{"id":"c6692863-d442-4981-b012-d0f307015e67","arxiv_id":"2506.04991","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Exact Sprague-Grundy formulas are proven for two new impartial games, PNim on Young diagrams and RNim on hyperrectangles, with a full description of partitions of value one.","lead":"This paper introduces two new versions of Nim, one played by deleting rows or columns from Young diagrams of integer partitions, and one played by shrinking sides of higher-dimensional boxes. It proves exact formulas for the Sprague-Grundy value of single boxes in both games and classifies which game positions are 'pet' in the Conway-Gurvich-Ho hierarchy.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2's proof of (i) contains a false construction: the prescribed column removal does not always land in S, e.g. for λ=J5,5,4K.","rationale":"The central claims are Theorems 1, 2, and 3. Theorems 1 and 3 have sound inductive proofs; Theorem 3 relies on Corollary 11, whose proof is omitted but follows directly from Lemma 5 and Observation 10. Theorem 2 is the only result whose proof contains a verifiable false statement: the column-removal construction in part (i) does not always land in S, as shown by J5,5,4K. This is load-bearing because Theorem 2 is a headline result and is used in Section 5.2 for optimal misère play. The reader's concern about a missing induction is real but secondary: once (i) and (ii) are established, the mex characterization follows by a standard well-founded induction on partition size. The concrete error in the construction of (i) is more serious and must be repaired. Nonetheless, the characterization appears true, since the counterexample to the construction still has a valid move into S, so the appropriate verdict remains conditional acceptance pending a corrected proof.","tokens_in":11496,"tokens_out":34840,"duration_ms":352241,"concrete_test":"Apply the construction in Theorem 2(i) to λ=J5,5,4K: compute the smallest i>1 with λ_i < λ1−i+2 (with zero-padding), remove the first λ1−i+1 columns, and verify the resulting partition is J3,3,1K, which is not in S. This directly refutes the proof step. To check whether the theorem itself survives, brute-force G(λ) for all partitions of size ≤20 and compare with S.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 2, the characterization of Sprague-Grundy value one, contains a false step in part (i). The proof defines S and claims that if λ=Jλ1,...,λrK∉S∪{JK}, then taking the smallest i>1 with λ_i<λ1−i+2 (after padding λ_i=0 for i>r) and removing the first λ1−i+1 columns produces a partition in S. This is not true. For λ=J5,5,4K we have λ1=5, r=3; after padding, the smallest violation is i=4 since λ4=0<3. Removing the first λ1−i+1=2 columns leaves columns 3,4,5 with heights 3,2,2; the resulting partition is J3,2,2K' = J3,3,1K. But J3,3,1K∉S: for r=3 the S-interval is between J3,3,2K and J3,3,3K, and for r=2 it is exactly J2,2K. Thus the claimed move does not land in S. (A different move from J5,5,4K, removing the first three columns, does reach J2,2K∈S, so the theorem itself may still be true.) The proof as written therefore fails to establish (i). The earlier 'equivalence' sentence is also misquantified: membership in S requires length(λ)=λ1 and all inequalities 2≤i≤λ1, not an equivalence for each i. This is a concrete correctness defect, not merely a missing induction.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces PNim, a game on Young diagrams where a move removes a positive number of rows or columns, and RNim, a game on hyperrectangles where a move reduces one side length. The main results are: Proposition 1/2 giving a tight longest-play upper bound for PNim; Theorem 1 giving G_P(Jc^rK) = ((r-1) XOR (c-1)) + 1 for rectangles; Theorem 2 characterizing partitions with Sprague-Grundy value one; Theorem 3 giving G_R(<k_1,...,k_d>) = ((k_1-1) XOR ... XOR (k_d-1)) + 1 for hyperrectangles; and a Conway-Gurvich-Ho classification together with misere-play consequences. The paper also contains computational data in appendices and formulates several conjectures.","tokens_in":11847,"tokens_out":7860,"duration_ms":87941,"significance":"The closed-form formulas for rectangles and hyperrectangles are elegant and connect the games to XOR arithmetic and Lucas's theorem in a way that does not arise from the disjunctive-sum structure. The CGH classification and the explicit misere strategy are useful additions to the study of impartial games on partitions. The paper is generally clearly written and includes concrete data supporting its conjectures. However, the proof of Theorem 2 currently contains a concrete false step in part (i), and Corollary 11, which is used in the proof of Theorem 3, is stated without proof. These issues must be repaired before the central claims can be regarded as established.","major_comments":[{"comment":"The construction claimed to produce a move into S is incorrect. For lambda = J5,5,4K, we have lambda_1 = 5; after padding lambda_4 = 0, the smallest violation is i = 4, so the prescribed move removes the first lambda_1 - i + 1 = 2 columns. The remaining columns have heights 3, 2, 2, giving the conjugate partition J3,3,1K, which is not in S: for r = 3 the S-interval is J3,3,2K <= . <= J3,3,3K, and S contains no partition J3,3,1K. A different move (removing the first three columns) does reach J2,2K in S, so the theorem may still be true, but the proof of (i) as written fails.","section":"Section 3, proof of Theorem 2(i)"},{"comment":"Even if part (i) were corrected, the two assertions (i) and (ii) are not by themselves sufficient to conclude the characterization. They do not force the mex to be exactly 1 for positions in S, nor do they show that positions outside S have mex different from 1. One needs a simultaneous induction on, say, lambda_1 + length(lambda) or the longest-play measure f, proving that every lambda in S has an option set containing 0 and avoiding 1, while every mu outside S has an option with Sprague-Grundy value 1. The manuscript should state this induction explicitly; the current proof omits it.","section":"Section 3, proof of Theorem 2"},{"comment":"Corollary 11 is stated with the proof omitted, but it is exactly the equality used in the final display of the proof of Theorem 3. Since Theorem 3 is a central result, this gap is load-bearing. The corollary follows from Lemma 5 and Observation 10, so it is not a false assertion, but the proof should be included rather than left as 'the proof, which we omit'.","section":"Section 4, Corollary 11"}],"minor_comments":[{"comment":"The reduction 'Without loss of generality assume lambda_1 >= r' should explicitly cite Observation 4 (conjugate invariance), since it is not otherwise clear why one may assume this inequality.","section":"Section 3, proof of Theorem 2"},{"comment":"The sentence 'Notice that for 2 <= i <= lambda_1 we have Jlambda_1,...,lambda_rK in S <=> lambda_i >= lambda_1 - i + 2' is misquantified: membership in S is a conjunction over all i in that range, together with the length constraint, and should be written with a universal quantifier.","section":"Section 3, proof of Theorem 2"},{"comment":"The expression 'G_P(J1K])' contains a stray closing bracket; it should read 'G_P(J1K)'.","section":"Section 3, proof of Proposition 3"},{"comment":"The phrase 'all partitions with of order at most 15' should read 'all partitions of order at most 15'.","section":"Section 6, paragraph after Question 1"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a solid CGT paper on two new partition games, with clean formulas for rectangles and hyperrectangles. The one real problem is the proof of Theorem 2, which contains a false construction in part (i); the theorem may still be true, but the proof as written does not establish it.\n\nWhat's new: PNim and RNim are natural variants, and the paper gives complete SG values for rectangles (Theorem 1) and all hyperrectangles (Theorem 3). The +1 shift relative to the Nim-sum is a nice, non-obvious fact, and the induction proofs for both are sound. The two-part case (Proposition 5) and the CGH classification are also good. The characterization of value-one partitions (Theorem 2) is plausible and well-motivated, and the paper uses it for misère play.\n\nWhere it gets shaky: the proof of Theorem 2. Part (i) claims that for any λ not in S∪{JK}, the smallest i>1 with λ_i < λ1−i+2 (padding zeros) gives a move to S by removing the first λ1−i+1 columns. That is false. For λ=J5,5,4K, after padding, the first violation is i=4, and removing the first 2 columns leaves J3,2,1K, which is not in S. The 'equivalence' sentence just before is also misquantified: membership in S requires all inequalities plus λ1 = length(λ), not a per-i condition. There is a different move from J5,5,4K to J1K ∈ S, so the theorem might well be true, but this proof does not establish (i). The reader's missing-induction point is real but secondary: if (i) and (ii) were true, a standard induction on partition size would finish the job; that induction is not stated, but that is an omission, not an error.\n\nCorollary 11 is stated without proof; it follows directly from Lemma 5 and Observation 10, so a one-line proof would suffice. Other than that, the rest of the paper is clean.\n\nWho it's for: combinatorial game theorists working on partition games or generalizations of Nim. Theorems 1 and 3 are citable results on their own. The paper deserves a serious referee, but the referee should be told to focus on Theorem 2. If the authors can repair that proof (and state the induction), the paper is acceptable. As it stands, it's conditionally acceptable.","headline":"Nice new games and solid rectangle/hyperrectangle formulas, but the proof of Theorem 2 has a real gap that needs fixing before I'd trust the paper.","tokens_in":12364,"tokens_out":10538,"would_cite":false,"duration_ms":105911,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91A46","05A17"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves exact Sprague–Grundy formulas for two Nim variants on partitions and hyperrectangles, and it characterizes all positions of value one in the partition game.","keywords":["impartial combinatorial game","Sprague–Grundy value","integer partition","Young diagram","hyperrectangle","Nim","misère play","Conway–Gurvich–Ho classification"],"falsifier":"Run an exhaustive recursive computation of $G_R$ on the box $\\langle 2,3,4\\rangle$: the formula predicts the value $1$, so any other value would refute Theorem 3. For Theorem 2, enumerate all partitions up to $n=26$, the range of the paper's appendix, by recursion and check whether any partition outside the stated intervals has value $1$, or any partition inside the intervals has a value different from $1$.","tokens_in":11313,"feed_emoji":"🧮","tokens_out":16223,"duration_ms":145378,"temperature":0.7,"pith_summary":"This paper introduces two impartial games: PNim, where a position is a Young diagram of an integer partition and a move deletes rows or columns (merging what remains), and RNim, where a move shortens one side of a hyperrectangle. The main result is a complete formula for RNim: a single box $\\langle k_1,\\ldots,k_d\\rangle$ has Sprague–Grundy value $((k_1-1)\\oplus\\cdots\\oplus(k_d-1))+1$, so any disjunctive sum of boxes is solved by XORing these values. For rectangular Young diagrams $Jc^rK$ in PNim the same shifted-XOR formula holds, and the paper proves a tight upper bound $\\lambda_1+r-1$ on the value of any partition. It also characterizes the partitions of value one: exactly $J1K$ and the partitions lying between $Jr,r,r-1,\\ldots,2K$ and $Jr^rK$ in Young's lattice for some $r\\ge 2$. These formulas and the placement of the games in the Conway–Gurvich–Ho hierarchy make optimal normal and misère play computable.","feed_headline":"Hyperrectangle Nim: one XOR formula solves every position","feed_subtitle":"The value of a d-dimensional box is the XOR of its reduced side lengths plus one; winning and misère moves follow.","key_machinery":"The argument runs through the mex recursion for Sprague–Grundy values, where mex is the least nonnegative integer not appearing among the values of reachable positions. The key identity is Lemma 5, $k+\\operatorname{mex}(S)=\\operatorname{mex}(\\{0,\\ldots,k-1\\}\\cup\\{s+k:s\\in S\\})$, which lets a shift by $1$ commute with taking the mex of reachable values. Applied to rectangles, whose moves are indexed by reducing one side, this produces $((r-1)\\oplus(c-1))+1$; Corollary 11 generalizes the same shifted-XOR mex identity to any number of dimensions and yields Theorem 3. For the value-one characterization, the load-bearing object is the family $S=\\{J1K\\}\\cup\\bigcup_{r\\ge2}\\{\\lambda: Jr,r,r-1,\\ldots,2K\\le\\lambda\\le Jr^rK\\}$; the proof uses the rank of a partition, the difference between its largest part and its number of parts, to show that no member of $S$ can move to another member. For heavy rectangles, Lucas's theorem on binomial coefficients modulo a prime decides when the upper bound is attained.","core_discovery":"The central claim, stated as Theorem 3, is that RNim has a closed form: for positive side lengths $k_1,\\ldots,k_d$, $G_R(\\langle k_1,\\ldots,k_d\\rangle)=((k_1-1)\\oplus\\cdots\\oplus(k_d-1))+1$. Theorem 1 gives the same formula for a rectangular partition $Jc^rK$ in PNim, namely $G_P(Jc^rK)=((r-1)\\oplus(c-1))+1$. For general partitions, Proposition 1 bounds $G_P(\\lambda)$ by $\\lambda_1+r-1$, and Theorem 2 says the value is exactly $1$ precisely for $J1K$ and for the partitions in the Young-lattice interval $Jr,r,r-1,\\ldots,2K\\le\\lambda\\le Jr^rK$ with $r\\ge2$. The paper also shows that $1$-PNim and $1$-RNim are pet and returnable but not forced, that PNim and RNim are miserable and returnable but not pet, and that these classifications make misère play follow from normal-play values through the tame-game theorem.","pith_inferences":["The shifted-XOR formula for RNim is likely a special case of a general principle: any impartial game whose positions are tuples, whose terminal positions are those with a zero coordinate, and whose moves reduce exactly one coordinate will have Sprague–Grundy value $1+\\bigoplus(k_i-1)$; checking other move sets, such as constrained reductions, would show where the principle breaks.","The value-$1$ partitions of Theorem 2 form intervals in Young's lattice, so the number of such partitions of each integer $n$ should have a generating function; the appendix data up to $n=26$ are a natural testbed for guessing and proving such a formula.","The rectangle-value matrix's stated property of being lexicographically first with no repeated row or column entry suggests that the XOR-plus-one array can be characterized purely order-theoretically, independent of game theory.","The conjectures on heavy partitions suggest that heavy positions may also form lattice intervals, analogous to the value-$1$ family; testing Conjecture 2 on the smallest non-heavy rectangles would either confirm the interval pattern or expose a sharper boundary."],"forward_implications":["Every position of RNim, including sums of boxes of different dimensions, has a Sprague–Grundy value computed by XORing one expression per box, so optimal moves can be read off directly from the side lengths.","Rectangular PNim is completely solved: the value of $Jc^rK$ is $((r-1)\\oplus(c-1))+1$, and the rectangle is heavy exactly when $\\binom{c+r-2}{r-1}$ is odd.","The value-$1$ partitions of Theorem 2 are exactly the losing positions of misère $1$-PNim, and the proof of the theorem supplies a concrete move into that set from every winning position.","Because PNim and RNim are pet and returnable, and their sum games are miserable and returnable, the tame-game theorem gives misère Grundy values of sums from the normal-play formulas.","Resolving PNim under normal play is equivalent to resolving it under misère play, so the same formulas govern both conventions."],"supporting_citations":[{"why":"Gives the Nim Sprague–Grundy recurrence that Observation 10 identifies with the mex of XOR sums over reduced side lengths.","marker":"[1]"},{"why":"Defines the rank of a partition, used in Theorem 2 to show no move inside the candidate family S stays in S.","marker":"[15]"},{"why":"Supplies the tame-game theorem used to compute misère Grundy values of disjunctive sums from normal-play values.","marker":"[18]"},{"why":"Provides the modulo-a-prime binomial theorem used to characterize heavy rectangles by oddness of a binomial coefficient.","marker":"[20]"},{"why":"Defines the pet, miserable, returnable, and forced classes and the closure properties used to classify PNim and RNim.","marker":"[22]"}],"fun_headline_variants":["XOR formula gives exact Grundy for every hyperrectangle in Nim","Hyperrectangle Nim: Grundy value is XOR of (side−1) plus one","One XOR equation solves all of hyperrectangle Nim","Partition and hyperrectangle Nim share a single closed form","Misère play for both Nim variants follows from tame games"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Theorem 2 rests on an unstated induction: the proof shows that every partition outside the candidate family can move into it and no member can move to another member, but it does not spell out the induction on partition size that would make those two facts force the Sprague–Grundy value to be exactly 1 for members and different from 1 for all other partitions.","fun_headline_variants_meta":{"raw":{"variants":["XOR formula gives exact Grundy for every hyperrectangle in Nim","Hyperrectangle Nim: Grundy value is XOR of (side−1) plus one","One XOR equation solves all of hyperrectangle Nim","Partition and hyperrectangle Nim share a single closed form","Misère play for both Nim variants follows from tame games"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000509,"raw_usage":{"total_tokens":2458,"prompt_tokens":904,"completion_tokens":1554,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":520,"completion_tokens_details":{"reasoning_tokens":1467}},"tokens_in":520,"tokens_out":1554,"duration_ms":16454,"temperature":1.0,"reasoning_tokens":1467,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:30:31.243547+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run an exhaustive recursive computation of $G_R$ on the box $\\langle 2,3,4\\rangle$: the formula predicts the value $1$, so any other value would refute Theorem 3. For Theorem 2, enumerate all partitions up to $n=26$, the range of the paper's appendix, by recursion and check whether any partition outside the stated intervals has value $1$, or any partition inside the intervals has a value different from $1$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Nim Sprague–Grundy recurrence that Observation 10 identifies with the mex of XOR sums over reduced side lengths."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the rank of a partition, used in Theorem 2 to show no move inside the candidate family S stays in S."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the tame-game theorem used to compute misère Grundy values of disjunctive sums from normal-play values."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the modulo-a-prime binomial theorem used to characterize heavy rectangles by oddness of a binomial coefficient."}],"review_version":1}