[Paper Review] Subtraction Division Games

Elizabeth J. Kupin|arXiv (Cornell University)|Dec 30, 2011

Artificial Intelligence in Games3 references3 citations

TL;DR

This paper studies a two-player combinatorial game called Subtraction-Division Games, where players alternately subtract a fixed value $ a $ or divide by a fixed divisor $ b $, aiming to reach 1. Using Sprague-Grundy theory, the authors fully characterize when the first player wins or loses for $ a=1 $ and even $ b=2d $, showing that the outcome depends on the base-$ 2d $ representation of $ n $, particularly the parity of blocks of even digits preceding the least significant odd digit.

ABSTRACT

A Subtraction-Division game is a two player combinatorial game with three parameters: a set S, a set D, and a number n. The game starts at n, and is a race to say the number 1. Each player, on their turn, can either move the total to n-s for some s in S or to \lceil\frac{n}{d} ceil, for some d in D (i.e. either subtract s or divide by d and round up). This paper examines in detail the Sprague-Grundy sequences for the games when |S|=|D|=1. For a large class of these games, the patterns in the sequences allow for a characterization of all the winning positions. The paper also proves that, for the same large class, these sequences are automatic. In the process, a large family of recurrences that completely determine their behavior is constructed.

Motivation & Objective

• To determine for which values of $ n $ the first player has a winning strategy in subtraction-division games with parameters $ a=1 $, $ b=2d $.
• To generalize the characterization of game outcomes beyond the special case $ S=\{1\}, D=\{2\} $.
• To apply Sprague-Grundy theory to analyze game states and classify positions as winning or losing.
• To derive a complete characterization of $ SG_{a,b}(n) = 0 $ (first player loss) based on the base-$ b $ representation of $ n $.

Proposed method

• Define the Sprague-Grundy function $ SG_{a,b}(n) $ recursively as the minimum excludant (mex) of the SG values of reachable positions: $ \text{mex}\{ SG(n-a), SG(\lceil n/b \rceil) \} $.
• Set $ SG(1) = 0 $, representing a terminal losing position for the first player.
• Use induction and structural lemmas to analyze game behavior for even $ b=2d $, focusing on parity and digit patterns in base $ 2d $.
• Analyze sequences of moves by repeatedly dividing by $ b $ and rounding up, tracking SG value alternation along such paths.
• Reduce complex game states to simpler ones using recursive reductions based on coefficient parity and digit blocks in base $ 2d $.
• Classify outcomes based on the block of even digits immediately before the least significant odd digit in the base-$ 2d $ expansion of $ n $.

Experimental results

Research questions

• RQ1 For $ a=1 $ and even $ b=2d $, what values of $ n $ result in a first player loss (i.e., $ SG_{a,b}(n) = 0 $)?
• RQ2 How does the base-$ 2d $ representation of $ n $ determine the Sprague-Grundy value and thus the game outcome?
• RQ3 Can the game outcomes be fully characterized using digit patterns in base $ b $, particularly blocks of even digits preceding odd digits?
• RQ4 What recursive reductions preserve the Sprague-Grundy value and allow for efficient classification of game states?
• RQ5 Under what conditions does the game exhibit periodic or alternating behavior in SG values along division paths?

Key findings

• For $ a=1 $ and $ b=2d $, $ SG_{1,2d}(n) = 0 $ if and only if the block of even digits immediately preceding the least significant odd digit in the base-$ 2d $ representation of $ n $ has even length.
• If $ n $ is even, then $ SG_{1,2d}(n) \neq 0 $, meaning the first player can always win by subtracting 1, provided $ n > 1 $.
• When $ n \equiv 1,3,\ldots,2d-1 \pmod{4d} $, the game outcome depends on the number of steps required to reach a number in $ \{2d+1, 2d+3, \ldots, 4d-1\} $, with odd-length paths leading to first player wins and even-length paths to losses.
• The Sprague-Grundy value of a number $ n $ with base-$ 2d $ expansion $ 2dc_1 + 4d^2c_2 + \cdots $ can be reduced recursively by analyzing the parity of coefficients and applying reductions that preserve the SG value.
• The reduction process shows that $ SG(2dc_1 + 4d^2c_2 + \cdots) = SG(c_1 + 2dc_2 + \cdots) $ when $ c_1 $ is even and non-zero, enabling recursive simplification.
• The analysis confirms that $ SG_{a,b}(n) $ is bounded and that no two adjacent states share the same SG value, a key property of such impartial games.