REVIEW 1 major objections 4 minor 7 references
Delete Nim
T0 review · 1 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read The G-value of every Delete Nim position $(x,y)$ is $v_2((x\vee y)+1)$, the exponent of $2$ in one plus the bitwise OR of the heap sizes.
desk verdict A small, correct, self-contained result: Delete Nim's Grundy value is v2((x∨y)+1), with a sound mex proof and a useful isomorphism to VDN. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the pair of heap sizes $(x,y)$ together with the function $v_2((x\vee y)+1)$, where $v_2(n)$ is the 2-adic valuation, meaning the exponent of the highest power of 2 dividing $n$. The argument's engine is the mex recursion for G-values (mex is the minimum excluded value): $G$ at a position is the smallest nonnegative integer not attained among the G-values of its options. To prove the formula, the paper shows (i) from a position with value $h$, no option has value $h$, because any move changes the bit pattern of $x \vee y$; and (ii) for each $h' < h$, the explicit construction $x' = x - 2^{h'}$, $y' = 2^{h'}-1$ (after choosing a heap whose $h'$-th bit is 1) gives a legal move to a position with value $h'$. The isomorphism $F((x,y)) = (x-1,y-1)$ then transfers the formula to the Variant of Delete Nim.
What would settle it
Compute $G((1,2))$ by hand: the theorem predicts $v_2((1\vee 2)+1)=v_2(4)=2$. Under the paper's convention the options are $(0,0)$, $(1,0)$, and $(0,1)$, whose G-values are $0$, $1$, and $1$, so the mex is $2$; if zero heaps are banned, only the move to the empty position remains, so the mex is $1$, a direct counterexample to the formula under the stricter convention.
Extended reading notes
Core claim
On the paper's own terms, the discovery is that the G-value of Delete Nim is exactly the 2-adic valuation of one plus the bitwise OR of the two heap sizes: $G((x,y)) = v_2((x\vee y)+1)$. Concretely, write $x$ and $y$ in binary, OR them bit by bit, add 1, and count how many times 2 divides the result; that count is the G-value, which in turn decides the winner in any disjoint sum. Because $v_2((x\vee y)+1)=0$ exactly when the OR is even, the losing positions are precisely those in which both heap sizes are even. The proof separates into two claims: no legal move preserves the value $h$, and for every $h' < h$ some legal move produces a position with value $h'$. A shift map $F((x,y))=(x-1,y-1)$ is then shown to be an isomorphism between the variant game VDN and Delete Nim, so VDN's G-values are $v_2(((x-1)\vee (y-1))+1)$.
Load-bearing premise
The load-bearing premise is that an empty heap counts as a legal heap, so positions like $(n,0)$ are allowed and the move that takes one stone without splitting is represented as $(n,0)$; without that convention the move set and the formula both change.
Editorial extensions
If this is right
- A Delete Nim position is a P-position (previous player wins) exactly when both heap sizes are even, because $v_2((x\vee y)+1)=0$ in precisely that case.
- The G-value of any position can be computed in time proportional to the number of bits: take the bitwise OR of the heap sizes, add one, and count trailing zero bits.
- The isomorphism $(x,y)\mapsto(x-1,y-1)$ gives the full G-value computation for the Variant of Delete Nim, whose P-positions were previously known.
- By the standard disjoint-sum theorem for impartial games, the value of any disjoint sum of Delete Nim positions is the bitwise XOR of the individual $v_2$ values, so sums can be analyzed once each position is evaluated.
- The formula is a rare case in which the OR operation, rather than XOR, controls the outcome values, as the paper notes.
Reading between the lines
- The theorem depends on allowing heap sizes of 0 in positions such as $(n,0)$; if empty heaps were forbidden and splitting made mandatory, the same recursion would give different values, so the empty-heap convention is part of the game's definition.
- Because the formula is governed by the highest power of 2 dividing $(x\vee y)+1$, the game has a recursive structure in which positions with large value reduce to a heap of size $2^{h'}-1$ and a remaining heap; this suggests a natural family of games where the subtracted number is a fixed $k$, with values perhaps given by variants of $v_2((x\vee y)+k)$.
- The fact that P-positions are exactly the even-even pairs hints that the outcome is invariant under scaling both heaps by a power of 2, a pattern that could be tested in a three-heap analogue.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper introduces Delete Nim, a two-heap impartial game in which a move consists of deleting one heap, removing one stone from the remaining heap, and optionally splitting that remaining heap into two heaps. The main result, Theorem 4, gives a closed formula for the Sprague-Grundy value of any position (x,y): G((x,y)) = v2((x∨y)+1), where ∨ is bitwise OR and v2 is the 2-adic valuation. The proof is a direct mex argument: it shows no option has the same v2 value and that every smaller value is attained by an explicit option. Theorem 5 then gives an isomorphism between Delete Nim and a previously studied variant, VDN, transferring the formula to that game. The paper is short, elementary, and self-contained apart from the standard Sprague-Grundy theory.
Significance. If the intended convention on empty heaps is made explicit, the result is a neat, verifiable closed form for the G-values of a natural impartial game. The proof is genuinely constructive: the option attaining each smaller G-value is given explicitly, and the obstruction to attaining the same G-value is shown directly from the 2-adic structure. There are no fitted parameters, no circular dependencies, and the isomorphism to VDN is clean. The novelty is modest, but the connection between bitwise OR and G-values is a pleasant addition to the small catalogue of games with simple G-value formulas.
major comments (1)
- [Section 2, Theorem 4 proof] The proof of the 'attaining smaller values' direction uses the position (x−2^{h'}, 2^{h'}−1). For h'=0 this is (x−1,0), and the un-split move is also represented as (x−1,0). The paper never states the convention that heap sizes may be zero and that a position may consist of one empty and one non-empty heap. This is load-bearing: if positions are required to be two positive heaps, the move set is different and Theorem 4 is false under that reading; for example, from (1,1) the move that removes the last stone cannot produce two positive heaps, so (1,1) would have no options and G((1,1))=0, whereas the formula gives v2(2)=1. Please state explicitly in the definition of Delete Nim that heap sizes are nonnegative integers and that a move that does not split is recorded as (n,0). The same convention is needed for the isomorphism in Theorem 5, since F(x,y)=(x−1,y−1) sends positive VDN positions to Delete Nim positions with possible zero entries.
minor comments (4)
- [Section 2, after Theorem 5] The expression v2((x−1)∨(y−1)+1) is ambiguous; write v2(((x−1)∨(y−1))+1) to make the scope of the +1 clear.
- [Section 2, Theorem 4 proof] In the h>0 case, the sentence 'Then x_h = 1 or y_h = 1, which is a contradiction' relies on the fact that h=v2((x∨y)+1) forces the h-th bit of x∨y to be 0; this should be stated explicitly.
- [Title page / Abstract] There are a few typographical errors: 'Aknowledgements' should be 'Acknowledgements', 'Cleary' should be 'Clearly', and the text says Bouton's result is from 1902 while the reference gives 1901.
- [Section 1, definitions] The case vp(0)=∞ in the definition of the p-adic valuation is never used in the formula, since (x∨y)+1 is always positive; it can be kept for completeness but is unnecessary.
Circularity Check
No circularity: the G-value formula is derived directly from the mex definition with no fitted parameters or load-bearing self-citations.
full rationale
The derivation chain is self-contained. Theorem 4 is proven directly from the recursive definition of G (Definition 2) by a mex argument: first the paper shows that no option (x',y') of (x,y) satisfies v2((x'∨y')+1)=h, and then it shows that for every h'<h an option with v2((x'∨y')+1)=h' exists by explicitly constructing x'=x−2^{h'}, y'=2^{h'}−1. These two facts are exactly what the mex definition requires, so the formula G((x,y))=v2((x∨y)+1) follows by induction without any fitted quantity or imported result. The only prior work cited for the game itself, [6], is used as background: the authors mention that VDN was introduced there and that its P-positions were shown, but Theorem 5 proves the isomorphism F((x,y))=(x−1,y−1) directly by a one-to-one move correspondence rather than relying on the cited P-position result. The Sprague-Grundy theorems are standard external machinery and are not being used to smuggle in the target formula. The zero-heap convention used implicitly in the construction (e.g., y'=0 for h'=0) is an unstated modeling assumption and a potential correctness caveat under a different convention, but it is not circularity: it does not make the derivation equivalent to its input. There are no self-citations of the authors' own unverified work, no fitted-parameter-then-predicted moves, and no renaming of a known result. The central claim is therefore independently derived from the game rules and the mex definition.
Assumptions & free parameters
assumptions (4)
- domain assumption Every play of the game ends in a finite number of moves.
- standard math Sprague-Grundy theorem: G(g)=mex({G(g')}) determines P-positions and disjoint sums via XOR.
- domain assumption Zero-size heaps are allowed as positions in the two-heap representation.
- standard math Bitwise OR is the usual binary OR operation on nonnegative integers.
Cite this review
Pith. "Pith review of Delete Nim." pith.science (2026). https://pith.science/paper/4DK2NF6U
@misc{pith2026190807763,
author = {Pith},
title = {Pith review of: Delete Nim},
year = {2026},
howpublished = {\url{https://pith.science/paper/4DK2NF6U}},
note = {Machine review of arXiv:1908.07763}
}
read the original abstract
In this paper, we study an impartial game called Delete Nim. In this game, there are two heaps of stones. The player chooses one of the heaps and delete the other heap. Next, she takes away one stone from the chosen heap and optionally splits it into two heaps. We show the way to calculate the G-value of this game by using the OR operation and 2-adic valuation.
Figures
Reference graph
Works this paper leans on
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[6]
Z. Stankova and T. Rike, editors. A Decade of the Berkeley Math Circle , volume 1, page 159. Mathematical Circles Library, 2008
work page 2008
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[1]
C. L. Bouton. Nim, a game with a complete mathematical theory. Annals of Mathematics , 3(1/4):35–39, 1901
work page 1901
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[2]
Alex Fink, Aviezri S Fraenkel, and Carlos Santos. Lim is not slim. International Journal of Game Theory , 43(2):269–281, 2014
work page 2014
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[3]
A. S. Fraenkel and M. Lorberbom. Nimhoff games. Journal of Combinatorial Theory, Series A , 58(1):1–25, 1991
work page 1991
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[4]
Patrick M. Grundy. Mathematics and games. Eureka, 2:6–9, 1939
work page 1939
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[5]
Roland P. Sprague. ¨Uber mathematische Kampfspiele. Tˆ ohoku Math. J., 41:438–444, 1935-36
work page 1935
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[7]
Cornelius P Welter. The theory of a class of games on a sequence of squares, in terms of the advancing operation in a special group. Indag. Math., 16:194–200, 1954. 4
work page 1954
Reviewed August 14, 2026 · model on record in the stance chip above.
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