and vice versa. It is miserable if every position satisfies one of three alternatives: it is itself a $i$4- or $i$5-position; or it has no move to either such position; or it has moves to both. It is pet if the only Conway pairs are

$i$6

RIT inherits its placement in this hierarchy from the Conway-pair identity with Nim. Because Nim is forced and miserable but not pet, the same is true of RIT.

Within the broader literature, RIT belongs to the class of impartial games on partitions and Young diagrams. A closely related but distinct example is LCTR, in which players remove either the top row or the left column of a partition; LCTR also admits a detailed Sprague–Grundy analysis, but its move rule and resulting structure differ sharply from the core–remnant reduction characteristic of RIT (Gottlieb et al., 2022). At a more abstract level, work on hypergraph combinations of impartial games studies when structured simultaneous moves reduce to generalized Nim positions; that framework is not specific to RIT, but it clarifies that reduction-to-Nim phenomena can arise from underlying structural decompositions rather than from ad hoc calculation alone (Boros et al., 2017).

The significance of RIT lies in the exactness of its reduction. The strategically relevant data are not the full row lengths but the odd-even row differences

$i$7

while the core functions as an inert scaffold. This suggests a broader research direction in partition games: identifying decompositions in which a geometrically natural diagram invariant carries the entire Conway-pair structure. The paper explicitly points to further study of special partition families such as staircases and quadrated partitions, and to the exploration of other partition games derived from or branching from RIT (Gottlieb et al., 3 Sep 2025).