Lexical Recall or Logical Reasoning: Probing the Limits of Reasoning Abilities in Large Language Models
Authors: Henrike Beyer, Chris Reed
Accepted at: acl25

Lexical Recall or Logical Reasoning: Probing the Limits of Reasoning Abilities in Large Language Models

Henrike Beyer and Chris Reed
Centre for Argument Technology
University of Dundee
Dundee DD1 4HN, United Kingdom
2579207@dundee.ac.uk; c.a.reed@dundee.ac.uk

Abstract

Despite the increasing interest in the reasoning abilities of Large Language Models (LLMs), existing work shows limitations in assessing logic abilities independently from lexical memory. We address this gap with Mystery-Zebra. This robust two-part benchmark (4,290 puzzles) challenges the logic abstraction abilities of LLMs in two setups: (1) a lexical obfuscation setup tests the dependence of LLMs on lexical content based on two canonical grid puzzles widely spread on the Internet; (2) a set of new grid puzzles in 42 different sizes and 12 difficulty levels tests how the formal difficulty degree of a puzzle affects LLMs. We test open and closed-weight LLMs on both parts of the benchmark. The results on part two suggest that model sizes up to 70B parameters have only a minor influence when solving newly generated puzzles, while performance mainly relates to the number of items in the puzzle. The results on the first part of the benchmark suggest that the applied obfuscation strategies help to mitigate effects of logic puzzles being part of LLM training data, showing a drastic drop in performance for obfuscated versions of well-known puzzles. In addition we conduct a case-study on the first part of the benchmark predicting the position of single items, unveiling that the reasoning abilities of LLMs are mainly limited to a few consecutive steps of reasoning.1

1 Introduction

Alongside the general improvement in Large Language Model (LLM) performance across NLP tasks, the trust in their reasoning abilities increases. Their deployment in real-world applications calls for a critically informed view on LLM reasoning since misconceptualising these abilities has severe consequences.

There are five different-coloured houses: red, green, ivory, yellow, blue
There live five residents of a different nationality: English, Spanish, Norwegian, Ukrainian, Japanese
Each resident owns a different pet: Dog, Zebra, Fox, Horse, Snails
Each prefers a different drink: coffee, tea, milk, orange-juice, water
Each smokes a different brand of cigarette: old-gold, kools, chesterfields, lucky-stripe, parliament
You’re given 15 other facts:
1. There are five houses.
2. The Englishman lives in the red house.
3. The Spaniard owns the dog.
4. Coffee is drunk in the green house.
5. The Ukrainian drinks tea.
6. The man who smokes chesterfields lives immediately to the right of the ivory house.
7. The Old Gold smoker owns snails.
8. Kools are smoked in the yellow house.
9. Milk is drunk in the middle house.
10. The man who smokes kools lives next to the house where the horse is kept.
11. The man who smokes chesterfields lives in the house next to the man with the fox.
12. Kools are smoked in a house next to the house where the horse is kept.
13. The Lucky Strike smoker drinks orange juice.
14. The Japanese smokes Parliaments.
15. The Norwegian lives next to the blue house.

Figure 1: Grid puzzles like the “Zebra” puzzle encode relationships between items in natural language clues. Through deduction and elimination, these clues can be interpreted to populate a unique solution grid.

While classic reasoning benchmarks for LLMs cover basic maths, rule-based, or inferential reasoning (Wan et al., 2024; Mirzadeh et al., 2024; Gui et al., 2024; Wang, 2024), another line of re-

1The code used to create obfuscated puzzle variants, run experiments, and evaluate results is available under: https://github.com/arg-tech/MysteryZebra
search assesses LLMs on games like Minecraft, card games, and logic puzzles (e.g. Sudoku) (Wang et al., 2023; Gupta, 2023; Guo et al., 2023; Li et al., 2024a; Shah et al., 2024; Saha et al., 2024).

Nevertheless, the intense scaffolding needed to apply LLMs in games blurs insights into their reasoning abilities. Further, abstract puzzles like Sudoku rely on more abstract non-linguistic representations offering a limited perspective on the link between language and reasoning in LLMs. In contrast, grid puzzles like the popular “Zebra”2 and “Einstein”3 puzzle offer a constrained task to evaluate LLM reasoning abilities in a natural-language setup. As illustrated in Figure 1, a typical grid puzzle contains a set of clues describing the relative position of items of the same syntactic domain (e.g. pets). The correct solution can be translated to a grid, where each row represents a domain, where the items are positions relatively to each other in columns. For a successful solution, the model needs to identify clue(s) leading to a valid elimination of positioning options or a correct positioning, keep track of the grid’s current status, and ensure a conflict-free solution grid.

Grid-puzzles are rule-based (Giadikiaroglou et al., 2024) and therefore have a strict set of constraints to be satisfied in the solution grid. These constraints can take the form of symbolic logic expressions, turning the puzzles into Constraint Satisfaction Problems (CSPs). This allows to decouple the logical structure of the the clue from its linguistic representation. In the solution process, it is irrelevant whether the clues involve pets or beverages as these are just variables.

Figure 2 details the workflow of this paper, which leverages the similarity to CSPs to propose Mystery Zebra. The two parts of this benchmark challenge the reasoning capabilities of LLMs necessary to solve logic puzzles.

Part one of the benchmark targets the dependence of LLMs on residual lexical clues from the training content when solving a complex logic puzzle. In order to make the logical structure independent from the linguistic representation, we turn each clue into its symbolic logical representation; e.g. The dog-owner drinks coffee → A = B. Similarly to Valmeekam et al. (2023), we apply different obfuscation techniques to manipulate the popular “Zebra” and “Einstein” puzzle instances, which are

2https://en.wikipedia.org/wiki/Zebra_Puzzle
3https://www.chessandpoker.com/einsteins-problem-solution.html

![Workflow diagram showing the Mystery-Zebra Benchmark, including Part 1 and Part 2, with steps for Zebra & Einstein Puzzle (original), Newly generated puzzles, Original Puzzles + Rephrasing + 18 variants, 4,250 Puzzles (12 levels, 42 sizes), Q&A case study (accuracy per solution step), Grid prediction (Grid accuracy), and Evaluation.](page_768_180_613_670.png)

Figure 2: Workflow of creating and testing the Mystery Zebra benchmark. In two distinct parts: (1) contains the two canonical puzzle variants (“Zebra” and “Einstein”) with lexical manipulations of them; (2) with 4,250 newly generated puzzles in 42 sizes and with 12 difficulty levels defined through formal difficulty of the puzzle clues. Both parts are tested in a grid prediction setup, while only part 1 is evaluated in a Q&A case-study.

widely spread on the Internet. We manipulate the phrasing of clues, but also the lexical content of the variables in the clues without changing the underlying symbolic structure. This helps to assess the dependence of LLMs on the exact lexical content of puzzles that are likely contained in their training data. Part two of the benchmark assesses the effect of grid size and formal difficulty of clues. It includes 4,250 newly generated puzzles in 12 formal difficulty classes and 42 sizes. While existing datasets control difficulty mainly based on the grid size or coarse difficulty categories, Mystery-Zebra uses the amount of information conveyed by the clues to judge difficulty on a fine-grained basis.
A range of open and closed weight LLMs is evaluated on both parts are evaluated via grid-accuracy when predicting the whole solution grid. In addition, part two is used to conduct a case study in a Q&A setup.

The contributions of this paper are 7-fold: (1) we introduce the first adaptable reasoning benchmark enabling multidimensional analysis of LLM reasoning; (2) we evaluate LLMs’ logical abstraction abilities from the lexical content of a well-known puzzle; (3) we show that our obfuscation techniques can mitigate data contamination from training; (4) we provide a formally motivated reference point for puzzle difficulty; (5) we find that LLMs struggle primarily with larger puzzles requiring longer reasoning chains; (6) we show that scaling models up to 70B parameters yields only limited gains in performance; and (7) we demonstrate, through a case study, that models can perform only a limited number of consecutive inferential reasoning steps.

2 Related Work

Puzzle solving  Formal logic and early machine learning started tackling automatic puzzle solving including CSPs like maths puzzles, Sudoku, Crossword puzzles or grid-puzzles (Wos, 1988; Shazeer et al., 1999; Goldberg et al., 2002; Jones et al.; Shapiro, 2011; Chesani et al., 2017; Valentine and Davis, 1987; Salavati et al., 2009).

Pre-LLM approaches in NLP tried solving grid puzzles with controlled or hybrid architectures (Schwitzer, 2013; Jabrayilzade and Tekir, 2020). As LLMs’ performance increased, a range of works explored their reasoning abilities with puzzles like Minesweeper (Li et al., 2024b). Results suggest very basic reasoning abilities and issues in performing consistent, coherent longer reasoning chains.

Live-Bench (White et al., 2024) includes Zebra-Puzzles in a Q&A format asking for one single item. Tyagi et al. (2024) develop a LLM grid-puzzle benchmark with 5 sizes and 3 difficulty levels, analysing reasoning chains in depth. They conclude that mistakes are more frequent in the second half of the reasoning process than in the initial part. Lin et al. (2024) provides puzzles in 36 sizes and rank puzzle difficulty based on puzzle size. They find that models between 7 and 10 billion parameters struggle with hard puzzles and show a low accuracy for smaller puzzles. Despite the advancements in these works, they offer only a few sizes and fail to provide fine-grained difficulty levels. Further, potential effects of residual lexical clues from the training data are ignored. This work aims to address this gap by offering a larger variety of sizes and difficulty levels based on a formal perspective and providing systematic obfuscation techniques to mitigate effects for puzzles that might be part of LLM training data.

Task obfuscation  Obfuscation techniques originate from adversarial training of image classification models to test the robustness against variations in the input data (Xu et al., 2018; Pezzementi et al., 2018; Woods et al., 2019; Dong et al., 2020; Zhang et al., 2020; Badjie et al., 2024). In language modelling, a wide range of techniques was developed to generate manipulated language input and test the robustness of language models (Xu et al., 2018; Wang et al., 2019; Zhang et al., 2019; Niewinski et al., 2019; Liu et al., 2020).

Further, obfuscation is used in adversarial studies to assess LLM-weaknesses (Schwinn et al., 2023; Zou et al., 2024; Kumar, 2024). Steindl et al. (2024); Liu et al. (2024); Yao et al. (2024); Ahmed and Angel Arul Jothi (2024) use this to evaluate the robustness of LLMs against jailbreak attacks that attempt to circumvent safeguarding mechanisms .

Recently, many other planning or reasoning tasks adopted obfuscation methods to probe the reasoning abilities of LLMs. Valmeekam et al. (2023) manipulate the Blocks-World planning task systematically replacing the actions required in the planning process. Nezhurina et al. (2024) propose the Alice in Wonderland (AIW) task, manipulating lexical items in a simple logic puzzle. Xie et al. (2024) use a similar approach on Knights & Knaves puzzles. All works report a breakdown in performance for puzzles from manipulated (training) data.

3 Grid Puzzles as CSPs

A CSP perspective on grid-puzzles helps to determine the difficulty of puzzles. By formulating the rules of the puzzle in pseudo-logical form, it can be determined how many variables are linked through a single clue. In addition, this angle helps to explore how tightly the clues condition the position of items in the grid. These properties are crucial for formal approaches as they influence how many options there are for placing an item in the grid after interpreting a clue. They also determine if a clue can be interpreted on its own or requires combined information from other clues.

This work uses this property to offer a fine-
Rephrased Zebra puzzle

1. the ☀️ is the person who likes 🍒
2. the 🇩🇪 is the owner of the 🐾
3. the ☀️ person drinking 🍹 is the person who likes 🍒
4. the ☀️ is the person drinking 🍹
5. the person who likes 🍒 is to the right of the person who likes 🍒
6. the 🇩🇪-smoker is the owner of the 🐾
7. the 🇩🇪-smoker is the person who likes 🍒
8. the 🇩🇪-smoker is in the middle
9. the 🇩🇪 is on the far left
10. the 🇩🇪-smoker is on the left or right of the owner of the 🐾
11. the 🇩🇪-smoker is to the left or right of the owner of the 🐾
12. the 🇩🇪-smoker is the person drinking 🍹
13. the 🍒 is the 🇩🇪-smoker
14. the 🍒 is to the left or right of the person who likes 🍒

5-Domain obfuscated Zebra puzzle

1. the ☀️ is the person who drives the 🚗
2. the ☀️ is the person whose hobby is 🎯
3. the person’s sport is 🏠 is the person who drives the 🚗
4. the ☀️ is the person who’s sport is 🏠
5. the person who drives the 🚗 is to the right of the person who drives the 🚗
6. the person eating 🍳 is the person who’s hobby is 🎯
7. the person eating 🍳 is the person who drives the 🚗
8. the person who drives the 🚗 is in the middle
9. the 🍳 is on the far left
10. the person eating 🍳 is on the left or right of the person who’s hobby is 🎯
11. the person eating 🍳 is to the left or right of the person who’s hobby is 🎯
12. the person eating 🍳 is the person who’s sport is 🏠
13. the 🍳 is the person eating the 🍳
14. this is to the left or right of the person who drives the 🚗

Figure 3: Example of the obfuscation process for the “Zebra” puzzle. On the right is the rephrased “Zebra” puzzle (see Figure 1 for the original phrasing). The items in the puzzle and solution grid are then systematically replaced in the 5-domain obfuscated puzzle so that the logical structure of deductions leading to the solution grid remains intact.

grained range of difficulty levels based on the properties of the clues of the riddle. The lowest level of difficulty uses the commonly employed clue types which link two variables with determined positions. With increasing difficulty, the clues link up to 3 variables in the grid and become less restricting, with the highest levels of difficulty including less strict position clues (A is somewhere to the right of B) and optional conditions (A ‘or’ B). The difficulty of correctly interpreting the advanced types of clues rises with the size of the grid and especially with the number of items per domain.

4 Dataset creation

Mystery-Zebra contains 4,290 puzzles in two parts: Part one (40 puzzles) contains the original “Einstein” and “Zebra”5 puzzles with 19 each, preserving the logical structure, while varying the lexical content. Part two includes 4,250 newly generated puzzles not found in any available corpus.4

4.1 Manipulated Puzzles

Part one of the benchmark includes the original ‘Einstein’ and the “Zebra”5 puzzle (hereafter canonical puzzles; see App. B for the original puzzles) These puzzles are solved in \(5 \times 5\) grids and their solutions can be found on various webpages, making it likely that they are part of LLM training data.

For each of the canonical puzzles, 19 obfuscated variants are created to test the ability to abstract from the lexical content to the logical structure. We opt for minimal manipulations in order to showcase the effect of minor changes on the model performance. We keep the symbolic structure of the canonical puzzles and vary the lexical representation of variables in the puzzle. in order

We obfuscate the puzzles in a 3-step process. (1) the puzzles are translated to a symbolic representation. (2) optionally, the content of the solution grid and corresponding clues is changed. This step uses the obfuscation strategies detailed below. (3) the puzzle clues are rephrased in a uniform way so that the clues are grammatically valid structures (see App. C for details). Mystery Zebra contains 3 versions per obfuscation strategy.

Rephrasing The canonical puzzles both vary the verb-phrases used for the domains. To allow automatic obfuscation, we adopt uniform verb-phrases within a domain of items (see App. C). In order to test the effect of this, we apply the uniform phrasing to the original puzzles, skipping step (2).

\(n\)-domain obfuscation This strategy replaces in step (2) \(n\) domains and their corresponding items in the solution grid and clues (e.g. beverage→hobby). As \(n\) increases, the lexical difference to the original puzzle increases. Figure 3 illustrates the dif-

4The benchmark is released under BY-NC-SA 4.0 and available here: https://huggingface.co/datasets/arg-tech/MysteryZebra
5We use the more commonly known version where the green house is to the right of the ivory house.
ference between the rephrased and the 5-domain obfuscated version of the “Zebra” puzzle.

in domain obfuscation Here, step (2) replaces all items of the solution grid with different items of the same domain, thus keeping the original domains intact (e.g. yellow→azure in color). This creates a smaller semantic difference to the original, also leaving the corresponding verb-phrases unchanged.

4.2 Generated Puzzles
The puzzles for part two are created systematically using a puzzle generator6. The difficulty of a puzzle is determined by the size of the solution-grid and the types of clues. The dataset includes puzzles in sizes between \(1 \times 2\) and \(7 \times 7\). The generator provides transparent fine-grained metrics for the formal difficulty of a puzzle. We use the first 12 difficulty levels provided. Each level is associated with specific formal clue-structures (e.g. \(A = B\); \(A = B \lor A = C\)) which are added to a growing pool from which the generator can choose (see App. A for all clue-structures). The difficulty of a structure is determined by how restrictive it would be as the only clue. With increasing difficulty, the number of possible solution grids matching the clue increases, meaning that the amount of information contained within the clue decreases.

At least one clue must have a structure from target level (e.g. all Level 6 puzzles have at least one clue in level 6 clue-structure). Since some structures require larger grids (e.g. Level 2: *A is between B and C*), not all grid sizes are available at all levels. Table 5 in App. A contains information on the level size combinations. We create 10 puzzles for each level size combination.

5 Methodology

5.1 Problem formulation
In our experiments, the models predict the solution grid \(G\) given a set of clues \(C = \{c_1 \ldots c_n\}\) and the empty target grid \(G\) with \(d\) domains and \(n\) items per domain. The corresponding prompt is in App. D.

5.2 Grading solutions
We require the model to provide the full solution grid. Because only one misplaced item can lead to a wrong answer, we refrain from a classical accuracy measure. Instead, a point scale (one point per correctly filled cell) is applied, which can be converted to an accuracy score reflecting the percentage of correct cells in the solution grid. This offers a more differentiated and comparable measure across sizes. In the following, this is called grid completion accuracy.

As a baseline, we use a random, uniformly distributed assignment of each item to a random position within the correct domain. This simulates a model that ignores the clues but still places each item exactly once and within the correct domain. Te expected value for correctly guessed cells in a grid with \(d\) domains and \(n\) items per domain is \(E = d\). Transferred to the measure of grid accuracy, this is: \(E = \frac{d}{n \cdot d} = \frac{1}{n}\)

6 Experimental setup
We assess open and closed-weight models for different puzzle sizes, clue difficulties and obfuscation techniques (see App. E for the hyper-parameters used). The first experiment includes Llama 3.1-8B, 3.3-70B, 3.3-70B (Touvron et al., 2023), Mistral-7B (Jiang et al., 2023), Qwen 2.5-7B, 72B7, GPT 4o-mini, GPT 4o (Achiam et al., 2023), and R1 (DeepSeek-AI et al., 2025). The second experiment uses only open-weight models between 7 and 72B parameters due to its higher cost. In a pre-study involving three example puzzles from the second part of the benchmark, we determined the models’ reactions to the prompt in App. D. No additional prompting strategies are used as Tyagi et al. (2024) report no improvement in puzzle solving over a variety of prompting strategies.

Experiment 1 The first experiment evaluates the open-weight and GPT models on the canonical puzzles and their obfuscated versions. For the original and rephrased versions, three solutions are predicted with different random initialisations. The *n-Domain* and *in domain* obfuscations three different obfuscated versions are tested.

Experiment 2 In this experiment, the models predict a solution for 10 different puzzles from the second part of the benchmark of each level size combination. As we expect no effect from training data in this case, no obfuscated versions are tested.

7 Results
The evaluation of the grid-format experiments presupposes a table format with the domains as rows and the items in the domains as columns. The
Einstein puzzle and obfuscations

Zebra puzzle and obfuscations

Figure 4: Average grid accuracy on the puzzles and their obfuscated versions of Deepseek R1, GPT 4o, GPT 4o-mini, Llama 3.1-8B, 3.1-70B, 3.3-70B, Qwen 2.5-7B, 2.5-70B, and Mistral 7B. The random baseline for 5 × 5-puzzles of 0.2 is indicated by the dashed black line. Star-shaped points indicate the 1% significance interval. Data points without a correctly formatted prediction are squares or triangles.

Q&A format is evaluated based on the target format defined in the prompt and accounts for minor variations through additional white spaces. All generated answers that do not contain the target formats are excluded from the analysis (see App. F). The significance intervals provided are calculated as described in App. G.

Experiment 1 Figure 4 reports the results of experiment 1. The performance of the GPT models on the “Einstein” puzzle (left) is nearly perfect on the original puzzle and degrades with the degree of obfuscation, being close to the random baseline (0.2) for in domain and 5 domain obfuscation. The performance of R1 is high for up to 2-domain obfuscations but degrades afterwards. For the in domain obfuscation, the performance of R1 is significantly above the random baseline. The performance of open-weight models is below the GPT models on the original puzzle. Qwen 2.5-72B is an exception and performs more similarly to the GPT models.

The results for the “Zebra” puzzle show an overall lower performance with the maximum achieved grid completion accuracy being 97% by 4o-mini on the rephrased version. All models except for R1 and Qwen 2.5-7B perform worse on the puzzle as found on the internet compared to the rephrased version. The performance still decreases as the level of obfuscation increases. No model except for R1 outperforms the random baseline significantly after the 2-domain obfuscations. R1’s performance is more unstable, bouncing between performances significantly above the random baseline and insignificant performances. Nevertheless, a clear tendency for degrading performance is visible. Mistral and Llama show a poor performance close to or below random baseline across the board of versions. These two models struggle significantly with producing the target format. The larger open-source models achieve a performance more similar to the GPT models with Llama 3.1-70B, even surpassing them for the original and rephrased version.

<table>
  <tr>
    <th rowspan="2">Lv</th>
    <th colspan="2">Qwen2.5</th>
    <th colspan="2">Llama 3.1</th>
    <th colspan="2">Llama 3.3</th>
    <th colspan="2">Mistral</th>
  </tr>
  <tr>
    <th>7B</th>
    <th>72B</th>
    <th>8B</th>
    <th>70B</th>
    <th>70B</th>
    <th>7B</th>
  </tr>
  <tr><td>Lv1</td><td>0.38</td><td>0.29</td><td>0.30</td><td>0.51*</td><td>0.57**</td><td>0.20</td></tr>
  <tr><td>Lv2</td><td>0.29</td><td>0.23</td><td>0.21</td><td>0.38</td><td>0.43*</td><td>0.15</td></tr>
  <tr><td>Lv3</td><td>0.35</td><td>0.27</td><td>0.25</td><td>0.44*</td><td>0.51*</td><td>0.18</td></tr>
  <tr><td>Lv4</td><td>0.37</td><td>0.28</td><td>0.29</td><td>0.45*</td><td>0.51*</td><td>0.21</td></tr>
  <tr><td>Lv5</td><td>0.36</td><td>0.27</td><td>0.30</td><td>0.46*</td><td>0.55**</td><td>0.22</td></tr>
  <tr><td>Lv6</td><td>0.29</td><td>0.22</td><td>0.22</td><td>0.38</td><td>0.48*</td><td>0.20</td></tr>
  <tr><td>Lv7</td><td>0.29</td><td>0.22</td><td>0.22</td><td>0.38</td><td>0.48*</td><td>0.15</td></tr>
  <tr><td>Lv8</td><td>0.32</td><td>0.28</td><td>0.27</td><td>0.40</td><td>0.47*</td><td>0.21</td></tr>
  <tr><td>Lv9</td><td>0.31</td><td>0.26</td><td>0.23</td><td>0.38</td><td>0.43</td><td>0.20</td></tr>
  <tr><td>Lv10</td><td>0.23</td><td>0.20</td><td>0.21</td><td>0.30</td><td>0.33</td><td>0.15</td></tr>
  <tr><td>Lv11</td><td>0.23</td><td>0.21</td><td>0.22</td><td>0.28</td><td>0.35</td><td>0.15</td></tr>
  <tr><td>Lv12</td><td>0.24</td><td>0.19</td><td>0.20</td><td>0.28</td><td>0.31</td><td>0.14</td></tr>
</table>

Table 1: Average table completion accuracy (for correctly formatted predictions) per level (Lv) for Model (Md) Qwen 2.5-7B, 2.5-72B, Llama 3.1-8B, 3.1-70B, 3.3-70B, and Mistral-7B. With respect to a random baseline, * indicates \( p \leq 0.05 \) and ** indicates \( p \leq 0.01 \).

Experiment 2 Table 1 focuses on the results per difficulty level, while averaging over grid sizes and
Table 2: Average table completion accuracy (for correctly formatted predictions) for each grid size (with d (domains) × n (items per domain)) for Qwen 2.5-7B, 72B, Llama 3.1-8B, 70B, Llama 3.3-70B, and Mistral-7B. With respect to a random baseline, * indicates \( p \leq 0.05 \) and ** indicates \( p \leq 0.01 \).

shows a slight decrease in performance for higher difficulty levels. Level 2 and 7 stand out with a relatively low performance.

Table 2 shows the results for each grid size when averaged over the difficulty levels. Here, a strong variation in performance across different grid sizes is visible. All models perform best on 1 × 2-grids. For the 7B models, as d increases, the performance drops to values around the random baseline, while the larger equivalents of Llama present a more stable performance for \( n = 2 \). Qwen 2.5 72B is an outlier, under-performing its smaller equivalent. Nevertheless, even the performance of best model in this experiment (Llama 3.3-70B) decreases as the grid size increases and reaches values close to the random baseline for larger puzzles.

To reveal underlying effects between the performance and model size, grid size, and level, a correlation analysis is conducted. The results in Table 3 suggest a weak positive effect between model size. In general, the number of items per domain (\( n \)) has the strongest negative effect on the performance, while all other factors have medium to low negative effects on the performance.

<table>
  <tr>
    <th rowspan="2">model</th>
    <th rowspan="2">d</th>
    <th colspan="6">n</th>
    <th>baseline</th>
  </tr>
  <tr>
    <th>1</th>
    <th>2</th>
    <th>3</th>
    <th>4</th>
    <th>5</th>
    <th>6</th>
    <th>7</th>
  </tr>
  <tr>
    <td rowspan="7">Qwen2.7B</td>
    <td>2</td>
    <td>0.95**</td>
    <td>0.59*</td>
    <td>0.66**</td>
    <td>0.51</td>
    <td>0.54</td>
    <td>0.54</td>
    <td>0.50</td>
    <td></td>
  </tr>
  <tr>
    <td>3</td>
    <td>0.41</td>
    <td>0.43**</td>
    <td>0.41*</td>
    <td>0.41*</td>
    <td>0.39*</td>
    <td>0.37</td>
    <td>0.33</td>
    <td></td>
  </tr>
  <tr>
    <td>4</td>
    <td>0.35*</td>
    <td>0.30</td>
    <td>0.29</td>
    <td>0.30*</td>
    <td>0.26</td>
    <td>0.27</td>
    <td>0.27</td>
    <td>0.25</td>
  </tr>
  <tr>
    <td>5</td>
    <td>0.28*</td>
    <td>0.23</td>
    <td>0.24</td>
    <td>0.23</td>
    <td>0.20</td>
    <td>0.21</td>
    <td>0.20</td>
    <td>0.20</td>
  </tr>
  <tr>
    <td>6</td>
    <td>0.22</td>
    <td>0.19</td>
    <td>0.20</td>
    <td>0.18</td>
    <td>0.17</td>
    <td>0.16</td>
    <td>0.16</td>
    <td>0.17</td>
  </tr>
  <tr>
    <td>7</td>
    <td>0.20</td>
    <td>0.16</td>
    <td>0.16</td>
    <td>0.13</td>
    <td>0.13</td>
    <td>0.13</td>
    <td>0.14</td>
    <td></td>
  </tr>
  <tr>
    <td></td>
    <td></td>
    <td></td>
    <td></td>
    <td></td>
    <td></td>
    <td></td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td rowspan="7">Qwen2.72B</td>
    <td>2</td>
    <td>0.95**</td>
    <td>0.55*</td>
    <td>0.55*</td>
    <td>0.43*</td>
    <td>0.51*</td>
    <td>0.51*</td>
    <td>0.50</td>
    <td></td>
  </tr>
  <tr>
    <td>3</td>
    <td>0.41</td>
    <td>0.34</td>
    <td>0.35</td>
    <td>0.34</td>
    <td>0.32</td>
    <td>0.34</td>
    <td>0.37</td>
    <td>0.33</td>
  </tr>
  <tr>
    <td>4</td>
    <td>0.27</td>
    <td>0.30</td>
    <td>0.24</td>
    <td>0.23</td>
    <td>0.26</td>
    <td>0.22</td>
    <td>0.24</td>
    <td>0.25</td>
  </tr>
  <tr>
    <td>5</td>
    <td>0.25</td>
    <td>0.21</td>
    <td>0.18</td>
    <td>0.20</td>
    <td>0.18</td>
    <td>0.18</td>
    <td>0.18</td>
    <td>0.25</td>
  </tr>
  <tr>
    <td>6</td>
    <td>0.20</td>
    <td>0.17</td>
    <td>0.17</td>
    <td>0.16</td>
    <td>0.16</td>
    <td>0.16</td>
    <td>0.17</td>
    <td>0.17</td>
  </tr>
  <tr>
    <td>7</td>
    <td>0.15</td>
    <td>0.13</td>
    <td>0.12</td>
    <td>0.14</td>
    <td>0.14</td>
    <td>0.13</td>
    <td>0.13</td>
    <td>0.14</td>
  </tr>
  <tr>
    <td></td>
    <td></td>
    <td></td>
    <td></td>
    <td></td>
    <td></td>
    <td></td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td rowspan="7">Llama3.1-8B</td>
    <td>2</td>
    <td>0.68**</td>
    <td>0.64**</td>
    <td>0.60**</td>
    <td>0.55*</td>
    <td>0.56*</td>
    <td>0.58*</td>
    <td>0.55</td>
    <td>0.50</td>
  </tr>
  <tr>
    <td>3</td>
    <td>0.44</td>
    <td>0.36</td>
    <td>0.40</td>
    <td>0.39</td>
    <td>0.38</td>
    <td>0.40*</td>
    <td>0.39*</td>
    <td>0.33</td>
  </tr>
  <tr>
    <td>4</td>
    <td>0.31</td>
    <td>0.26</td>
    <td>0.26</td>
    <td>0.27</td>
    <td>0.22</td>
    <td>0.28</td>
    <td>0.26</td>
    <td>0.25</td>
  </tr>
  <tr>
    <td>5</td>
    <td>0.28</td>
    <td>0.23</td>
    <td>0.23</td>
    <td>0.23</td>
    <td>0.19</td>
    <td>0.23</td>
    <td>0.23</td>
    <td>0.23</td>
  </tr>
  <tr>
    <td>6</td>
    <td>0.14</td>
    <td>0.08</td>
    <td>0.19</td>
    <td>0.19</td>
    <td>0.19</td>
    <td>0.19</td>
    <td>0.19</td>
    <td>0.17</td>
  </tr>
  <tr>
    <td>7</td>
    <td>0.18</td>
    <td>0.06</td>
    <td>0.15</td>
    <td>0.16</td>
    <td>0.15</td>
    <td>0.15</td>
    <td>0.14</td>
    <td>0.14</td>
  </tr>
  <tr>
    <td></td>
    <td></td>
    <td></td>
    <td></td>
    <td></td>
    <td></td>
    <td></td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td rowspan="7">Llama3.1-70B</td>
    <td>2</td>
    <td>0.73**</td>
    <td>0.90**</td>
    <td>0.82**</td>
    <td>0.76**</td>
    <td>0.77**</td>
    <td>0.74**</td>
    <td>0.50</td>
    <td></td>
  </tr>
  <tr>
    <td>3</td>
    <td>0.54**</td>
    <td>0.52**</td>
    <td>0.49**</td>
    <td>0.55**</td>
    <td>0.50**</td>
    <td>0.48**</td>
    <td>0.43**</td>
    <td>0.35</td>
  </tr>
  <tr>
    <td>4</td>
    <td>0.47**</td>
    <td>0.41**</td>
    <td>0.31**</td>
    <td>0.31**</td>
    <td>0.30**</td>
    <td>0.27**</td>
    <td>0.20</td>
    <td>0.20</td>
  </tr>
  <tr>
    <td>5</td>
    <td>0.39**</td>
    <td>0.31**</td>
    <td>0.25**</td>
    <td>0.25**</td>
    <td>0.23**</td>
    <td>0.23**</td>
    <td>0.20</td>
    <td>0.17</td>
  </tr>
  <tr>
    <td>6</td>
    <td>0.35**</td>
    <td>0.25**</td>
    <td>0.25**</td>
    <td>0.22**</td>
    <td>0.21**</td>
    <td>0.18**</td>
    <td>0.14</td>
    <td>0.14</td>
  </tr>
  <tr>
    <td>7</td>
    <td>0.30**</td>
    <td>0.27**</td>
    <td>0.25**</td>
    <td>0.22**</td>
    <td>0.21**</td>
    <td>0.19**</td>
    <td>0.14</td>
    <td>0.14</td>
  </tr>
  <tr>
    <td></td>
    <td></td>
    <td></td>
    <td></td>
    <td></td>
    <td></td>
    <td></td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td rowspan="7">Mistral-7B</td>
    <td>2</td>
    <td>0.66**</td>
    <td>0.62**</td>
    <td>0.59**</td>
    <td>0.54**</td>
    <td>0.54**</td>
    <td>0.57**</td>
    <td>0.53</td>
    <td>0.50</td>
  </tr>
  <tr>
    <td>3</td>
    <td>0.40*</td>
    <td>0.70**</td>
    <td>0.65**</td>
    <td>0.58*</td>
    <td>0.54*</td>
    <td>0.54*</td>
    <td>0.57*</td>
    <td>0.33</td>
  </tr>
  <tr>
    <td>4</td>
    <td>0.68**</td>
    <td>0.58**</td>
    <td>0.47**</td>
    <td>0.42**</td>
    <td>0.41*</td>
    <td>0.39*</td>
    <td>0.37*</td>
    <td>0.25</td>
  </tr>
  <tr>
    <td>5</td>
    <td>0.71**</td>
    <td>0.47**</td>
    <td>0.38*</td>
    <td>0.36*</td>
    <td>0.34*</td>
    <td>0.29*</td>
    <td>0.29*</td>
    <td>0.20</td>
  </tr>
  <tr>
    <td>6</td>
    <td>0.56**</td>
    <td>0.33**</td>
    <td>0.27**</td>
    <td>0.25**</td>
    <td>0.22**</td>
    <td>0.21**</td>
    <td>0.19*</td>
    <td>0.14</td>
  </tr>
  <tr>
    <td>7</td>
    <td>0.43</td>
    <td>0.28</td>
    <td>0.29</td>
    <td>0.32</td>
    <td>0.28</td>
    <td>0.25</td>
    <td>0.23*</td>
    <td>0.33</td>
  </tr>
  <tr>
    <td>8</td>
    <td>0.29</td>
    <td>0.15</td>
    <td>0.20</td>
    <td>0.20</td>
    <td>0.18*</td>
    <td>0.18*</td>
    <td>0.18*</td>
    <td>0.25</td>
  </tr>
  <tr>
    <td>9</td>
    <td>0.27</td>
    <td>0.14</td>
    <td>0.14</td>
    <td>0.14</td>
    <td>0.14</td>
    <td>0.16</td>
    <td>0.16</td>
    <td>0.25</td>
  </tr>
  <tr>
    <td>10</td>
    <td>0.22</td>
    <td>0.14</td>
    <td>0.13</td>
    <td>0.15</td>
    <td>0.12</td>
    <td>0.12</td>
    <td>0.12</td>
    <td>0.17</td>
  </tr>
  <tr>
    <td>11</td>
    <td>0.17</td>
    <td>0.11</td>
    <td>0.09</td>
    <td>0.12</td>
    <td>0.13</td>
    <td>0.12</td>
    <td>0.12</td>
    <td>0.14</td>
  </tr>
</table>

Table 3: Correlation analysis of grid accuracy by n (items per domain), d (number of domains), model size (only across all models), difficulty level. Red reflects negative effects, green indicates positive effects. Saturation indicates effect strength.

<table>
  <tr>
    <th>n</th>
    <th>d</th>
    <th>model size</th>
    <th>level</th>
  </tr>
  <tr>
    <td>all models</td>
    <td>-0.70</td>
    <td>-0.22</td>
    <td>0.25</td>
    <td>-0.17</td>
  </tr>
  <tr>
    <td>Llama 3.1-8B</td>
    <td>-0.83</td>
    <td>-0.03</td>
    <td>–</td>
    <td>-0.10</td>
  </tr>
  <tr>
    <td>Llama 3.1-70B</td>
    <td>-0.82</td>
    <td>-0.29</td>
    <td>–</td>
    <td>-0.26</td>
  </tr>
  <tr>
    <td>Llama 3.3-70B</td>
    <td>-0.82</td>
    <td>-0.29</td>
    <td>–</td>
    <td>-0.26</td>
  </tr>
  <tr>
    <td>Qwen 2.5-7B</td>
    <td>-0.75</td>
    <td>-0.38</td>
    <td>–</td>
    <td>-0.21</td>
  </tr>
  <tr>
    <td>Qwen 2.5-72B</td>
    <td>-0.82</td>
    <td>-0.11</td>
    <td>–</td>
    <td>-0.15</td>
  </tr>
  <tr>
    <td>Mistral-7B</td>
    <td>-0.71</td>
    <td>-0.19</td>
    <td>–</td>
    <td>-0.10</td>
  </tr>
</table>

7.1 Error analysis

To gain a deeper insight into the reasoning chains of the models, a manual qualitative error analysis is conducted. For open weight models, 50 randomly selected, incorrectly solved puzzles up to a size of 5 × 5 are assessed. For the GPT models and R1, 20 incorrectly solved answers are considered. This analysis also includes outputs that did not adhere to the target format. A list of the error sources and their distribution can be found in App. H.

All models frequently produce pseudo-reasoning (41%); i.e. instances where linguistic markers of reasoning are used to combine premises and conclusions that make no sense. Further, the free-text often contradicts the produced solution grid (37%), while no model is able to detect and correct errors in the predicted solution grid. The only exceptions are Qwen 2.5-7B, Mistral-7B and R1, the former two detect errors in 2% and 4% of the analysed outputs, incorrectly concluding that the puzzle was unsolvable. R1 however, claims to fix detected errors, but fails to accomplish this.

Further, smaller 7-8B parameter models produce the highest amount of (partially) empty or malformed solution grids (44%). This effect is strongest in Llama 3.1-8B (52%) and Mistral-7B (58%). Another prevalent source of errors in these models are contradictions between the free-text reasoning and the solution grid (43%).

70-72B parameter models show a positioning bias that prefers to place items in the leftmost free position (35%) and place items twice in the solution grid (31%). Further, these models ignore clues in the produced free-text reasoning more often than others (17% vs. 0.29%). These models also struggle with clues that contain a negation (13%). Qwen 2.5-72B is the only model that produces grids without free-text reasoning (82%).
The GPT models show the highest rate of pseudo-reasoning (78%) and tend to misinterpret placement clues (28%). This especially includes cases where clues are interpreted more loosely than intended (e.g. “to left” or “to the right” is interpreted as “somewhere to the left/right”). Further, these models are more prone to placing before the drawn conclusions are sufficient for a definitive positioning (23%) and also tend to place items in the leftmost available position (30%).

In \( n \)-domain obfuscated puzzles, R1 solves not obfuscated rows perfectly, while making mistakes in obfuscated rows. This suggests a strong influence from training material. Further, this model produces the longest free-text reasoning among all tested models, resulting in self-repetitions (75%) and as a result overwrites formerly correct predictions (35%). This model has as well a bias for positioning items in the leftmost position (35%).

7.2 Case study

In addition, we conduct a case study on the canonical puzzles to analyse model-specific behaviours in different reasoning steps of the solution. This is done in a Q&A setup, which asks for the position \( P \) of a specific item \( I \) in a row \( R \) of the solution grid \( G \) given a set of clues \( C = \{c_1 \ldots c_n\} \). The corresponding prompt is shown in App. D. This format is evaluated based on the accuracy of correct answers. Since this setup focuses on the position within a single domain, the random baseline for a puzzle with \( n \) items per domain is: \( E = \frac{1}{n} \).

We test the GPT and open-weight models from Experiment 1 on the original, rephrased, *in domain*, and 5-*domain* obfuscations of the canonical puzzles to avoid cases where only a part of the puzzle is manipulated. We run 10 predictions for each item with open-weight models and 3 predictions per item with GPT models. The items are ordered by the reasoning depth in the canonical puzzles in accordance with solution guides available on the Internet. App. I.1 details the step-by-step solutions.

A per-item analysis shows that 4o, 4o-mini and Llama 3.3 are mostly consistent in their predictions for the same item (see Tables 16-20 in App. I). Since we test only one *in domain* and one 5-*domain* obfuscation, the resulting accuracies are close to 0 or 1. Hence, the random baseline is unsuitable and no significance can be reported.

Figures 5 and 6 in App. I report the accuracy at each reasoning step. In all assessed puzzle variations and models, the first two reasoning steps are performed successfully. For smaller models of 7-8B parameters, the performance degrades afterwards regardless of the obfuscation of the puzzle. On the original an rephrased puzzles, Llama 3.3-70B and Qwen 2.5-72B recover from this drop in performance and reach relatively high performance in the last step. The GPT models keep a relatively high performance across the original grid versions with medium drops in performance from the 3rd reasoning step. On the obfuscated versions, the performance drops also for bigger models from the 3rd step onward. Nevertheless, peaks of medium to high accuracy can be found at specific steps in the reasoning process (4th, 7th, and 8th for “Einstein”; 5th and 7th for “Zebra”).

8 Discussion

In both experiments, the reasoning abilities of smaller models (7-8B) are limited. They are successful in one-step reasoning in \( d \times 2 \) puzzles. For bigger puzzles, their performance approaches the random baseline. The case study in 7.2 and Lin et al. (2024) support this.

In general, models around 70B parameters outperform the random baseline by a statistically significant but very small margin. This indicates that easier clues are solved but not the harder ones.

The correlation analysis reveals only a modest performance gain when scaling models up to 70B parameters, suggesting diminishing returns at even larger scales (e.g. 400B), especially given the substantial computational and environmental costs. The formal difficulty of clues is also a limited predictor. The observed irregularities in Levels 2 and 7 might be related to clues containing the relation *between* introduced in these levels. Further effects may be invisible due to the correlation of difficult clues with more items per domain (see Table 5, App. A), which is the strongest predictor in this analysis.

The correlation between grid size and performance is supported by the case study, which reveals a decrease in performance for longer reasoning chains in fully obfuscated canonical puzzles. Irregular performance peaks in the case study occur in specific later reasoning steps, where the last 1-2 positions in a row are filled (see App. I.1, which increases the likelihood of successful guessing. This effect is further reinforced as the models with these peaks are more deterministic in their predictions.

The error analysis suggests severe difficulties in detecting contradictions in the produced solution
grids for all models, while they produce a lot of pseudo-reasoning and are unable to align free-text reasoning and solution grids.

For the original “Einstein” puzzle, the open-weight models show performances significantly above the random baseline, suggesting that the existence of this puzzle in the training data is beneficial for the performance on this puzzle. For the original “Zebra” puzzle, only Llama and GPT models show this effect. For any obfuscated version, the open-weight models show similar performances to newly generated puzzles of the same size, indicating that the obfuscation has the intended effect.

The closed weight models show nearly perfect performance on the original version of the “Einstein” puzzle. But their performance degrades drastically with the gradual manipulation of the lexical content of the solution grid, finally not outperforming the random baseline. The error analysis suggests a high degree of pseudo-reasoning for these models and a tendency to place items prematurely in the leftmost available position.

The observed decrease in performance of larger models with the increase of obfuscation suggests that they are able to cope with slightly larger variations than smaller models. Nevertheless, this trend suggests that the dependence on lexical items in the input is not overcome in models with more parameters. R1’s relatively high performance in the Einstein-in domain is composed of a singular trial with high performance and two low performances, signalling high instability.

On the “Zebra” puzzle, some models improve performance on the rephrased compared to the original puzzle. This could be explained by a dominance of passive voice in the original “Zebra” puzzle (see App. B), which the rephrased version turns into active. This is supported by work reporting that LLMs use passive voice in written text less frequently than humans (Reinhart et al., 2024).

9 Conclusion

This paper develops Mystery-Zebra, the first adaptable systematic benchmark to evaluate on multiple levels. By evaluating a range of open and closed weight LLMs, we demonstrate their dependence on the lexical content from training material for the successful solution of complex grid puzzles.

In comparison to the original puzzles, the performance drops significantly for lexically manipulated puzzles and even reaches the random baseline when all noun phrases in the puzzle are altered. The reasoning abilities of smaller LLMs break down with the slightest variation of the lexical context of a puzzle that is part of their training data.

Our obfuscation strategies can easily be transferred to other tasks to mitigate the effect of logic puzzle benchmarks being part of the training data of the LLMs. This will help to preserve the significance of benchmark results in the future.

An extensive analysis of open weight models on the newly generated puzzles in the second part of the benchmark demonstrated that the grid size has a stronger influence on the performance than the clue difficulty or model size.

In the error analysis, models proved to be incapable to detect and correct contradictions in their output. In addition, our detailed case study of reasoning steps in the solution process demonstrated that the assessed models were only able to perform a very limited number of consecutive reasoning steps correctly.

The results from this paper can direct future research in LLM reasoning by pointing out that improvements through traditional fine-tuning are no clear indicator of improvements in the task itself (see Experiment 1). Hybrid solutions, however, might provide a more sustainable direction to successful LLM reasoning.

Limitations

Due to limits in the compute budget, this work only assesses open-weight models up to a size of 72B parameters on the second part of the benchmark. For this reason, the paper can make only limited claims with regard to the influence of very large parameter sizes on the performance on unknown puzzles of various sizes and difficulty levels.

Due to the substantial compute time of Deepseek R1 (404 seconds on average), this model is only evaluated in the first experiment and the error analysis, but not considered for the case-study. This leads to limited insights into the influence of reasoning depth on the performance of this model. Nevertheless, the error analysis suggests a strong dependence of this model on training material. Future work should consider analysing the reasoning abilities of this model in depth.

Further, the benchmark obfuscates only the nouns and related verb-phrases of the clues and leaves the structures referring to the positions in the solution grid untouched. The influence of such manipulations should be investigated in future re-
search.

Finally, the provided explanation for the improvement in performance observed on the rephrased “Zebra” puzzle relies only on a possible linguistic explanation and evidence from free-text writing studies. Future work should consider the influence of passive voice on performance in more detail.

Ethics statement

The authors didn’t identify immediate ethical concerns around the work presented in this paper. Nevertheless, any work around the reasoning abilities of Language Models should be considered with care due to the societal impact of inaccurate assumptions on these abilities.

Acknowledgements

The authors would like to thank Jonathan Pampel for the very helpful discussions on the evaluation and significance testing done in this paper. This work was funded in part by Volkswagen Stiftung Foundation under grant 98 543, “Deliberation Laboratory”.

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A  Puzzle Levels

<table>
  <tr>
    <th>Level</th>
    <th>Semi-Formal expression</th>
    <th>Description</th>
  </tr>
  <tr>
    <td rowspan="4">Level 1</td>
    <td>A == B</td>
    <td>An object that has attribute A has attribute B.</td>
  </tr>
  <tr>
    <td>A is on the left of B</td>
    <td>An object with attribute A is next to the left of an object with attribute B (A-B).</td>
  </tr>
  <tr>
    <td>A is on the right of B</td>
    <td>An object with attribute A is next to the right of an object with attribute B (B-A).</td>
  </tr>
  <tr>
    <td>A is on the far left</td>
    <td>An object with attribute A is on the far left (A...).</td>
  </tr>
  <tr>
    <td></td>
    <td>A is on the far right</td>
    <td>An object with attribute A is on the far right (...A).</td>
  </tr>
  <tr>
    <td></td>
    <td>A is in the middle</td>
    <td>An object with attribute A is in the middle.</td>
  </tr>
  <tr>
    <td>Level 2</td>
    <td>A is between B and C</td>
    <td>An object with attribute A is between an object with attribute B, and an object with attribute C (any order: B-A-C, C-A-B).</td>
  </tr>
  <tr>
    <td>Level 3</td>
    <td>A is on the left or right of B</td>
    <td>An object with attribute A is next to the left or right of an object with attribute B (A-B or B-A).</td>
  </tr>
  <tr>
    <td></td>
    <td>A is on the far left or far right</td>
    <td>An object with attribute A is on the far left or far right. (A... or ...A)</td>
  </tr>
  <tr>
    <td>Level 4</td>
    <td>A is in an odd position</td>
    <td>An object with attribute A is in an odd position (odd positions : 1, 3, 5, ...).</td>
  </tr>
  <tr>
    <td></td>
    <td>A is in an even position</td>
    <td>An object with attribute A is in an even position (even positions : 2, 4, 6, ...).</td>
  </tr>
  <tr>
    <td>Level 5</td>
    <td>A is somewhere to the left of B</td>
    <td>An object with attribute A is somewhere to the left of an object with attribute B (any number of intermediates, including 0 : A...--B).</td>
  </tr>
  <tr>
    <td></td>
    <td>A is somewhere to the right of B</td>
    <td>An object with attribute A is somewhere to the right of an object with attribute B (any number of intermediates, including 0 : B...--A).</td>
  </tr>
  <tr>
    <td>Level 6</td>
    <td>A != B</td>
    <td>An object that has attribute A does not have attribute B.</td>
  </tr>
  <tr>
    <td>Level 7</td>
    <td>A is somewhere between B and C</td>
    <td>An object with attribute A is somewhere between an object with attribute B, and an object with attribute C (any number of intermediates, including 0 : B...--C, C...--A, ...B).</td>
  </tr>
  <tr>
    <td>Level 8</td>
    <td>A is not to the left of B</td>
    <td>An object with attribute A is not to the left of an object with attribute B.</td>
  </tr>
  <tr>
    <td></td>
    <td>A is not to the right of B</td>
    <td>An object with attribute A is not to the right of an object with attribute B.</td>
  </tr>
  <tr>
    <td>Level 9</td>
    <td>A and B have different parity positions</td>
    <td>An object with attribute A and an object with attribute B have different parity positions.</td>
  </tr>
  <tr>
    <td></td>
    <td>A and B have the same parity positions</td>
    <td>An object with attribute A and an object with attribute B have the same parity positions (positions may be the same or different, but the parity is always the same).</td>
  </tr>
  <tr>
    <td>Level 10</td>
    <td>A == B or A == C, but not both</td>
    <td>An object that has attribute A has attribute B, or an object that has attribute A has attribute C, but not both.</td>
  </tr>
  <tr>
    <td></td>
    <td>A == B or B == C, but not both</td>
    <td>An object that has attribute A has attribute B, or an object that has attribute B has attribute C, but not both.</td>
  </tr>
  <tr>
    <td>Level 11</td>
    <td>A == B or A == C or both</td>
    <td>An object that has attribute A has attribute B, or an object that has attribute A has attribute C, or both.</td>
  </tr>
  <tr>
    <td></td>
    <td>A == B or B == C or both</td>
    <td>An object that has attribute A has attribute B, or an object that has attribute B has attribute C, or both.</td>
  </tr>
  <tr>
    <td>Level 12</td>
    <td>A != B or A != C or both</td>
    <td>An object that has attribute A has not attribute B, or an object that has attribute A has not attribute C, or both.</td>
  </tr>
  <tr>
    <td></td>
    <td>A != B or B != C or both</td>
    <td>An object that has attribute A has not attribute B, or an object that has attribute B has not attribute C, or both.</td>
  </tr>
</table>

Table 4: Puzzle clue types per level taken from https://github.com/quint-t/Puzzle-Generator-and-Solver/tree/master

<table>
  <tr>
    <th>Level</th>
    <th>Puzzle sizes (domains × items per domain)</th>
    <th>number</th>
  </tr>
  <tr>
    <td>Level 1</td>
    <td>all sizes from 1 × 2 to 7 × 7</td>
    <td>420</td>
  </tr>
  <tr>
    <td>Level 2</td>
    <td>sizes with at least 3 items per domain</td>
    <td>350</td>
  </tr>
  <tr>
    <td>Level 3</td>
    <td>all sizes from 1 × 2 to 7 × 7</td>
    <td>420</td>
  </tr>
  <tr>
    <td>Level 4</td>
    <td>all sizes from 1 × 2 to 7 × 7</td>
    <td>420</td>
  </tr>
  <tr>
    <td>Level 5</td>
    <td>all sizes from 1 × 2 to 7 × 7</td>
    <td>420</td>
  </tr>
  <tr>
    <td>Level 6</td>
    <td>sizes with at least 2 domains</td>
    <td>360</td>
  </tr>
  <tr>
    <td>Level 7</td>
    <td>sizes with at least 3 items per domain</td>
    <td>350</td>
  </tr>
  <tr>
    <td>Level 8</td>
    <td>all sizes from 1 × 2 to 7 × 7</td>
    <td>420</td>
  </tr>
  <tr>
    <td>Level 9</td>
    <td>sizes with at least 2 domains</td>
    <td>360</td>
  </tr>
  <tr>
    <td>Level 10</td>
    <td>sizes with at least 2 domains and 3 items per domain</td>
    <td>250</td>
  </tr>
  <tr>
    <td>Level 11</td>
    <td>sizes with at least 2 domains and 3 items per domain</td>
    <td>250</td>
  </tr>
  <tr>
    <td>Level 12</td>
    <td>sizes with at least 2 domains and 3 items per domain</td>
    <td>250</td>
  </tr>
</table>

Table 5: Combinations of difficulty levels and sizes available in the benchmark of newly generated puzzles, since some puzzle sizes are not compatible with some types of clues (e.g. Level 2: A is between B and C.)
B Original Puzzle

Einstein Puzzle
There are 5 houses (in a row) painted 5 different colors: Blue, Green, Red, White and Yellow. In each house there lives a person of a different nationality: Brit, Dane, German, Norwegian or Swede. These 5 owners each drink a certain beverage: Beer, Coffee, Milk, Tea or Water. They also smoke a certain brand of cigar: Bluemaster, Dunhill, Pall Mall, Prince or Blend. Additionally, they keep a certain type of pet: Cats, Birds, Dogs, Fish or Horses. The owners DO NOT have the same pet, smoke the same brand of cigar or drink the same beverage.

1. The Brit lives in a Red house.
2. The Swede keeps Dogs as pets.
3. The Dane drinks Tea.
4. The Green house is on the left of the White house.
5. The Green house owner drinks Coffee.
6. The owner who smokes Pall Mall rears Birds.
7. The owner of the Yellow house smokes Dunhill.
8. The owner living in the center house drinks Milk.
9. The Norwegian lives in the first house.
10. The owner who smokes Blend lives next to the one who keeps Cats.
11. The owner who keeps horses lives next to the man who smokes Dunhill.
12. The owner who smokes Bluemaster drinks Beer.
13. The German smokes Prince.
14. The Norwegian lives next to the Blue house.
15. The owner who smokes Blend has a neighbor who drinks Water.

Zebra Puzzle
There are five different-colored houses: red, green, ivory, yellow, blue There live five resident of a different nationality: english, spanish, ukrainian, norwegian, japanese Each resident owns a different pet: dog, fox, zebra, horse, snails Each one prefers a different drink: coffee, tea, milk, orange-juice, water Each one smokes a different brand of cigarettes: old-gold, kools, chesterfields, lucky-strike, parlaments

<table>
  <tr>
    <th>Domain</th>
    <th>Phrasing in obfuscated / generator puzzles</th>
    <th>Phrasing in canonical Einstein</th>
    <th>Phrasing in canonical Zebra</th>
  </tr>
  <tr>
    <td>Color</td>
    <td>the person who likes [color]</td>
    <td>[color] house</td>
    <td>[color] house</td>
  </tr>
  <tr>
    <td>Beverage</td>
    <td>the person drinking [beverage]</td>
    <td>drinks [beverage]</td>
    <td>[beverage] is drunk / drinks [beverage]</td>
  </tr>
  <tr>
    <td>Pet</td>
    <td>the owner of the [pet]</td>
    <td>keeps [pet] as pets / rears [pet] / keeps [pet]</td>
    <td>owns the [pet] / owns [pet] / the [pet] is kept / the man with the [pet]</td>
  </tr>
  <tr>
    <td>Nationality</td>
    <td>the [nationality]</td>
    <td>the [nationality]</td>
    <td>the [nationality]</td>
  </tr>
  <tr>
    <td>Cigar</td>
    <td>the [cigar]-smoker</td>
    <td>the owner/man who smokes [cigar] / smokes [cigar]</td>
    <td>[cigar] are smoked / the [cigar] smoker / smokes [cigar]</td>
  </tr>
  <tr>
    <td>Food</td>
    <td>the person eating [food]</td>
    <td>–</td>
    <td>–</td>
  </tr>
  <tr>
    <td>House</td>
    <td>the [house] house</td>
    <td>–</td>
    <td>–</td>
  </tr>
  <tr>
    <td>Job</td>
    <td>the [job]</td>
    <td>–</td>
    <td>–</td>
  </tr>
  <tr>
    <td>Transport</td>
    <td>the person driving the [transport name]</td>
    <td>–</td>
    <td>–</td>
  </tr>
  <tr>
    <td>Music-Genre</td>
    <td>the fan of [music-genre]</td>
    <td>–</td>
    <td>–</td>
  </tr>
  <tr>
    <td>Movie-Genre</td>
    <td>the person watching [movie-genre]</td>
    <td>–</td>
    <td>–</td>
  </tr>
  <tr>
    <td>Sport</td>
    <td>the person who’s sport is [sport]</td>
    <td>–</td>
    <td>–</td>
  </tr>
  <tr>
    <td>Hobby</td>
    <td>the person who’s hobby is [hobby]</td>
    <td>–</td>
    <td>–</td>
  </tr>
  <tr>
    <td>Flower</td>
    <td>the person who grows [flower]</td>
    <td>–</td>
    <td>–</td>
  </tr>
  <tr>
    <td>Birthday</td>
    <td>the person who’s birthday is in [birthday month]</td>
    <td>–</td>
    <td>–</td>
  </tr>
  <tr>
    <td>Game</td>
    <td>the person playing [game]</td>
    <td>–</td>
    <td>–</td>
  </tr>
</table>

Table 6: Original “Einstein” and “Zebra” puzzle

Table 7: Phrasing patterns used in the obfuscated versions of the canonical puzzles and the generator-puzzles in the second part of Mystery-Zebra. For reproducibility of the rephrasing obfuscation, the equivalents in the canonical puzzles are detailed in the two rightmost columns.
C Obfuscation Details

To facilitate the obfuscation process, the canonical puzzles are first translated to the same pseudo-logical language as used by the puzzle-generator, where items are stated as domain:item, equivalence is expressed as “==” and negation is expressed as “!=”. Any plural forms are replaced with their singular equivalents for this step.

Once the obfuscation is complete, the items are translated back to natural language using the patterns in Table 7. Tables 9 and 10 show the full rephrased versions in comparison to the original phrasing. The same phrasing patterns are applied as well to translate the puzzles from the generator to natural language input. For simplicity, only one phrasing pattern is used for each domain. The pseudo-logical formulas from the generator are translated to natural language according to Table 8.

<table>
  <tr>
    <th>pseudo-logical sign</th>
    <th>phrasing in obfuscated / generator-puzzles</th>
  </tr>
  <tr>
    <td>!=</td>
    <td>is not</td>
  </tr>
  <tr>
    <td>=</td>
    <td>is</td>
  </tr>
</table>

Table 8: Translation patterns for the pseudo-logical structures created by the puzzle generator and used to obfuscate canonical puzzles.

<table>
  <tr>
    <th>original phrasing</th>
    <th>rephrased</th>
  </tr>
  <tr>
    <td>1. The Brit lives in a Red house.</td>
    <td>1. The british is the person who likes red</td>
  </tr>
  <tr>
    <td>2. The Swede keeps Dogs as pets.</td>
    <td>2. the swedish is the owner of the dog</td>
  </tr>
  <tr>
    <td>3. The Dane drinks Tea.</td>
    <td>3. the danish is the person drinking tea</td>
  </tr>
  <tr>
    <td>4. The Green house is on the left of the White house.</td>
    <td>4. The person who likes green is on the left of the person who likes white</td>
  </tr>
  <tr>
    <td>5. The Green house owner drinks Coffee.</td>
    <td>5. The person who likes green is the person drinking coffee</td>
  </tr>
  <tr>
    <td>6. The owner who smokes Pall Mall rears Birds.</td>
    <td>6. The pall-mall-smoker is the owner of the bird</td>
  </tr>
  <tr>
    <td>7. The owner of the Yellow house smokes Dunhill.</td>
    <td>7. The person who likes yellow is the dunhill-smoker</td>
  </tr>
  <tr>
    <td>8. The owner living in the center house drinks Milk.</td>
    <td>8. The person drinking milk is in the middle</td>
  </tr>
  <tr>
    <td>9. The Norwegian lives in the first house.</td>
    <td>9. The norwegian is on the far left</td>
  </tr>
  <tr>
    <td>10. The owner who smokes Blend lives next to the one who keeps Cats.</td>
    <td>10. The blend-smoker is on the left or right of the owner of the cat</td>
  </tr>
  <tr>
    <td>11. The owner who keeps horses lives next to the man who smokes Dunhill.</td>
    <td>11. The owner of the horses is on the left or right of the dunhill-smoker</td>
  </tr>
  <tr>
    <td>12. The owner who smokes Bluemaster drinks Beer.</td>
    <td>12. The bluemaster-smoker is the person drinking beer</td>
  </tr>
  <tr>
    <td>13. The German smokes Prince.</td>
    <td>13. The german is the prince-smoker</td>
  </tr>
  <tr>
    <td>14. The Norwegian lives next to the Blue house.</td>
    <td>14. The norwegian is on the left or right of the person who likes blue</td>
  </tr>
  <tr>
    <td>15. The owner who smokes Blend has a neighbor who drinks Water.</td>
    <td>15. The blend-smoker is on the left or right of the person drinking water</td>
  </tr>
</table>

Table 9: Original “Einstein” puzzle in comparison to the rephrased obfuscation.

<table>
  <tr>
    <th>original phrasing</th>
    <th>rephrased</th>
  </tr>
  <tr>
    <td>1. There are five houses.</td>
    <td>1. the english is the person who likes red</td>
  </tr>
  <tr>
    <td>2. The Englishman lives in the red house.</td>
    <td>2. the spanish is the owner of the dog</td>
  </tr>
  <tr>
    <td>3. The Spaniard owns the dog.</td>
    <td>3. the person drinking coffee is the person who likes green</td>
  </tr>
  <tr>
    <td>4. Coffee is drunk in the green house.</td>
    <td>4. the ukrainian is the person drinking tea</td>
  </tr>
  <tr>
    <td>5. The Ukrainian drinks tea.</td>
    <td>5. the person who likes green is to the right of the person who likes ivory</td>
  </tr>
  <tr>
    <td>6. The green house is immediately to the right of the ivory house (to your right as you stand facing the row of five houses).</td>
    <td>6. the old-gold-smoker is the owner of the snail</td>
  </tr>
  <tr>
    <td>7. The Old Gold smoker owns snails.</td>
    <td>7. the kools-smoker is the person who likes yellow</td>
  </tr>
  <tr>
    <td>8. Kools are smoked in the yellow house.</td>
    <td>8. the person drinking milk is in the middle</td>
  </tr>
  <tr>
    <td>9. Milk is drunk in the middle house.</td>
    <td>9. the norwegian is on the far left</td>
  </tr>
  <tr>
    <td>10. The Norwegian lives in the first house.</td>
    <td>10. the chesterfield-smoker is on the left or right of the owner of the fox</td>
  </tr>
  <tr>
    <td>11. The man who smokes Chesterfields lives in the house next to the man with the fox.</td>
    <td>11. the kools-smoker is to the left or right of the owner of the horse</td>
  </tr>
  <tr>
    <td>12. Kools are smoked in a house next to the house where the horse is kept.</td>
    <td>12. the lucky-strike-smoker is the person drinking orange-juice</td>
  </tr>
  <tr>
    <td>13. The Lucky Strike smoker drinks orange juice.</td>
    <td>13. the japanese is the parliament-smoker</td>
  </tr>
  <tr>
    <td>14. The Japanese smokes Parliaments.</td>
    <td>14. the norwegian is to the left or right of the person who likes blue</td>
  </tr>
  <tr>
    <td>15. The Norwegian lives next to the blue house.</td>
    <td></td>
  </tr>
</table>

Table 10: Original “Zebra” puzzle in comparison to the rephrased obfuscation.

D Model Prompting

This work uses a basic prompting strategy since the work by Tyagi et al. (2024) suggests that elaborate prompting strategies like self-consistency have no beneficial effects on the grid-puzzle solving abilities of LLMs.

Grid-setup The grid-setup prompt specifies an empty solution grid to improve the consistency in formatting in the outputs.

> Please solve the following logic puzzle in the following table:

[EMPTY SOLUTION GRID]

Puzzle:
[PUZZLE CLUES].

Think step by step when solving the puzzle. Please put ’###########’ around the final solution table.

Q&A-setup The Q&A-setup prompts the model first to solve the puzzle and then to indicate the position of a specific item in the grid in the form of item:Num.

> Solve the following logic puzzle:
[PUZZLE CLUES]

After solving tell me where is **[TARGET ITEM]**. Give the answer in the format **[TARGET ITEM]:Num**.

E Hyper-Parameters

The hyper parameters of the models were left at their default if not mentioned explicitly.

<table>
  <tr>
    <th>model</th>
    <th>hyper-parameter</th>
    <th>setting</th>
  </tr>
  <tr>
    <td rowspan="3">LLama 3.1-8B</td>
    <td>max_new_tokens</td>
    <td>10000</td>
  </tr>
  <tr>
    <td>temperature</td>
    <td>0.1</td>
  </tr>
  <tr>
    <td>top_p</td>
    <td>0.9</td>
  </tr>
  <tr>
    <td rowspan="3">LLama 3.1-70B</td>
    <td>max_new_tokens</td>
    <td>10000</td>
  </tr>
  <tr>
    <td>temperature</td>
    <td>0.1</td>
  </tr>
  <tr>
    <td>top_p</td>
    <td>0.9</td>
  </tr>
  <tr>
    <td rowspan="3">LLama 3.3-70B</td>
    <td>max_new_tokens</td>
    <td>10000</td>
  </tr>
  <tr>
    <td>temperature</td>
    <td>0.1</td>
  </tr>
  <tr>
    <td>top_p</td>
    <td>0.9</td>
  </tr>
  <tr>
    <td>Qwen 2.5-7B</td>
    <td>max_new_tokens</td>
    <td>10000</td>
  </tr>
  <tr>
    <td>Qwen 2.5-72B</td>
    <td>max_new_tokens</td>
    <td>10000</td>
  </tr>
  <tr>
    <td>Mistral-7B</td>
    <td>max_new_tokens</td>
    <td>10000</td>
  </tr>
</table>

Table 11: Hyper-parameters for open weight models

F Pre-processing

Grid-setup The open-ended generations of the models were pre-processed according to this requirement. Since this study focuses on the logic capabilities and not on formatting abilities non-adhering answers were excluded from the evaluation process. This filtering was done checking for two criteria: First, the result needs to be a table with boundaries marked by “|”. Second, the leftmost column needs to contain all expected domain names for the respective puzzle.\footnote{Any row with an invalid domain name (e.g. a header row) is not considered in the evaluation.} This method does not account for all possible formatting issues (see the error analysis in 7.1), but to in order to ensure reproducibility of this pre-processing, no more fine-grained case distinctions were included. Table 12 reports the percentage of wrong output format for the evaluated models, indicating that especially some of the smaller open-weight models struggle significantly with the target output format.

Q&A-setup The prompt for the Q&A specifies the target format used for the pre-processing as item:Num. The pre-processing filters the open-ended generations by the models for this format and slight variations containing additional white-space characters, the word position or house, or one letter in front of the integer indicating the position in the grid. All other versions or formats are excluded from the analysis in Section 7. The percentage of outputs excluded from the analysis is reported in Table 13.

<table>
  <tr>
    <th>Model</th>
    <th>Excluded from evaluation</th>
  </tr>
  <tr>
    <td>RJ</td>
    <td>0.00%</td>
  </tr>
  <tr>
    <td>GPT 4o</td>
    <td>0.00%</td>
  </tr>
  <tr>
    <td>GPT 4o-mini</td>
    <td>0.00%</td>
  </tr>
  <tr>
    <td>Qwen 2.5-7B</td>
    <td>5.72%</td>
  </tr>
  <tr>
    <td>Qwen 2.5-72B</td>
    <td>13.55%</td>
  </tr>
  <tr>
    <td>Llama 3.1-8B</td>
    <td>14.39%</td>
  </tr>
  <tr>
    <td>Llama 3.1-70B</td>
    <td>5.30%</td>
  </tr>
  <tr>
    <td>Llama 3.3-70B</td>
    <td>3.88%</td>
  </tr>
  <tr>
    <td>Mistral</td>
    <td>43.28%</td>
  </tr>
</table>

Table 12: Percentage of outputs per model excluded from the evaluation in the grid-setup. The evaluation requires tables with domains in the rows and items in the columns in the model output.

<table>
  <tr>
    <th>Model</th>
    <th>Excluded from evaluation</th>
  </tr>
  <tr>
    <td>GPT 4o</td>
    <td>0.00%</td>
  </tr>
  <tr>
    <td>GPT 4o-mini</td>
    <td>0.00%</td>
  </tr>
  <tr>
    <td>Llama 3.1 8B</td>
    <td>40.75%</td>
  </tr>
  <tr>
    <td>Llama 3.1-70B</td>
    <td>14.45%</td>
  </tr>
  <tr>
    <td>Llama 3.3-70B</td>
    <td>13.95%</td>
  </tr>
  <tr>
    <td>Mistral 7B</td>
    <td>39.85%</td>
  </tr>
  <tr>
    <td>Qwen 2.5-72B</td>
    <td>13.40%</td>
  </tr>
  <tr>
    <td>Qwen 2.5 7B</td>
    <td>5.55%</td>
  </tr>
</table>

Table 13: Percentage of outputs per model excluded from the evaluation in the Q&A-setup. The evaluation requires the format item:num.

G Significance Testing

Grid-setup The grid-setup in Experiments 1 and 3 does not allow to calculate significance values straightforward since the grid-score is not a binomially distributed variable. For this reason, we use the following approximation to calculate significances. The expected values E and Variances V can be calculated for the case \( d = 1 \) as follows:

\[
E = \frac{1}{n}, \quad V = \frac{2n - 1}{n^2}
\]

By approximating the case for \( d > 1 \) as a
Gauss-Distribution, we get:

\[
E = \frac{1}{n}, \quad V = \frac{2n-1}{dn^2}
\]

As the experiments are repeated \( k \)-times, we get:

\[
E = \frac{1}{n}, \quad V = \frac{2n-1}{dkn^2}
\]

The average Variance over a grid size or level is then:

\[
V \left( \frac{X_1 + X_2 + \ldots + X_j}{j} \right) = \frac{V(X_1) + \ldots + V(X_j)}{j^2}
\]

The significance for the measured average normalized grid-score \( m \) then calculated in a one-sided T-test:

\[
p = 2 \cdot P(X > m) = 2 \cdot P \left( \mathcal{N} > \frac{(m - E)}{\sqrt{V}} \right)
\]
H  Error Analysis

<table>
  <tr>
    <th>model</th>
    <th>R1</th>
    <th>4o</th>
    <th>4o-mini</th>
    <th>Qwen2.5</th>
    <th>Llama3.1</th>
    <th>Llama3.3</th>
    <th>Mistral</th>
    <th>Average</th>
  </tr>
  <tr>
    <td></td>
    <td></td>
    <td></td>
    <td></td>
    <td>7B</td>
    <td>72B</td>
    <td>8B</td>
    <td>70B</td>
    <td>7B</td>
  </tr>
  <tr>
    <td>E1</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>15.00%</td>
    <td>2.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>14.00%</td>
    <td>8.00%</td>
    <td>0.00%</td>
    <td>4.17%</td>
  </tr>
  <tr>
    <td>E2</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>4.00%</td>
    <td>2.00%</td>
    <td>4.00%</td>
    <td>1.39%</td>
  </tr>
  <tr>
    <td>E3</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>6.00%</td>
    <td>0.00%</td>
    <td>0.83%</td>
  </tr>
  <tr>
    <td>E4</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>6.00%</td>
    <td>0.00%</td>
    <td>4.00%</td>
    <td>4.00%</td>
    <td>8.00%</td>
    <td>6.00%</td>
    <td>3.89%</td>
  </tr>
  <tr>
    <td>E5</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>8.00%</td>
    <td>2.00%</td>
    <td>2.00%</td>
    <td>2.00%</td>
    <td>0.00%</td>
    <td>16.00%</td>
    <td>4.17%</td>
  </tr>
  <tr>
    <td>E6</td>
    <td>0.00%</td>
    <td>25.00%</td>
    <td>20.00%</td>
    <td>0.00%</td>
    <td>2.00%</td>
    <td>6.00%</td>
    <td>20.00%</td>
    <td>6.00%</td>
    <td>16.00%</td>
    <td>9.44%</td>
  </tr>
  <tr>
    <td>E7</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>22.00%</td>
    <td>4.00%</td>
    <td>52.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>58.00%</td>
    <td>18.89%</td>
  </tr>
  <tr>
    <td>E8</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>2.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>4.00%</td>
    <td>0.00%</td>
    <td>0.83%</td>
  </tr>
  <tr>
    <td>E9</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>8.00%</td>
    <td>2.00%</td>
    <td>22.00%</td>
    <td>30.00%</td>
    <td>40.00%</td>
    <td>40.00%</td>
    <td>14.17%</td>
  </tr>
  <tr>
    <td>E10</td>
    <td>35.00%</td>
    <td>0.00%</td>
    <td>20.00%</td>
    <td>2.00%</td>
    <td>2.00%</td>
    <td>6.00%</td>
    <td>2.00%</td>
    <td>4.00%</td>
    <td>14.00%</td>
    <td>9.44%</td>
  </tr>
  <tr>
    <td>E12</td>
    <td>50.00%</td>
    <td>30.00%</td>
    <td>25.00%</td>
    <td>4.00%</td>
    <td>40.00%</td>
    <td>30.00%</td>
    <td>58.00%</td>
    <td>46.00%</td>
    <td>36.67%</td>
    <td></td>
  </tr>
  <tr>
    <td>E13</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>15.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>2.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>8.00%</td>
    <td>2.22%</td>
  </tr>
  <tr>
    <td>E13</td>
    <td>0.00%</td>
    <td>50.00%</td>
    <td>15.00%</td>
    <td>2.00%</td>
    <td>2.00%</td>
    <td>12.00%</td>
    <td>14.00%</td>
    <td>36.00%</td>
    <td>10.00%</td>
    <td>14.17%</td>
  </tr>
  <tr>
    <td>E14</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>2.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.28%</td>
  </tr>
  <tr>
    <td>E15</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>82.00%</td>
    <td>0.00%</td>
    <td>12.00%</td>
    <td>2.00%</td>
    <td>0.00%</td>
    <td>13.33%</td>
    <td></td>
  </tr>
  <tr>
    <td>E16</td>
    <td>75.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>4.17%</td>
  </tr>
  <tr>
    <td>E17</td>
    <td>35.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>0.00%</td>
    <td>1.94%</td>
  </tr>
</table>

Table 14: Distribution of Error sources (Err) per Model (Md) in the Error-Analysis of Experiments 1 and 2. The column average gives the average of each error source for each model. For an explanation of the error sources, see Table 15.

<table>
  <tr>
    <th>Error code</th>
    <th>Explanation</th>
  </tr>
  <tr>
    <td>E1</td>
    <td>Issues with placing items in inter-domain relations</td>
  </tr>
  <tr>
    <td>E2</td>
    <td>Negations in clues</td>
  </tr>
  <tr>
    <td>E3</td>
    <td>Grid size not respected</td>
  </tr>
  <tr>
    <td>E4</td>
    <td>Abbreviations</td>
  </tr>
  <tr>
    <td>E5</td>
    <td>Item placed prematurely (before it was possible to deduce the correct position)</td>
  </tr>
  <tr>
    <td>E6</td>
    <td>(partially) empty grid / columns and rows swapped / other issues with the grid</td>
  </tr>
  <tr>
    <td>E7</td>
    <td>Puzzle classified as unsolvable</td>
  </tr>
  <tr>
    <td>E9</td>
    <td>Pseudo-Reasoning</td>
  </tr>
  <tr>
    <td>E10</td>
    <td>Position bias (items placed from left to right without justifiable indication)</td>
  </tr>
  <tr>
    <td>E11</td>
    <td>Inconsistency between produced free-text reasoning and solution grid</td>
  </tr>
  <tr>
    <td>E12</td>
    <td>Items positioned twice in the grid</td>
  </tr>
  <tr>
    <td>E13</td>
    <td>Clue misinterpreted</td>
  </tr>
  <tr>
    <td>E14</td>
    <td>Not all clues used</td>
  </tr>
  <tr>
    <td>E15</td>
    <td>No reasoning provided</td>
  </tr>
  <tr>
    <td>E16</td>
    <td>Repeats itself more than necessary</td>
  </tr>
  <tr>
    <td>E17</td>
    <td>Overwrites already correct solutions</td>
  </tr>
</table>

Table 15: Reference table for error codes and explanation.
1 Case Study

1.1 Solution paths
The solution paths used to analyse the stepwise accuracy of in the solution process are based on solution paths widely found on the internet.9 The solution processes are relatively constrained for both puzzles due to their setup and refers to the clues as given in Table 6 in App. B. In the following the two solution paths will be described referring to the original item names. For the equivalents in the in domain and in domain obfuscations, refer to Tables 17 and 20.

I.1.1 Einstein solution path
Step 1: Norwegian, Milk  This step places “Norwegian” and “Milk” directly to positions 1 and 3 based on clues 8) and 9). Hence, this step does not require further deduction and only basic understanding of the clues’ semantics.

Step 2: Blue  In this step, “blue” is placed to position 2 via a 1-step inference based on clue 14) stating that the Norwegian lives next to the blue house.

Step 3: Green, Coffee, White  This step places “green”, “white”, and “coffee” based on clues 4) and 5). Restricted by the deduction from clue 5) which places Milk in the centre and from clue 14), the only remaining positions are 4 for “green” and “coffee” and 5 for “white”.

Step 4: British, Red, Yellow  Here, we place “British” and “Red” based on clue 1). Given the former deductions, the only remaining position for a colour and a nationality is 3. “yellow” fits then in the remaining spot for a colour: 1. Note that it is irrelevant for the correct position of yellow, whether “green”, “white” and “red” were placed in the correct positions, as long as “blue” and “Norwegian” were placed correctly.

Step 5: Dunhill, Horses  This step fills in “Dunhill” to position 1 and “Horses” to 2 based on the position of “yellow” and clues 7) and 11).

Step 6: Dane, Tea, Cat, Beer, Bluemaster, Water, Blend  This step considers the possible positions for the groups [Dane + Tea] → 2 or 5 (clue 3), [Bluemaster + Beer] → 2 or 5 (clue 12), [Blend next to Cats and next to Water] → [2, 1, 1], [4, 3, 5], [5, 5] (clues 10 and 15) in the context of the former deductions in the respective domains. By elimination, the positions can be determined as follows. Blend:2, Water:1, Cat:1, Dane:2, Tea:2, Bluemaster:5, Beer:5.

Step 7: German, Prince, Swede, Dog  Based on the former deductions and clues 2) and 13), there remains only one option to place the pairs [German + Prince] → 4 and [Swede + Dog] → 5. Also note that a former wrong placing of the other items into potential other positions still allows to place at least German and Prince correctly.

Step 8: Birds, Pall-Mall  The only remaining position based on clue 6) and former deductions for “Pall-Mall” and “Birds” is 3. Also for these items, there is an increased probability of correct positioning since there is only one potential positioning before that would make position 3 unavailable.

Step 9: Fish  In this step, the last remaining spot for a pet is filled, which is 4.

I.1.2 Zebra solution path
An important property of the Zebra puzzle is its branched reasoning path. The reasoning path reflects the order in which items can be placed without a doubt. For this reason, the path places the possible correct deductions during the branching before the deductions that can be made only after determining the correct branch.

Step 1: Norwegian, Milk  This step places “Norwegian” and “Milk” directly to positions 1 and 3 based on clues 9) and 10). Hence, this step does not require further deduction and only basic understanding of the clues’ semantics.

Step 2: Blue  In this step, “blue” is placed to position 2 via a 1-step inference based on clue 15) stating that the Norwegian lives next to the blue house.

Step 3: Yellow, Kools, Horse  From the third step, based on clues 6) and 4), there are two possible positions for “green”+“coffee”, “ivory” and “red”, which are either 4, 3, 5 or 5, 4, 3. This cannot be resolved until step 6, but the options leave only one option to position “yellow ” correctly, which is 1 since blue occupies 2. Based on this and clue 8) and 12), “Kools” goes to position 1 and “Horse” goes to 1.
Step 4: Ukrainian, Tea   Another conclusion that is independent from the branched reasoning is the positioning of “Ukrainian” and “Tea” based on clue 5). Independently from former conclusions, 2 is the only valid position for this. This is entirely independent from the deductions made in step 3 and the branch of the reasoning path that considers “green”, “ivory” and “red”.

Step 5: Water   Based on Step 4 and clue 13), which demands to place “orange-juice” and “Lucky-Strike” together and blocks positions 4 or 5 depending on the branch taken for “green”+“coffee”. This means that only one option remains free for “water”, which is 1.

Step 6: Ivory, Green, Coffee   At this point, it can be observed that only the reasoning path, where “green”+“coffee”, “ivory” and “red” are at 4, 5, 3 allows further valid deductions.

Step 7: Orange-juice, Lucky-Strike   Following from the step before, the only remaining position for “orange-juice” and “lucky-strike” is 4.

Step 8: Japanese, Spanish, Red, English, Snails, Dog, Old-Gold, Parliaments   In the next step clues 1), 3), 14), and 7) are evaluated. From the former steps “Spanish”+“Red” go into position 3. Since [Parliaments + Japanese] can only go to 5, the further possible options are limited to [Spanish + Dog] → 5, [Old-Gold + Snails] → 3.

Step 9: Fox, Chesterfields   Finally, based on clue 11), there is only one option left to place “Fox” and “Chesterfields”, which are “fox”:1 and “Chesterfields”:2.

Step 10: Zebra   This step fills the last remaining position for a pet, which is 4, with “Zebra”.
1.2 Per-step accuracy plots

Einstein original
4o   4o-mini   Qwen2.5-7B   Qwen2.5-72B
Llama3.1-8B   Llama3.1-70B   Llama3.3-70B
Mistral-7B 7B   Random baseline

Einstein 5-domain
4o   4o-mini   Qwen2.5-7B   Qwen2.5-72B
Llama3.1-8B   Llama3.1-70B   Llama3.3-70B
Mistral-7B 7B   Random baseline

Einstein rephrased
4o   4o-mini   Llama3.1-8B   Llama3.1-70B
Llama3.3-70B   Qwen2.5-7B   Qwen2.5-72B
Mistral-7B 7B   Random baseline

Einstein in domain
4o   4o-mini   Llama3.1-8B   Llama3.1-70B
Llama3.3-70B   Qwen2.5-7B   Qwen2.5-72B
Mistral-7B 7B   Random baseline

Figure 5: Q&A accuracy per reasoning step on Einstein puzzles with unchanged grids (original, rephrased) and manipulated grids (5-domain, in domain). The dashed line indicates the random baseline.
Zebra original

Zebra 5-domain

Zebra rephrased

Zebra in domain

Figure 6: Q&A accuracy per reasoning step on Zebra puzzles with unchanged grids (original, rephrased) and manipulated grids (5-domain, in domain). The dashed line indicates the random baseline.
1.3 Per-item accuracy tables

<table>
  <tr>
    <th rowspan="2">item</th>
    <th rowspan="2">step</th>
    <th colspan="7">Einstein original Q&A per item</th>
  </tr>
  <tr>
    <th>4o</th>
    <th>4o-mini</th>
    <th>Qwen2.5</th>
    <th>Llama3.1</th>
    <th>Llama3.3</th>
    <th>Mistral</th>
  </tr>
  <tr><td>milk</td><td>1</td><td>1.00</td><td>0.00</td><td>0.80</td><td>1.00</td><td>0.67</td><td>1.00</td><td>1.00</td><td>0.56</td></tr>
  <tr><td>norwegian</td><td>1</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.86</td><td>1.00</td><td>1.00</td><td>0.75</td></tr>
  <tr><td>blue</td><td>2</td><td>1.00</td><td>1.00</td><td>0.44</td><td>1.00</td><td>0.00</td><td>1.00</td><td>1.00</td><td>0.50</td></tr>
  <tr><td>coffee</td><td>3</td><td>1.00</td><td>0.00</td><td>0.30</td><td>0.43</td><td>0.33</td><td>0.11</td><td>0.30</td><td>0.00</td></tr>
  <tr><td>green</td><td>3</td><td>1.00</td><td>1.00</td><td>0.44</td><td>0.75</td><td>0.25</td><td>0.30</td><td>0.20</td><td>0.00</td></tr>
  <tr><td>white</td><td>3</td><td>1.00</td><td>1.00</td><td>0.40</td><td>0.88</td><td>0.22</td><td>0.13</td><td>0.25</td><td>0.14</td></tr>
  <tr><td>brit</td><td>4</td><td>1.00</td><td>1.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>red</td><td>4</td><td>1.00</td><td>1.00</td><td>0.30</td><td>0.59</td><td>0.22</td><td>0.33</td><td>0.67</td><td>0.33</td></tr>
  <tr><td>yellow</td><td>4</td><td>1.00</td><td>1.00</td><td>0.10</td><td>0.76</td><td>0.22</td><td>0.20</td><td>0.20</td><td>0.00</td></tr>
  <tr><td>dunhill</td><td>5</td><td>1.00</td><td>1.00</td><td>0.10</td><td>0.67</td><td>0.38</td><td>0.00</td><td>0.10</td><td>0.25</td></tr>
  <tr><td>horses</td><td>5</td><td>1.00</td><td>1.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>beer</td><td>6</td><td>1.00</td><td>1.00</td><td>0.30</td><td>0.44</td><td>0.17</td><td>0.40</td><td>0.30</td><td>0.56</td></tr>
  <tr><td>blend</td><td>6</td><td>1.00</td><td>1.00</td><td>0.40</td><td>0.33</td><td>0.13</td><td>0.00</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>bluemaster</td><td>6</td><td>1.00</td><td>1.00</td><td>0.10</td><td>0.75</td><td>0.25</td><td>0.57</td><td>0.22</td><td>0.43</td></tr>
  <tr><td>cats</td><td>6</td><td>0.00</td><td>1.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>dane</td><td>6</td><td>0.00</td><td>1.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>tea</td><td>6</td><td>0.00</td><td>1.00</td><td>0.00</td><td>0.57</td><td>0.00</td><td>0.22</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>water</td><td>6</td><td>1.00</td><td>1.00</td><td>0.10</td><td>1.00</td><td>0.13</td><td>0.43</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>dogs</td><td>7</td><td>1.00</td><td>1.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>german</td><td>7</td><td>1.00</td><td>1.00</td><td>0.60</td><td>0.78</td><td>0.56</td><td>0.56</td><td>0.70</td><td>0.38</td></tr>
  <tr><td>prince</td><td>7</td><td>1.00</td><td>1.00</td><td>0.30</td><td>0.44</td><td>0.43</td><td>0.44</td><td>0.90</td><td>0.33</td></tr>
  <tr><td>swede</td><td>7</td><td>1.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>birds</td><td>8</td><td>1.00</td><td>1.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>pall mall</td><td>8</td><td>0.00</td><td>0.00</td><td>0.50</td><td>0.38</td><td>0.60</td><td>0.29</td><td>0.80</td><td>0.33</td></tr>
  <tr><td>fish</td><td>9</td><td>1.00</td><td>1.00</td><td>0.40</td><td>0.67</td><td>0.00</td><td>0.25</td><td>0.89</td><td>0.00</td></tr>
</table>

<table>
  <tr>
    <th rowspan="2">item</th>
    <th rowspan="2">step</th>
    <th colspan="7">Einstein rephrased Q&A per item</th>
  </tr>
  <tr>
    <th>4o</th>
    <th>4o-mini</th>
    <th>Qwen2.5</th>
    <th>Llama3.1</th>
    <th>Llama3.3</th>
    <th>Mistral</th>
  </tr>
  <tr><td>milk</td><td>1</td><td>1.00</td><td>1.00</td><td>0.80</td><td>1.00</td><td>0.38</td><td>0.90</td><td>1.00</td><td>0.43</td></tr>
  <tr><td>norwegian</td><td>1</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.86</td><td>0.89</td><td>1.00</td><td>0.75</td></tr>
  <tr><td>blue</td><td>2</td><td>1.00</td><td>1.00</td><td>0.20</td><td>0.78</td><td>0.17</td><td>0.40</td><td>0.90</td><td>0.17</td></tr>
  <tr><td>coffee</td><td>3</td><td>1.00</td><td>0.00</td><td>0.11</td><td>0.63</td><td>0.20</td><td>0.11</td><td>0.20</td><td>0.00</td></tr>
  <tr><td>green</td><td>3</td><td>1.00</td><td>1.00</td><td>0.11</td><td>0.33</td><td>0.00</td><td>0.20</td><td>0.10</td><td>0.00</td></tr>
  <tr><td>white</td><td>3</td><td>1.00</td><td>0.00</td><td>0.56</td><td>0.13</td><td>0.40</td><td>0.40</td><td>0.29</td><td>0.00</td></tr>
  <tr><td>british</td><td>4</td><td>1.00</td><td>1.00</td><td>0.22</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>red</td><td>4</td><td>1.00</td><td>1.00</td><td>0.10</td><td>0.67</td><td>0.14</td><td>0.10</td><td>0.10</td><td>0.29</td></tr>
  <tr><td>yellow</td><td>4</td><td>1.00</td><td>1.00</td><td>0.10</td><td>0.67</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.20</td></tr>
  <tr><td>dunhill</td><td>5</td><td>1.00</td><td>0.67</td><td>0.00</td><td>0.75</td><td>0.00</td><td>0.33</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>horse</td><td>5</td><td>1.00</td><td>1.00</td><td>0.22</td><td>0.25</td><td>0.00</td><td>0.11</td><td>0.10</td><td>0.25</td></tr>
  <tr><td>beer</td><td>6</td><td>0.00</td><td>0.00</td><td>0.30</td><td>0.56</td><td>0.29</td><td>0.30</td><td>0.00</td><td>0.13</td></tr>
  <tr><td>blend</td><td>6</td><td>1.00</td><td>1.00</td><td>0.20</td><td>0.40</td><td>0.14</td><td>0.30</td><td>0.20</td><td>0.00</td></tr>
  <tr><td>bluemaster</td><td>6</td><td>0.00</td><td>0.67</td><td>0.35</td><td>0.10</td><td>0.40</td><td>0.00</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>cat</td><td>6</td><td>0.00</td><td>1.00</td><td>0.00</td><td>0.20</td><td>0.00</td><td>0.20</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>danish</td><td>6</td><td>0.00</td><td>1.00</td><td>0.30</td><td>0.80</td><td>0.40</td><td>0.14</td><td>0.80</td><td>0.00</td></tr>
  <tr><td>tea</td><td>6</td><td>1.00</td><td>0.00</td><td>0.22</td><td>0.40</td><td>0.00</td><td>0.00</td><td>0.60</td><td>0.40</td></tr>
  <tr><td>water</td><td>6</td><td>0.33</td><td>1.00</td><td>0.00</td><td>0.80</td><td>0.17</td><td>0.44</td><td>0.00</td><td>0.14</td></tr>
  <tr><td>dog</td><td>7</td><td>1.00</td><td>1.00</td><td>0.33</td><td>0.56</td><td>0.25</td><td>0.50</td><td>0.22</td><td>0.17</td></tr>
  <tr><td>german</td><td>7</td><td>1.00</td><td>0.00</td><td>0.75</td><td>0.50</td><td>0.25</td><td>0.30</td><td>0.60</td><td>0.00</td></tr>
  <tr><td>prince</td><td>7</td><td>1.00</td><td>1.00</td><td>0.33</td><td>0.11</td><td>0.00</td><td>0.63</td><td>0.70</td><td>0.00</td></tr>
  <tr><td>swedish</td><td>7</td><td>0.00</td><td>0.00</td><td>0.40</td><td>0.80</td><td>0.29</td><td>0.63</td><td>0.30</td><td>0.40</td></tr>
  <tr><td>bird</td><td>8</td><td>1.00</td><td>1.00</td><td>0.10</td><td>0.33</td><td>0.22</td><td>0.80</td><td>0.29</td><td>0.00</td></tr>
  <tr><td>pall-mall</td><td>8</td><td>1.00</td><td>1.00</td><td>0.10</td><td>0.14</td><td>0.33</td><td>0.22</td><td>0.80</td><td>0.29</td></tr>
  <tr><td>fish</td><td>9</td><td>1.00</td><td>1.00</td><td>0.30</td><td>0.30</td><td>0.50</td><td>0.38</td><td>0.50</td><td>0.20</td></tr>
</table>

Table 16: Q&A results on “Einstein original” and “Einstein rephrased” as per-item accuracy.
<table>
  <tr>
    <th colspan="12">Einstein in domain Q&amp;A per item</th>
  </tr>
  <tr>
    <th>item</th>
    <th>orig item</th>
    <th>step</th>
    <th>4o</th>
    <th>4o-mini</th>
    <th>Qwen2.5</th>
    <th>Llama3.1</th>
    <th>Llama3.3</th>
    <th>Mistral</th>
    <th>7B</th>
    <th>72B</th>
    <th>8B</th>
    <th>70B</th>
    <th>7B</th>
  </tr>
  <tr><td>dutch</td><td>norwegian</td><td>1</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.89</td><td>1.00</td><td>1.00</td><td>0.50</td></tr>
  <tr><td>mizinda</td><td>milk</td><td>1</td><td>1.00</td><td>1.00</td><td>0.75</td><td>1.00</td><td>0.71</td><td>0.67</td><td>1.00</td><td>0.38</td></tr>
  <tr><td>chestnut</td><td>blue</td><td>2</td><td>1.00</td><td>0.00</td><td>0.50</td><td>0.25</td><td>0.20</td><td>0.57</td><td>1.00</td><td>0.33</td></tr>
  <tr><td>7up</td><td>coffee</td><td>3</td><td>0.00</td><td>0.00</td><td>0.10</td><td>0.29</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.33</td></tr>
  <tr><td>aquamarine</td><td>white</td><td>3</td><td>0.00</td><td>0.00</td><td>0.30</td><td>0.50</td><td>0.17</td><td>0.11</td><td>0.20</td><td>0.14</td></tr>
  <tr><td>grey</td><td>green</td><td>3</td><td>0.00</td><td>0.33</td><td>0.10</td><td>0.13</td><td>0.20</td><td>0.00</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>azure</td><td>yellow</td><td>4</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.14</td><td>0.00</td><td>0.00</td><td>0.20</td><td>0.00</td></tr>
  <tr><td>black</td><td>red</td><td>4</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.44</td><td>0.40</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>japanese</td><td>english</td><td>4</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.25</td><td>0.29</td><td>0.13</td><td>0.70</td><td>0.13</td></tr>
  <tr><td>game</td><td>dunhill</td><td>5</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>turtle</td><td>horse</td><td>5</td><td>1.00</td><td>0.00</td><td>0.11</td><td>0.10</td><td>0.00</td><td>0.13</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>almond-milk</td><td>tea</td><td>6</td><td>0.00</td><td>0.00</td><td>0.57</td><td>0.11</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>australian</td><td>dane</td><td>6</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.20</td><td>0.00</td><td>0.20</td><td>0.40</td><td>0.00</td></tr>
  <tr><td>chamomile</td><td>blend</td><td>6</td><td>0.00</td><td>1.00</td><td>0.00</td><td>0.13</td><td>0.00</td><td>0.13</td><td>0.10</td><td>0.33</td></tr>
  <tr><td>davidoff</td><td>blue/master</td><td>6</td><td>0.00</td><td>1.00</td><td>0.33</td><td>0.38</td><td>0.00</td><td>0.33</td><td>1.00</td><td>0.60</td></tr>
  <tr><td>fanta</td><td>beer</td><td>6</td><td>0.00</td><td>1.00</td><td>0.56</td><td>0.30</td><td>0.50</td><td>0.25</td><td>1.00</td><td>0.00</td></tr>
  <tr><td>ferret</td><td>cat</td><td>6</td><td>0.00</td><td>0.00</td><td>0.11</td><td>0.11</td><td>0.00</td><td>0.29</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>hot-chocolate</td><td>water</td><td>6</td><td>0.00</td><td>1.00</td><td>0.00</td><td>0.60</td><td>0.14</td><td>0.00</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>french</td><td>german</td><td>7</td><td>1.00</td><td>0.67</td><td>0.30</td><td>0.00</td><td>0.00</td><td>0.13</td><td>0.80</td><td>0.00</td></tr>
  <tr><td>guinea-pig</td><td>dog</td><td>7</td><td>0.00</td><td>0.00</td><td>0.20</td><td>0.29</td><td>0.50</td><td>0.80</td><td>0.00</td><td>0.43</td></tr>
  <tr><td>havana</td><td>prince</td><td>7</td><td>0.00</td><td>1.00</td><td>0.44</td><td>0.38</td><td>0.66</td><td>0.57</td><td>1.00</td><td>0.00</td></tr>
  <tr><td>spanish</td><td>swedish</td><td>7</td><td>0.00</td><td>0.00</td><td>0.33</td><td>0.00</td><td>0.33</td><td>0.00</td><td>0.80</td><td>0.00</td></tr>
  <tr><td>baccarat</td><td>pall-mall</td><td>8</td><td>0.00</td><td>0.00</td><td>0.33</td><td>0.00</td><td>0.33</td><td>0.00</td><td>0.80</td><td>0.00</td></tr>
  <tr><td>chinchilla</td><td>bird</td><td>8</td><td>0.00</td><td>1.00</td><td>0.50</td><td>0.22</td><td>0.40</td><td>0.43</td><td>0.70</td><td>0.11</td></tr>
  <tr><td>lizard</td><td>fish</td><td>9</td><td>0.00</td><td>0.00</td><td>0.40</td><td>0.25</td><td>0.25</td><td>0.10</td><td>0.20</td><td>0.50</td></tr>
  <tr><td>Variance</td><td></td><td></td><td>0.44</td><td>0.47</td><td>0.26</td><td>0.27</td><td>0.25</td><td>0.28</td><td>0.43</td><td>0.20</td></tr>
</table>

<table>
  <tr>
    <th colspan="12">Einstein in domain Q&amp;A per item</th>
  </tr>
  <tr>
    <th>item</th>
    <th>orig item</th>
    <th>step</th>
    <th>4o</th>
    <th>4o-mini</th>
    <th>Qwen2.5</th>
    <th>Llama3.1</th>
    <th>Llama3.3</th>
    <th>Mistral</th>
    <th>7B</th>
    <th>72B</th>
    <th>8B</th>
    <th>70B</th>
    <th>7B</th>
  </tr>
  <tr><td>folk</td><td>norwegian</td><td>1</td><td>1.00</td><td>1.00</td><td>0.90</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.50</td></tr>
  <tr><td>gothic-revival</td><td>milk</td><td>1</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.60</td><td>1.00</td><td>1.00</td><td>0.38</td></tr>
  <tr><td>motorbike</td><td>blue</td><td>2</td><td>0.00</td><td>0.67</td><td>0.00</td><td>0.40</td><td>0.50</td><td>0.70</td><td>0.90</td><td>0.50</td></tr>
  <tr><td>snowmobile</td><td>white</td><td>3</td><td>0.33</td><td>1.00</td><td>0.30</td><td>0.63</td><td>0.00</td><td>0.11</td><td>0.30</td><td>0.00</td></tr>
  <tr><td>subway</td><td>green</td><td>3</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.20</td><td>0.25</td><td>0.10</td><td>0.10</td><td>0.00</td></tr>
  <tr><td>wooden</td><td>coffee</td><td>3</td><td>0.00</td><td>0.00</td><td>0.20</td><td>0.13</td><td>0.29</td><td>0.22</td><td>0.10</td><td>0.00</td></tr>
  <tr><td>quad-bike</td><td>yellow</td><td>4</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>train</td><td>red</td><td>4</td><td>1.00</td><td>1.00</td><td>0.22</td><td>0.50</td><td>0.25</td><td>0.50</td><td>0.80</td><td>0.00</td></tr>
  <tr><td>trance</td><td>english</td><td>4</td><td>1.00</td><td>0.63</td><td>0.40</td><td>0.25</td><td>0.38</td><td>0.60</td><td>0.25</td></tr>
  <tr><td>july</td><td>horse</td><td>5</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.11</td><td>0.00</td></tr>
  <tr><td>mechanic</td><td>dunhill</td><td>5</td><td>0.00</td><td>0.00</td><td>0.13</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.25</td><td>0.00</td></tr>
  <tr><td>futuristic</td><td>water</td><td>6</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.13</td><td>0.50</td><td>0.13</td><td>0.11</td><td>0.17</td></tr>
  <tr><td>june</td><td>cat</td><td>6</td><td>0.00</td><td>0.00</td><td>0.30</td><td>0.00</td><td>0.33</td><td>0.00</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>librarian</td><td>blend</td><td>6</td><td>0.00</td><td>0.00</td><td>0.10</td><td>0.20</td><td>0.00</td><td>0.25</td><td>0.20</td><td>0.40</td></tr>
  <tr><td>motorbikemaster</td><td>bluemaster</td><td>6</td><td>0.00</td><td>0.00</td><td>0.50</td><td>0.13</td><td>0.00</td><td>0.13</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>palace</td><td>tea</td><td>6</td><td>0.00</td><td>0.00</td><td>0.10</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.17</td><td>0.00</td></tr>
  <tr><td>ranch-style</td><td>beer</td><td>6</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>rock</td><td>dane</td><td>6</td><td>0.00</td><td>0.00</td><td>0.13</td><td>0.50</td><td>0.00</td><td>0.13</td><td>0.00</td><td>0.29</td></tr>
  <tr><td>ambient</td><td>german</td><td>7</td><td>1.00</td><td>0.00</td><td>0.30</td><td>0.40</td><td>0.00</td><td>0.20</td><td>0.70</td><td>0.20</td></tr>
  <tr><td>bartender</td><td>prince</td><td>7</td><td>0.00</td><td>0.00</td><td>0.50</td><td>0.30</td><td>0.00</td><td>0.13</td><td>0.10</td><td>0.33</td></tr>
  <tr><td>reggae</td><td>swedish</td><td>7</td><td>0.67</td><td>1.00</td><td>0.22</td><td>0.56</td><td>0.00</td><td>0.86</td><td>0.60</td><td>0.00</td></tr>
  <tr><td>september</td><td>dog</td><td>7</td><td>1.00</td><td>1.00</td><td>0.44</td><td>0.50</td><td>0.33</td><td>0.78</td><td>0.75</td><td>0.14</td></tr>
  <tr><td>architect</td><td>pall-mall</td><td>8</td><td>1.00</td><td>0.00</td><td>0.44</td><td>0.50</td><td>0.17</td><td>0.44</td><td>0.00</td><td>0.33</td></tr>
  <tr><td>may</td><td>bird</td><td>8</td><td>1.00</td><td>0.00</td><td>0.10</td><td>0.30</td><td>0.60</td><td>0.56</td><td>0.33</td><td>0.00</td></tr>
  <tr><td>february</td><td>fish</td><td>9</td><td>0.00</td><td>1.00</td><td>0.33</td><td>0.11</td><td>0.20</td><td>0.13</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>Variance</td><td></td><td></td><td>0.47</td><td>0.48</td><td>0.27</td><td>0.28</td><td>0.26</td><td>0.33</td><td>0.36</td><td>0.18</td></tr>
</table>

Table 17: Q&amp;A results on “Einstein” in domain and Einstein in domain as per-item accuracy.
<table>
  <tr>
    <th rowspan="2">item</th>
    <th rowspan="2">step</th>
    <th colspan="7">Zebra original Q&amp;A per item</th>
  </tr>
  <tr>
    <th>4o</th>
    <th>4o-mini</th>
    <th>Qwen2.5</th>
    <th>Llama3.1</th>
    <th>Llama3.3</th>
    <th>Mistral</th>
  </tr>
  <tr>
    <td>milk</td><td>1</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.56</td><td>1.00</td><td>1.00</td><td>0.67</td>
  </tr>
  <tr>
    <td>norwegian</td><td>1</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.78</td><td>1.00</td><td>1.00</td><td>0.50</td>
  </tr>
  <tr>
    <td>blue</td><td>2</td><td>1.00</td><td>1.00</td><td>0.10</td><td>0.78</td><td>0.38</td><td>0.90</td><td>1.00</td><td>0.50</td>
  </tr>
  <tr>
    <td>fox</td><td>3</td><td>0.00</td><td>0.00</td><td>0.11</td><td>0.20</td><td>0.38</td><td>0.63</td><td>0.00</td><td>0.00</td>
  </tr>
  <tr>
    <td>lucky-strike</td><td>3</td><td>1.00</td><td>1.00</td><td>0.10</td><td>0.60</td><td>0.13</td><td>0.00</td><td>0.00</td><td>0.11</td>
  </tr>
  <tr>
    <td>yellow</td><td>3</td><td>1.00</td><td>1.00</td><td>0.10</td><td>0.67</td><td>0.00</td><td>0.11</td><td>0.00</td><td>0.13</td>
  </tr>
  <tr>
    <td>tea</td><td>4</td><td>1.00</td><td>0.00</td><td>0.20</td><td>0.90</td><td>0.13</td><td>0.20</td><td>0.11</td><td>0.38</td>
  </tr>
  <tr>
    <td>ukrainian</td><td>4</td><td>1.00</td><td>1.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td>
  </tr>
  <tr>
    <td>water</td><td>5</td><td>1.00</td><td>1.00</td><td>0.11</td><td>0.25</td><td>0.86</td><td>0.35</td><td>0.40</td><td>0.40</td>
  </tr>
  <tr>
    <td>coffee</td><td>6</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.25</td><td>0.33</td><td>0.10</td><td>0.11</td><td>0.13</td>
  </tr>
  <tr>
    <td>green</td><td>6</td><td>1.00</td><td>0.00</td><td>0.00</td><td>0.20</td><td>0.33</td><td>0.30</td><td>0.50</td><td>0.00</td>
  </tr>
  <tr>
    <td>ivory</td><td>6</td><td>1.00</td><td>0.00</td><td>0.22</td><td>0.11</td><td>0.57</td><td>0.80</td><td>0.80</td><td>0.13</td>
  </tr>
  <tr>
    <td>horse</td><td>7</td><td>0.00</td><td>0.33</td><td>0.44</td><td>0.25</td><td>0.38</td><td>0.50</td><td>1.00</td><td>0.25</td>
  </tr>
  <tr>
    <td>orange-juice</td><td>7</td><td>0.67</td><td>1.00</td><td>0.10</td><td>0.43</td><td>0.00</td><td>0.22</td><td>0.57</td><td>0.13</td>
  </tr>
  <tr>
    <td>dog</td><td>8</td><td>0.00</td><td>0.00</td><td>0.10</td><td>0.29</td><td>0.17</td><td>0.33</td><td>0.44</td><td>0.00</td>
  </tr>
  <tr>
    <td>english</td><td>8</td><td>1.00</td><td>1.00</td><td>0.00</td><td>0.63</td><td>0.17</td><td>0.11</td><td>0.67</td><td>0.00</td>
  </tr>
  <tr>
    <td>japanese</td><td>8</td><td>1.00</td><td>0.00</td><td>0.56</td><td>0.63</td><td>0.38</td><td>0.30</td><td>0.80</td><td>0.71</td>
  </tr>
  <tr>
    <td>old-gold</td><td>8</td><td>1.00</td><td>1.00</td><td>0.00</td><td>0.25</td><td>0.30</td><td>0.33</td><td>0.40</td><td>0.25</td>
  </tr>
  <tr>
    <td>parliaments</td><td>8</td><td>1.00</td><td>1.00</td><td>0.40</td><td>0.80</td><td>0.38</td><td>0.70</td><td>0.89</td><td>0.67</td>
  </tr>
  <tr>
    <td>red</td><td>8</td><td>1.00</td><td>0.33</td><td>0.50</td><td>0.20</td><td>0.13</td><td>0.22</td><td>0.10</td><td>0.00</td>
  </tr>
  <tr>
    <td>snails</td><td>8</td><td>1.00</td><td>1.00</td><td>0.00</td><td>0.10</td><td>0.29</td><td>0.50</td><td>0.00</td><td>0.00</td>
  </tr>
  <tr>
    <td>spanish</td><td>8</td><td>0.67</td><td>0.00</td><td>0.10</td><td>0.56</td><td>0.33</td><td>0.63</td><td>0.89</td><td>0.00</td>
  </tr>
  <tr>
    <td>kools</td><td>9</td><td>1.00</td><td>1.00</td><td>0.33</td><td>0.63</td><td>0.33</td><td>0.20</td><td>0.00</td><td>0.00</td>
  </tr>
  <tr>
    <td>zebra</td><td>9</td><td>0.00</td><td>1.00</td><td>0.11</td><td>0.20</td><td>0.00</td><td>0.13</td><td>0.33</td><td>0.14</td>
  </tr>
  <tr>
    <td>chesterfields</td><td>10</td><td>1.00</td><td>1.00</td><td>0.30</td><td>0.60</td><td>0.00</td><td>0.29</td><td>0.14</td><td>0.29</td>
  </tr>
</table>

<table>
  <tr>
    <th rowspan="2">item</th>
    <th rowspan="2">step</th>
    <th colspan="7">Zebra rephrased Q&amp;A per item</th>
  </tr>
  <tr>
    <th>4o</th>
    <th>4o-mini</th>
    <th>Qwen2.5</th>
    <th>Llama3.1</th>
    <th>Llama3.3</th>
    <th>Mistral</th>
  </tr>
  <tr>
    <td>milk</td><td>1</td><td>1.00</td><td>1.00</td><td>0.50</td><td>0.90</td><td>1.00</td><td>0.89</td><td>1.00</td><td>0.17</td>
  </tr>
  <tr>
    <td>norwegian</td><td>1</td><td>1.00</td><td>1.00</td><td>0.78</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.00</td>
  </tr>
  <tr>
    <td>blue</td><td>2</td><td>0.00</td><td>0.33</td><td>0.43</td><td>0.70</td><td>1.00</td><td>0.20</td><td>0.56</td><td>0.20</td>
  </tr>
  <tr>
    <td>horse</td><td>3</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.22</td><td>0.00</td><td>0.00</td><td>0.20</td><td>0.00</td>
  </tr>
  <tr>
    <td>kools</td><td>3</td><td>0.00</td><td>1.00</td><td>0.13</td><td>0.00</td><td>0.00</td><td>0.25</td><td>0.00</td><td>0.00</td>
  </tr>
  <tr>
    <td>yellow</td><td>3</td><td>0.00</td><td>0.00</td><td>0.22</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td>
  </tr>
  <tr>
    <td>tea</td><td>4</td><td>1.00</td><td>1.00</td><td>0.40</td><td>0.22</td><td>0.30</td><td>0.00</td><td>0.89</td><td>0.00</td>
  </tr>
  <tr>
    <td>ukrainian</td><td>4</td><td>0.00</td><td>1.00</td><td>0.13</td><td>0.20</td><td>0.40</td><td>0.00</td><td>0.67</td><td>0.00</td>
  </tr>
  <tr>
    <td>water</td><td>5</td><td>1.00</td><td>1.00</td><td>0.00</td><td>0.25</td><td>0.00</td><td>0.00</td><td>1.00</td><td>0.17</td>
  </tr>
  <tr>
    <td>coffee</td><td>6</td><td>0.00</td><td>0.33</td><td>0.13</td><td>0.25</td><td>0.00</td><td>0.10</td><td>0.30</td><td>0.20</td>
  </tr>
  <tr>
    <td>green</td><td>6</td><td>0.33</td><td>0.00</td><td>0.00</td><td>0.22</td><td>0.00</td><td>0.10</td><td>0.40</td><td>0.00</td>
  </tr>
  <tr>
    <td>ivory</td><td>6</td><td>0.00</td><td>0.00</td><td>0.29</td><td>0.33</td><td>0.20</td><td>0.11</td><td>0.14</td><td>0.00</td>
  </tr>
  <tr>
    <td>lucky-strike</td><td>7</td><td>1.00</td><td>0.67</td><td>0.17</td><td>0.50</td><td>0.78</td><td>0.29</td><td>0.33</td><td>0.00</td>
  </tr>
  <tr>
    <td>orange-juice</td><td>7</td><td>0.00</td><td>1.00</td><td>0.00</td><td>0.60</td><td>0.11</td><td>0.44</td><td>0.44</td><td>0.29</td>
  </tr>
  <tr>
    <td>dog</td><td>8</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.22</td><td>0.30</td><td>0.00</td><td>0.30</td><td>0.13</td>
  </tr>
  <tr>
    <td>english</td><td>8</td><td>1.00</td><td>1.00</td><td>0.57</td><td>0.13</td><td>0.20</td><td>0.20</td><td>0.25</td><td>0.00</td>
  </tr>
  <tr>
    <td>japanese</td><td>8</td><td>1.00</td><td>0.00</td><td>0.25</td><td>0.25</td><td>0.00</td><td>0.10</td><td>0.14</td><td>0.25</td>
  </tr>
  <tr>
    <td>old-gold</td><td>8</td><td>0.33</td><td>1.00</td><td>0.25</td><td>0.25</td><td>0.00</td><td>0.10</td><td>0.14</td><td>0.25</td>
  </tr>
  <tr>
    <td>parliament</td><td>8</td><td>0.33</td><td>1.00</td><td>0.38</td><td>0.75</td><td>1.00</td><td>0.80</td><td>0.60</td><td>0.75</td>
  </tr>
  <tr>
    <td>red</td><td>8</td><td>0.00</td><td>0.67</td><td>0.57</td><td>0.20</td><td>0.10</td><td>0.30</td><td>0.25</td><td>0.00</td>
  </tr>
  <tr>
    <td>snails</td><td>8</td><td>1.00</td><td>1.00</td><td>0.13</td><td>0.10</td><td>0.00</td><td>0.00</td><td>0.22</td><td>0.17</td>
  </tr>
  <tr>
    <td>spanish</td><td>8</td><td>0.00</td><td>0.00</td><td>0.20</td><td>0.57</td><td>0.70</td><td>0.10</td><td>0.20</td><td>0.29</td>
  </tr>
  <tr>
    <td>chesterfield</td><td>9</td><td>0.33</td><td>0.00</td><td>0.00</td><td>0.11</td><td>0.20</td><td>0.50</td><td>0.33</td><td>0.33</td>
  </tr>
  <tr>
    <td>fox</td><td>9</td><td>1.00</td><td>0.67</td><td>0.00</td><td>0.22</td><td>0.00</td><td>0.00</td><td>0.10</td><td>0.00</td>
  </tr>
  <tr>
    <td>zebra</td><td>10</td><td>0.0</td><td>0.67</td><td>0.40</td><td>0.13</td><td>0.40</td><td>0.40</td><td>0.63</td><td>0.50</td>
  </tr>
</table>

Table 18: Q&A results on “Zebra original” and “Zebra rephrased” as per-item accuracy.
Table 19: Generated by Spread-LaTeX

<table>
  <tr>
    <th rowspan="2">item</th>
    <th rowspan="2">orig item</th>
    <th colspan="2">4o</th>
    <th colspan="2">4o-mini</th>
    <th colspan="2">Qwen2.5</th>
    <th colspan="2">Llama3.1</th>
    <th colspan="2">Llama3.3</th>
    <th>Mistral</th>
  </tr>
  <tr>
    <th>step</th>
    <th>7B</th>
    <th>7B</th>
    <th>7B</th>
    <th>7B</th>
    <th>8B</th>
    <th>7B</th>
    <th>7B</th>
    <th>7B</th>
    <th>7B</th>
    <th>7B</th>
  </tr>
  <tr><td>german</td><td>norwegian</td><td>1</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.83</td><td>1.00</td><td>1.00</td><td>0.14</td></tr>
  <tr><td>lemonade</td><td>milk</td><td>1</td><td>0.00</td><td>1.00</td><td>0.80</td><td>1.00</td><td>0.00</td><td>1.00</td><td>1.00</td><td>0.50</td></tr>
  <tr><td>coral</td><td>blue</td><td>2</td><td>1.00</td><td>1.00</td><td>0.33</td><td>0.50</td><td>0.00</td><td>0.78</td><td>1.00</td><td>0.20</td></tr>
  <tr><td>chestnut</td><td>yellow</td><td>3</td><td>0.00</td><td>0.00</td><td>0.22</td><td>0.10</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>havana</td><td>kools</td><td>3</td><td>0.00</td><td>0.00</td><td>0.10</td><td>0.11</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>mouse</td><td>horse</td><td>3</td><td>0.00</td><td>0.00</td><td>0.22</td><td>0.22</td><td>0.20</td><td>0.00</td><td>0.00</td><td>0.44</td></tr>
  <tr><td>almond-lemonade</td><td>tea</td><td>4</td><td>1.00</td><td>0.00</td><td>0.22</td><td>0.13</td><td>0.33</td><td>0.00</td><td>1.00</td><td>0.00</td></tr>
  <tr><td>dated</td><td>ukrainian</td><td>4</td><td>1.00</td><td>0.00</td><td>0.22</td><td>0.38</td><td>0.00</td><td>0.33</td><td>1.00</td><td>0.00</td></tr>
  <tr><td>hot-chocolate</td><td>water</td><td>5</td><td>0.00</td><td>1.00</td><td>0.20</td><td>0.13</td><td>0.00</td><td>0.33</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>aquamarine</td><td>ivory</td><td>6</td><td>0.00</td><td>0.00</td><td>0.20</td><td>0.11</td><td>0.20</td><td>0.22</td><td>0.30</td><td>0.13</td></tr>
  <tr><td>black</td><td>green</td><td>6</td><td>0.00</td><td>0.00</td><td>0.40</td><td>0.29</td><td>0.33</td><td>0.25</td><td>0.60</td><td>0.00</td></tr>
  <tr><td>cola</td><td>coffee</td><td>6</td><td>0.00</td><td>0.00</td><td>0.11</td><td>0.29</td><td>0.33</td><td>0.20</td><td>1.00</td><td>0.00</td></tr>
  <tr><td>baccarat</td><td>lucky-strike</td><td>7</td><td>1.00</td><td>0.00</td><td>0.56</td><td>0.38</td><td>0.33</td><td>0.57</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>iced-tea</td><td>orange-juice</td><td>7</td><td>1.00</td><td>0.00</td><td>0.44</td><td>0.29</td><td>0.40</td><td>0.50</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>bird</td><td>snails</td><td>8</td><td>0.00</td><td>0.00</td><td>0.20</td><td>0.11</td><td>0.25</td><td>0.22</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>fonseca</td><td>old-gold</td><td>8</td><td>1.00</td><td>0.00</td><td>0.00</td><td>0.11</td><td>0.25</td><td>0.22</td><td>0.00</td><td>0.13</td></tr>
  <tr><td>goldfish</td><td>dog</td><td>8</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.33</td><td>0.25</td><td>0.10</td><td>0.25</td></tr>
  <tr><td>italian</td><td>spanish</td><td>8</td><td>0.00</td><td>0.00</td><td>0.22</td><td>0.40</td><td>0.22</td><td>0.30</td><td>0.00</td><td>0.13</td></tr>
  <tr><td>malaysian</td><td>english</td><td>8</td><td>0.00</td><td>0.00</td><td>0.10</td><td>0.00</td><td>0.14</td><td>0.22</td><td>0.10</td><td>0.20</td></tr>
  <tr><td>mexican</td><td>japanese</td><td>8</td><td>1.00</td><td>0.67</td><td>0.70</td><td>0.29</td><td>0.38</td><td>0.89</td><td>1.00</td><td>0.33</td></tr>
  <tr><td>orange</td><td>red</td><td>8</td><td>0.00</td><td>0.33</td><td>0.00</td><td>0.00</td><td>0.50</td><td>0.11</td><td>0.00</td><td>0.14</td></tr>
  <tr><td>tiparillo</td><td>parliaments</td><td>8</td><td>0.00</td><td>1.00</td><td>0.67</td><td>1.00</td><td>0.80</td><td>0.00</td><td>0.25</td></tr>
  <tr><td>lizard</td><td>fox</td><td>9</td><td>0.00</td><td>0.00</td><td>0.27</td><td>0.10</td><td>0.14</td><td>0.00</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>Pall-mall</td><td>chesterfields</td><td>9</td><td>0.00</td><td>0.00</td><td>0.30</td><td>0.11</td><td>0.20</td><td>0.25</td><td>0.60</td><td>0.00</td></tr>
  <tr><td>cat</td><td>zebra</td><td>10</td><td>0.00</td><td>0.00</td><td>0.22</td><td>0.11</td><td>0.20</td><td>0.33</td><td>0.11</td><td>0.20</td></tr>
  <tr><th>Variance</th><td></td><td></td><td>0.48</td><td>0.44</td><td>0.26</td><td>0.27</td><td>0.25</td><td>0.32</td><td>0.46</td><td>0.21</td></tr>
</table>

Table 20: Q&A results on Zebra in domain and Zebra in domain as per-item accuracy.

<table>
  <tr>
    <th rowspan="2">item</th>
    <th rowspan="2">orig item</th>
    <th colspan="2">4o</th>
    <th colspan="2">4o-mini</th>
    <th colspan="2">Qwen2.5</th>
    <th colspan="2">Llama3.1</th>
    <th colspan="2">Llama3.3</th>
    <th>Mistral</th>
  </tr>
  <tr>
    <th>step</th>
    <th>7B</th>
    <th>7B</th>
    <th>7B</th>
    <th>7B</th>
    <th>8B</th>
    <th>7B</th>
    <th>7B</th>
    <th>7B</th>
    <th>7B</th>
    <th>7B</th>
  </tr>
  <tr><td>electronic</td><td>norwegian</td><td>1</td><td>1.00</td><td>1.00</td><td>0.89</td><td>1.00</td><td>0.57</td><td>1.00</td><td>1.00</td><td>0.57</td></tr>
  <tr><td>nonopoly</td><td>milk</td><td>1</td><td>1.00</td><td>1.00</td><td>0.80</td><td>1.00</td><td>0.50</td><td>1.00</td><td>1.00</td><td>0.29</td></tr>
  <tr><td>lettuce</td><td>blue</td><td>2</td><td>0.67</td><td>1.00</td><td>0.33</td><td>0.33</td><td>1.00</td><td>1.00</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>orchid</td><td>horse</td><td>3</td><td>0.67</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.33</td><td>0.29</td></tr>
  <tr><td>radish</td><td>yellow</td><td>3</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.25</td><td>0.00</td><td>0.00</td><td>0.33</td></tr>
  <tr><td>victorian</td><td>kools</td><td>3</td><td>0.00</td><td>0.00</td><td>0.10</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>domino</td><td>tea</td><td>4</td><td>0.33</td><td>1.00</td><td>0.00</td><td>0.13</td><td>0.20</td><td>0.00</td><td>0.20</td><td>0.33</td></tr>
  <tr><td>techno</td><td>ukrainian</td><td>4</td><td>0.00</td><td>0.33</td><td>0.11</td><td>0.13</td><td>0.00</td><td>0.20</td><td>0.00</td><td>0.13</td></tr>
  <tr><td>chess</td><td>water</td><td>5</td><td>0.33</td><td>1.00</td><td>0.11</td><td>0.00</td><td>0.25</td><td>0.11</td><td>0.00</td><td>0.25</td></tr>
  <tr><td>carrot</td><td>ivory</td><td>6</td><td>0.67</td><td>0.33</td><td>0.20</td><td>0.33</td><td>0.25</td><td>0.20</td><td>0.40</td><td>0.00</td></tr>
  <tr><td>go</td><td>coffee</td><td>6</td><td>0.00</td><td>0.33</td><td>0.10</td><td>0.33</td><td>0.14</td><td>0.14</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>zucchini</td><td>green</td><td>6</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.29</td><td>0.25</td><td>0.00</td><td>0.20</td><td>0.17</td></tr>
  <tr><td>mah-jongg</td><td>orange-juice</td><td>7</td><td>1.00</td><td>0.00</td><td>0.30</td><td>0.33</td><td>0.14</td><td>0.56</td><td>0.78</td><td>0.33</td></tr>
  <tr><td>townhouse</td><td>lucky-strike</td><td>7</td><td>1.00</td><td>0.33</td><td>0.20</td><td>0.29</td><td>0.43</td><td>0.33</td><td>0.90</td><td>0.13</td></tr>
  <tr><td>azalea</td><td>dog</td><td>8</td><td>0.00</td><td>1.00</td><td>0.11</td><td>0.11</td><td>0.25</td><td>0.40</td><td>0.50</td><td>0.13</td></tr>
  <tr><td>bellflower</td><td>snails</td><td>8</td><td>0.33</td><td>0.00</td><td>0.14</td><td>0.00</td><td>0.25</td><td>0.00</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>colonial</td><td>old-gold</td><td>8</td><td>0.00</td><td>0.00</td><td>0.20</td><td>0.00</td><td>0.14</td><td>0.00</td><td>0.30</td><td>0.17</td></tr>
  <tr><td>futuristic</td><td>parliaments</td><td>8</td><td>1.00</td><td>0.00</td><td>0.78</td><td>0.78</td><td>1.00</td><td>1.00</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>indie</td><td>english</td><td>8</td><td>0.00</td><td>0.00</td><td>0.11</td><td>0.44</td><td>0.14</td><td>0.00</td><td>0.00</td><td>0.00</td></tr>
  <tr><td>onion</td><td>red</td><td>8</td><td>0.00</td><td>1.00</td><td>0.00</td><td>0.13</td><td>0.17</td><td>0.00</td><td>0.00</td><td>0.11</td></tr>
  <tr><td>pop</td><td>spanish</td><td>8</td><td>1.00</td><td>0.00</td><td>0.20</td><td>0.11</td><td>0.29</td><td>0.25</td><td>0.10</td><td>0.00</td></tr>
  <tr><td>trance</td><td>japanese</td><td>8</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.43</td></tr>
  <tr><td>gothic-revival</td><td>chesterfields</td><td>9</td><td>0.33</td><td>0.00</td><td>0.20</td><td>0.60</td><td>0.10</td><td>0.44</td><td>0.90</td><td>0.00</td></tr>
  <tr><td>marigold</td><td>fox</td><td>9</td><td>0.00</td><td>0.00</td><td>0.11</td><td>0.20</td><td>0.00</td><td>0.13</td><td>0.00</td><td>0.14</td></tr>
  <tr><td>dahlia</td><td>zebra</td><td>10</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.50</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.60</td></tr>
  <tr><th>Variance</th><td></td><td></td><td>0.43</td><td>0.47</td><td>0.29</td><td>0.33</td><td>0.19</td><td>0.38</td><td>0.41</td><td>0.18</td></tr>
</table>

Reviewer paper summaries: This work presents a reasoning benchmark on Zebra and Einstein puzzles for LLMs. It explains the creation of the dataset using an existing generator and then how various obfuscation techniques were applied. Finally, it presents and discusses results and error analysis of bigger closed models (4o, 3.5-Turbo) and smaller open models (Llama 3.1-8B, Qwen 2-7B, and Mistral 7B).
Summary of strengths: - It is a passionate exploration of a relatively new test of LLM reasoning skills. The motivations and strengths of why this should be considered are presented very clearly.
- The authors also put in commendable effort in designing the tasks for a difficulty level-based evaluation, which can be spawned to make more lexical contamination-free puzzles in the domain.
- Finally there is a good amount of qualitative analysis of errors that could be relevant to the community with some more backing.
Summary of weaknesses: 1. There are some missing crucial details in the making of the dataset: (a) apart from in-domain obfuscation, the noun classes would have to match with the verbs in the phrases, e.g. a person who drinks a drink cannot be replaced by a person who drinks a house. This kind of consequence of rule-based obfuscation is not discussed clearly. (b) Similarly, the procedure of rephrasing is not explained enough for reproducibility.
2. There seems to be some bias in the analysis as about 40% of data from Llama 3.1-8B and 85% from Mistral were removed due to ill-formatting, and the same is not taken into statistical consideration. Significance values could have helped here. Similarly, line 583 claims that GPT models are better at the task, while only bigger GPT models were compared against much smaller open-sourced models.
3. A tighter statistical analysis could have helped; following from the above point and adding correlation values or line plots for experiment 2 would have made it very clear and reliable. Similarly, lines 440-462 are good qualitative analyses but could use more statistical backing, i.e. how many of the examples indicated the X might be happening due to Y.
Comments, suggestions and typos: - Typos:
-- Some latex issue: "r, ightarrow" line 266
-- Lines 504-505 “Nevertheless, also”
-- Line 648 has an empty Acknowledgements section

- Other work:
-- Shah et al. were cited once, but as it is a closely linked work, comparing scores and baselines would have added to the reliability of this work. (Similarly, LiveBench also includes Zebra puzzles without rephrasing, baselines could be compared)
-- It would have been interesting to explore grid size/position-based difficulties better or analyze lexical items/classes-based difficulties. (ref: "Embers of autoregression show how large language models are shaped by the problem they are trained to solve")
-- Places like line 460 on negation and LLMs, could use citations from other existing linguistic studies in the domain.

Finally, this was touched upon slightly in the discussions section, but I feel that it is very important to the paper to discuss the huge drop in Einstein puzzles on rephrasing (indicates a lot of contamination) as compared to a rise in rephrasing in Zebra puzzles (which could be stemming from the underspecified (pointed in the Weaknesses above) rephrasing strategy employed)
Overall assessment scores: 3
Metareview score: 3
Metareview: This paper needs major revision, including more baselines, analysis, clarification.
Summary of reasons to publish: N/A