The First Impression Problem: Internal Bias Triggers Overthinking in Reasoning Models
Authors: Renfei Dang
Zhening Li
Shujian Huang
Jiajun Chen
Venue: ICLR 2026 Poster (2026)
Keywords: Large Language Model
Large Reasoning Model
Overthinking
Primary area: foundation or frontier models, including LLMs

Abstract: Reasoning models often exhibit overthinking, characterized by redundant reasoning steps. We identify \emph{internal bias} elicited by the input question as a key trigger of such behavior. Upon encountering a problem, the model immediately forms a preliminary guess about the answer, which we term an internal bias since it may not be explicitly generated, and it arises without systematic reasoning. When this guess conflicts with its subsequent reasoning, the model tends to engage in excessive reflection, resulting in wasted computation. We validate the association between internal bias and overthinking across multiple models and diverse reasoning tasks. To demonstrate the causal relationship more rigorously, we conduct two counterfactual interventions, showing that removing the input question after the model reduces the redundant reasoning across various complex reasoning tasks, and manually injecting bias affects overthinking accordingly. Further interpretability experiments suggest that excessive attention to the input question serves as a key mechanism through which internal bias influences subsequent reasoning trajectories. Finally, we evaluated several methods aimed at mitigating overthinking, yet the influence of internal bias persisted under all conditions.

Published as a conference paper at ICLR 2026
THE FIRST IMPRESSION PROBLEM: INTERNAL BIAS TRIGGERS OVERTHINKING IN REASONING MODELS
Renfei Dang, Zhening Li, Shujian Huang!, JJajuun Chen National Key Laboratory for Novel Software Technology, Nanjing University (dangrf, 11 1izn) @ma i n ju. edu. en, (huangs j, chenj j edu.
ABSTRACT
Reasoning models often exhibit overthinking, characterized by redundant reason- ing steps. We identify internal bias elicited by the input question as a key trigger of such behavior. Upon encountering problem. the model immediately forms a preliminary guess about the answer, which we term an internal bias since it may not be explicitly generated, and it arises without systematic reasoning. When this guess conflicts with its subsequent reasoning, the model tends to engage in excessive reflection, resulting in wasted computation. We validate the associa- tion between internal bias and overthinking across multiple models and diverse reasoning tasks. To demonstrate the causal relationship more rigorously, we con- duct two counterfactual interventions, showing that removing the input question after the model reduces the redundant reasoning across various complex reason- ing tasks, and manually injecting bias affects overthinking accordingly. Further interpretability experiments suggest that excessive attention to the input question serves as key mechanism through which internal bias influences subsequent rea- soning trajectories. Finally, we evaluated several methods aimed at mitigating overthinking, yet the influence internal bias persisted under all conditions.
INTRODUCTION
Current o 1/R1-ty reasoning models (OpenAl et al. 2024 DeepSeck-Al et al. 2025a have demon- strated outstanding performance with their ability to spontaneously reflect and correct errors (Xu let al. 2025a). However, the reasoning models tend to overthink (Chen et al., (025b). which is characterized by behavioral patterns such as repeatedly reaching the same conclusion without con- tributing to the final answer. This redundant thinking results in significant waste in computation.
Despite efforts reducing overthinking through training (Chen al., 20256 Su al. 2025 Team et al. 2025) Shen et al.] [025 Arora and Zanette 2025 manual intervention during decoding (Chen et al. 2025a Zhang et al.) 2025a Ma et al. 2025), an important question remains underex-
plored: what drives the reasoning models to exhibit such overthinking tendency ?
We identify the internal bias of reasoning models as one important reason for their overthinking. Our hypothesis is twofold: first, upon encountering a problem, the model forms preliminary and intuitive guess, before engaging in formal reasoning: second, when this initial guess conflicts with the outcome of deliberate reasoning, the model is more likely to enter state of excessive reflection. We refer to this guess as an internal bias, to emphasize that it may not be explicitly output by the model, and is not derived from rigorous reasoning but originates solely from the input question. This entire process is illustrated in Figure
Extensive statistical experiments across different model families (DeepSeek (DeepSeek-Al et al., 2025b), Qwen (Qwen et al.) 2025)), model sizes (14B, 32B and 671B), downstream tasks (character operations, logical reasoning and mathematical reasoning) and questioning languages (English and Chinese) consistently demonstrate that internal bias has a widespread influence on reasoning be- havior. Specifically, the greater a model's internal bias deviates from its reasoning result, the more
Corresponding author.
'lll responses in these examples generated by DeepSeek-RI (2025/01/20). The complete model re-
sponses and details of internal bias detection can be found in Appendix A. and A.2

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Published as a conference paper at ICLR 2026
Internal Bias w/o reasoning
Internal Bias w/O reasoning
Question
Question
coutitul Integer. Call positive inoegern in digit
How many letters are in the word strawberry?
base and these twoe dights suis to For example, ts 13-eas iiful becatuee Gi 6&s and i 1- 3 S1. add the east imaeer 2 for which thene are more integers.
Reasoning Chunks
Reasoning Chunks
Ckay, so need to figure out how many mas the witer "I" eppears in the word "ttawcberry" too Walt, but that seound make chese
Okiy, so ieed to find the les integer is Therelone, the smallest such b-1 would the product the fine four primees
2"3"5">-210. hhreffree b-111
7927 bokems
But thought strawberry has two Rs. Naybe I'm miscounting..... So the Rss letters.
may larger necessary...
product
lcading
488 talkens
Wait, but aaways thought strawberry has two Ra. Koybe confusingit th another word.... So positi no 3.5 B That's throe Rs.
But may a smaller number with four distinct prime factors
exists... .... Therefore, 2n211 is cancidate.
2326 tokens
tokens
2629 tokers
But I'm second guessing because maybe I'm including extra R. Warr, let me lookup the word Te., Rt' S-T T-R-A-W B-E R----.. So three Rs. Therelone. the correct answer
Walt, but check another number with four distinct prime factors but smaller product.... Thorefore, the anlsoor (bcckod]221..
975 tollene
Two examples illustrating the existence of internal bias. Green texts denote correct an- swers derived from reasoning, while the navy-colored portions show the influence of internal bias. In the left example (a simpler case), the model develops an internal bias of "2", conflicting with the rea- soning result "3". In the more complex example (from AIME 2024) on the right, the model predicts the answer to be approximately "20", significantly deviating from the correct value "21 1" obtained via reasoning. For both examples, the reasoning process are manually separated into chunks for bet- ter illustration, where the model obtains the correct answer in the first chunk, but internal bias still triggers later reflection. The number in the bottom- rightt corner of each chunk indicates the length of it. It is clear that the model spends much more tokens in reflection due to internal bias.
Figure I:
likely it is to engage in excessive reflection. In extreme cases, this tendency can even lead to the parroting behavior (Xu et al. (022).
To establish causal link between internal bias and overthinking in reasoning models, propose two counterfactual validations: a reasoning trajectory intervention and a bias injection. The first intervention removes the input question after the model generates answer during its reasoning steps, forcing it to decide whether to continue reasoning based solely on its own past reasoning steps, thereby preventing the reactivation of input-dependent biases. This intervention reduces re- dundant reasoning length by 31% 31% to 53% across both synthetic simple tasks and complex logical and mathematical reasoning benchmarks, while largely maintaining or even improving accuracy, indicat- ing that the reduced reasoning steps are largely redundant. The second intervention, bias injection, deliberately manipulates the model's internal bias through controlled training signals, demonstrating that erroneous biases exacerbate overthinking, whereas accurate ones suppress it.
We further conducted attention analysis along the reasoning process, to further understand how internal bias affects reasoning. Interestingly, we found that after completing reasoning step, the model tends to excessively focus (attention) on the question description, which may introduce its internal bias into the decision-making process whether further reflection is needed. This suggests that models might implicitly compare their reasoning results with its internal bias, leading to extra reasoning in some cases despite having already arrived at the correct answer multiple times.
Finally, we tested several existing methods for mitigating overthinking. We found that, despite reducing average reasoning length, these methods fail to eliminate the influence of internal bias, demonstrating its resilience in model reasoning.
Our contributions are as follows:
Identifying internal bias as one important reason for overthinking in reasoning models, and validate the universality of this phenomenon across various experiment settings.
Demonstrating the causal relationship between internal bias and overthinking with rigorous experimental evidence.
Discovering that internal bias influences the reasoning process through the model's exces- sive attention the input question.
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Testing several methods of mitigating overthinking, and revealing they are ineffective in eliminating the influence of internal bias.
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RELATED WORK
2.1
MITIGATING OVERTHINKING IN REASONING MODELS
With the emergence of DeepSeek-RI (DeepSeek-Al al. 2025a) and OpenAl-ol (OpenAl al. 2024) style reasoning models, the overthinking phenomenon, in which models produce exces- sively long and unnecessary chains of thought, has become a widely recognized issue. This has spurred growing interest in efficient reasoning approaches aimed at mitigating such behavior (Yue) al. 2025; Sui al. 2025), which can be broadly categorized into three families: training-time, inference-time and prompting approaches.
Training- time approaches constrain reasoning length by encouraging shorter reasoning chains Chen et al. [2225b] Yu et al. [2255] Cheng et al., (025), or by incorporating difficult-dependee
length penalties that allow the model to adaptively select reasoning paths (Shen et al. 2025 Liu et al. 2025
Luo et al.) 2025a). Inference-tine approaches reduce redundant thinking through decoding interven-
tions, such as confidence- or consistency-bads based dynamic early exit Zhang et al. (2025a); Yang et al.
2025) representation--evel steering of model behavior (Chen et al. 2025a Huang et al. 2025
Eisenstadt et al. 2025). Prompting-based approaches control reasoning overhead through input de-
sign, for example by imposing token budgets (Han et al. (025), using draft-style prompting (Xu
ct al. 2025b), or even explicitly skipping reasoning when appropriate (Ma et al. (2025).
These methods aim to control model behavior without investigating the root cause of overthinking.
Our work addresses this gap by showing that internal bias is one of the key underlying factors driving overthinking.
2.2 BIAS AND UNFAITHFULNESS IN LANGUAGE MODELS
I. Language models often develop various biases or priors. One prominent category involves system- atic biases sensitive domains such as law and social ethics (Gallegos et al. 2024)) which may raise fairness concerns in practical applications. More broadly, models exhibit a wide range of behavioral biases, including a tendency to produce the most "common" rather than the correct answers for certain question types (Zhang et al.) 2025b), susceptibility to irrelevant contextual cues format- specific patterns (Minder et al. 2025] Weston and Sukhbaatar 2023), and preference priors induced inevitably by reinforcement learning reward (Greenblatt et al. 2024 Denison et al.) 2024). Ameisen et al. 2025) found that language models may have separate neural pathways that lead to a rough es- timate of the answer when solving simple addition tasks, providing additional mechanistic evidence for the presence of such biases. The internal bias discussed in this paper is closely related to this type of bias, and we focus on how such direct estimations of the answer can trigger overthinking.
Unfaithfulness is closely intertwined with biases. When a model produces Chain-of-Thought (CoT), the generated steps may not necessarily reflect the model's true reasoning process. Paul et al. (2024) shows that interfering with the CoT sentences typically has little effect on its final answer, sug- gesting spurious reasoming phenomenon during reasoning. [Trrpnn et al. (2023) and Chen et al. (2025c) demonstrate that models often produce explanations for pre-selected answer. Arcuschin al. (2025) extend this line of evidence by identifying bias-driven implicit post-hoe rationalization: systematic biases shape the model's internal decision, and explicit CoT is then used to rationalize the resulting biased answer, producing a divergence between surface-level reasoning and underly- under natural, non-adversarial prompts. These unfaithfulness works involve models deviating from normal reasoning processes. However, our work finds that even when mod- els successfully follow correct reasoning procedures, they can still be influenced by bias at critical reflection steps, leading to overthinking.
ing computation
3 MEASURING INTERNAL BIAS IN REASONING MODELS
We now introduce several definitions that will form the basis for quantifying and analyzing internal bias in the later sections.
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Direct Answer As introduced in conceptualize the model's internal bias its prelim- inary guess to a question, formally denoted as Abias = flg; f), where is the input question, represents the model parameters and Ohias denotes the biased answer. To observe this implicit guess, we force the model to skip reasoning and immediately output an answer. We name this answer as direct answer. Specifically, we templates containing no-reasoning prompt followed by the spe- cial token < </think> to indicate termination of thought. The content generated after this prompt reflects the preliminary judgment of the model formed prior to detailed reasoning. An example of such a template is provided below. In contrast, the final answer after full reasoning is denoted as
Gdial-
User > (question)< Assistant >>thhin>> Let me answer him without thinking more.</think Answer:
Internal Bias a Distribution As in probabilistic generation process, the generated direct answer may also be affected by the sampling strategy, e.g. the prompt or the temperature. On the other hand, the model's internal bias may also exist as distribution over possible answers rather than a single deterministic guess. So we use Ubias to denote the internal bias distribution. To approximate this distribution, we adopt multi-sampling approach with different templates. Empirically, we collect 64 direct answers for each question in our experiments, which is a better approximation of the model's internal bias by observing its behavior under varied conditions. All templates and decoding details can be found in Appendix
Deviation Degree The internal bias of model may deviate from its reasoning process, and different deviation degree may have varying effects on its behavior. We define the bias deviation degree Driss dist (dias: Qfinal,, where dist(:;.) .) is task-specific distance function. For tasks with numerical answers, we compute the mean absolute error (MAE) between the direct answers and Arinal- For tasks with categorical answers, such multiple-choice questions, using inconsistency rate, defined as the proportion of direct answers that differ from Géial- These serve approximate quantitative estimates of the gap between the model's 's direct guesses and the reasoning output.
approximate Dbins
IMPACT OF INTERNAL BIAS ON THE REASONING PROCESS
In this section, we take a macroscopic view the impact of internal bias on reasoning models. Our analysis includes the examination the direct answer and its relation reasoning length, well as the understanding of the parroting phenomenon.
4.1 SETUPS
Tasks. We design controllable character manipulation dataset: CharCount (zh) and CharCount (en): The model is tasked with counting how many specific letters are in a given word. To account for linguistic variation, the dataset is split into Chinese and English subsets, based on prompting language. More details are presented in Appendix C We also utilize several open-source reasoning datasets: KnowLogic (Zhan al., 2025): A complex dataset synthesized using real-world knowl- edge and logical reasoning rules, where each question provides four possible answers A, B, C, and D, and more than one may correct. AIME 2024 and AIME 2025 Challenging mathematical reasoning datasets.
Models. To demonstrate the universality of internal bias, conduct experiments different model families and sizes. Specifically, we select DeepSeek-R1 (DeepSeek-Al et al.) 2025a), R1- distill-Qwen-14B, and QwQ-32B (Qwen'Tean 2025).
II Appendix verify that direct answers consistent with latent representations, demonstrating that they can fairly and accurately capture internal bias.
https:: http:///oooooooooolmmoo
ng. ccm wi ki www.kkiiiinxxxpp///a.. Probl Problems, ems_and_
Sol Solutions

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Experiment Details. For each question, we let the model to generate 64 direct answers and one normally reasoned response. Similar to many other works (Chen et al. 2025a| Zhang et al. 2025a), count reflection-related keywords (e.g. "wait") roughly illustrate the number of the model's reflections. The full keyword lists are in Appendix D All experiments were conducted on NVIDIA RTX A6000 GPUs.
4.2
OVERALL RESULTS
Table 1: Results CharCount (zh).
KnowLogic and AIME 2024
datasets.
CharCount (zh)
Acc
AcCdirect
Liow
Lhigh
RA
DeepSeek-RI
99.2%
55.8%
556.7
735.6
32. %
QwQ-32B
92.6%
36.8%
715.6
866.0
21.0%
RI RI-Distill-Qwen- 14B
73.3%
17.3%
944.4
1228.7
30.1%
Logic
Acc
AcCdirett
Ljow
Leigh
RA
DeepSeek-RI
54.0%
29.4%
2965.7
4214.3
42.1% %
QwQ-32B
51.6%
29.7%
5694.7 5713.8
7167.1
25.9%
RI-Distil--Qwen- 14B
27.2%
24.9%
6927.6
21.2%
AIME 2024
Acc
AcCdirect
LLow
LHigh
RA
DeepSeck-RB
76.7%
3.3%
7934.8
9764.2
23.6%
QwQ-32B 73.3%
0.0%
10239.7
13521.6
32.1%
R1-Distill-Qwen 14B
63.3%
0.0%
8882.1
12709.9
43.1%
Table presents results on CharCount (zh), KnowLogic and AIME 2024 datasets, with the rest results provided in Appendix F.T In each table, "Acc" accuracy after reasoning: "AcCairet" the accuracy of direct answers computed by majority votes; Llow and high are the average reasoning length (number tokens) for cases with lower and higher half of deviation degree; R& represents the relative length increase of the high-deviation group compared with the low-deviation group.
Direct answer's accuracy is relatively low. The model exhibits a high error rate in its direct answers, indicating that it often fails to perform correct guess, which is why define such answers as internal bias. The result also suggests that bias is largely detrimental most cases, especially as the complexity of the questions increases.
The reasoning length of the high-deviation group is longer than that of the low-deviation group. The increase in length is least 21.0%, and reaches 42.1% 1% in cases. This obser- vation intuitively demonstrates that the degree of internal bias deviation has a significant impact on the model's reasoning length. The trends are consistent across all models and datasets.
4.3 FINE-GRAINED ANALYSES
To conduct a more fine-grained analysis of the impact of internal bias, we further divide the deviation degree into four smaller intervals. [ the results of R-disttll Seetill-Qwen--44 14B on CharCount (zh), representing tasks with the numerical answers, and QwQ-32B on KnowLogic, representing tasks with the categorical answers. More results are presented in Appendix F2
Reasoning models may naturally engage in reflection, so even for cases with low deviation bias (MAE< 0.5 Inconsistency Rate 25%), reflective behavior can still be observed following the initial reasoning step. The main reason for this reflection may be to check whether the result is correct. For these cases, the number of reflection keywords usually small.
The greater the deviation of the internal bias is, the more severely the model overthinks. As internal bias deviation increases, both the average output length and the count of reflection keywords rise sharply, which aligns with the earlier observation that internal bias triggers overthinking. For these thinking processes, the model tend to generate much more reflection keywords, because is switching between different thoughts. These steps do not involve thorough examination of the reasoning process but instead result primarily in increased token consumption without meaningful self-evaluation. This trend is consistently observed across all experiments.
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CharCount (zh) Qwen-14B
KnowLogic QwQ-328
Keyword
Reyword Count
1200
$223
5000
Aastage
3033
3139
10
0-0
15-382%
MAX
Inconalatency Rate
Figure 2: Correlation between deviation degree and reasoning behavior. The light-colored bars represent the full reasoning length, while the dark-colored bars indicate the position which the model first provides an answer. The orange line shows the number of reflection keywords. Qwen- 14B here is short for RI R1-distill-Qwen-Iee 14Bn-14e Appendix E describes the method used to identify the position at which the model first provides an answer during its reasoning process.
Internal bias rather than question complexity causes the observed frends. One might think that more difficult questions need longer thinking. Thus we collect the first reasoning length of each question, i.c. the length of the first complete thought that leads to an answer, which could be seen as indicator of the complexity of the problem. As shown in Figure 2 the first reasoning lengths are almost the same for examples within the same task, indicating that the complexity of questions are almost the same different groups the same task. Therefore, internal bias is more likely to be the reason for overthinking.
In extreme cases, internal bias can lead a parroting phenomenon. "Parroting" refers to sit- uations in which the model repeatedly generates exactly the same content, thus fails to provide final answer. Taking parroting cases, the direct answer accuracy only 4.4%, compared 17.3% across all cases (Ta- ble D. Conversely, we estimate MAE between the last numeric value in the model's ' output and direct answers, and split all cases into groups based MAE, the high-MAE group exhibits parroting probability of 6.6%, while the low-MAE group shows a significantly lower rate of only 2.8%, suggesting a correlation between deviation degree and the parroting behavior. This highlights the underlying reason behind the parroting phenomenon: driven by strong internal bias, the model engages in excessive reflection early in the reasoming process, which further reinforces its tendency to reflect. This self- perppterttn cycle ultimately leads to repetitive output during the later stages.
DeittillQween-teee performing CharCount (zh) as an example, among all
5 CAUSAL EFFECTS OBSERVED THROUGH COUNTERFACTUAL INTERVENTIONS
In this section, employ distinct counterfactual intervention methods to establish the causal relationship between internal bias and overthinking in reasoning models. We use E1-DDitill-Qwenn 14B model for controllable analysis.
5.1 REMOVING QUESTION FROM PROMPT
When the model first generates an answer during its reasoning steps, we immediately remove the input question from the prompt and allow the model to continue generating without access to the original task. Since complete reasoning process has already incorporated the necessary information from the question, this forces the model to decide solely based on its own prior reasoning trajectory whether to engage in further reflection terminate the thinking process. Under this intervention, we examine w whether the model reduces its reasoning length and assess how its final performance on answering the question is affected.
We validate this on the full KnowLogic, AIME 2024, and AIME 2025 datasets, as well as random selected 1, e,000eexample subsets from CharCount (en) and CharCount (zh). We introduce the length

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reduction ratio (Loe - (Lor (una) to measure the length reduction after the first reasoning answer is obtained, where L denotes the output length, the subscript ori/rem denotes the original and crestion-removed situation, and Pirsa denotes the position where the first reasoning answer is obtained, i.e. where we start to remove the input question. Results are shown Table
Table 2: Results of the removing question intervention.
Dataset
ACCon AcCrem
Luri Leen
P Piist
r
CharCount (en)
93.8 93.2
541.0/453.3
258.8
31.1
CharCount (zh)
73.4 72.9
1081.9/782.5
245.1
35.8
KnowLogic
27.2 29.3
6320.7 / 5857.5
4921.8
33.1
AIME 2024
63.3/66.7 66.7
10796.0/ 7706.
5022.7
53.5
AIME 2025
36.7/466
13428.9/ 10303.0
4323.4
34.3
Removing question consistently leads to reduction in redundant reasoning. Across all datasets, removing the question leads to a reduction ratio r ranging from 31.1% to 53.5%, causally confirming that the model's excessive focus on the question section is key driver of overthinking.
Removing question leads to better performance in complex tasks. We observe performance im- provements complex datasets. Case studies reveal that these improvements largely attributed to previously parroting behaviors now being resolved correctly, which, as discussed earlier in $4.3) are also strongly linked the influence of internal bias. The observed accuracy decrease in simpler tasks is slight and likely stems from the short reasoning chains, which may be more easily disrupted by our coarse-grand garse-grained intervention. These indicate that the reduced reasoning length primarily sists of redundant thoughts that do not contribute to the model's final answer.
5.2
BIAS INJECTION
We further conduct a bias injection experiment, deliberately manipulating the model's ' internal bias toward specific questions and examine whether its reasoning behavior changes accordingly.
Specifically, we select 500 samples each with the the lowest and highest deviation degree (Drias) from the CharCount (zh) dataset. Based on this, we design a bias injection method: we construct 50 rephrased declarative statements for each sample and then fine-tune the model on these statements to perform a sample-wise bias injection, and then test on the exact same sample. We design two training regimes: a) Low2Wrong, construct wrong declarative statements with the lowest Drias samples; b) High2Correct: construct correct statements with the highest Doias samples; c) Random2 test on high Daas samples, but construct random correct statements that differ from the testing sample; d) 1onndom2rrong test low Daia: samples, but construct random wrong statements. The last two serve as baselines to rule out influence of the bias injection training itself. An example of the declarative statement, additional data and training details provided in Appendix
As shown in Table 3 when comparing each set- ting with its corresponding Random baseline, we observe that in the Low2Wrong setting, the reason- ing length increases from 355.5 to 454.9, whereas in the High2Correct setting it decreases from 600.8 to 412.1. These changes indicate that injecting in- correct biases makes the model more prone to re- flection, while injecting correct ones reduces unnec- essary deliberation, providing strong evidence for causal relationship between the internal bias and the model's 's reflective reasoning patterns. This may in- dicate that the model's 's incorrect internal knowledge distribution is of the origins of internal bias.
Table 3: Bias injection results on sub- sampled CharCount(Zh) dataset.
Setting
Length
Acc
Random2Wrong
355.5
89.5%
Low2Wrong
454.9
83.4%
Rorret2Correct
600.8
67.0%
High2Correct
412.1
76.6%
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6 INTERPRETABILITY ANALYSIS
As shown Figure I although with the existence of internal bias, LLMs can indeed successfully derive the correct answer with its reasoning. An interesting question is: why the model keeps the internal bias in mind even after long steps of reasoning ? We observe that the model pays excessive attention to the input question when deciding whether engage in further reflection. Intuitively, the model may be referring the question to decide whether the current reasoning is correct. But this heightened attention also reactivates the internal bias, influencing the decision reflect. In Appendix we conduct probing study that verifies the hidden states from the question segment are highly correlated with the direct answers and thus encode the internal bias. This supports that bias could be reactivated through attention to the input. We now examine attention behavior through both an illustrative example and quantitative analysis.
6.1 THE "STRAWBERRY" EXAMPLE
Baarin_of__setterre < User
How many letters
are there in word str berry '? Answer
directly with Arabic number Assistant
<think>
Okay so need to figure out how many times the letter
appears in the word str aw berry
Let me
start by writing out the word to visualize it better
S -T
W
-B
-E
-R
-R
-Y Wail that 's not
quite right Let me make sure spell it correctly
S T
W
Hmm
actually
think
might adding extra there Let check again
The word is str aw berry Breaking it down letter by letter
S T
A W E R. R. Y Wait
that 's 0 letters But 'm trying to count the 'r 's's So starting from the beginning
S
[ no T no J.
yes that
). (
no
that 's two ). R that three Y
So count three 't's.
Wait. but I think I might be making a mistake here...
Figure 3: The "strawberry" example as an illustration of abnormally high attention scores on ques- tion part when reflection token is about to be output. Color intensity is used to represent the ratio of attention scores assigned to each preceding tokens at the following two steps: generating "three" and generating "Wait". Darker red indicates higher relative attention at the reflection point when "Wait" is generated, while darker blue reflects higher relative attention when generating "three".
As direct illustration, we revisit the "strawberry" example from Figure We compare the attention score for the context tokens at two different steps during the reasoning: the answer point, where the model generates an intermediate result; and the reflection point, where the model decides whether to perform a further reflection.
The visualization in Figure reveals clear shift in attention at the reflection point: Compared the the answer point, the model focuses more heavily on the question part (the first segment/paragraph) its own paraphrasing of the question, and earlier superficial reasoning steps with reflection tokens (the second segment), while reducing its attention to the core reasoning process that actually obtains the answer (the third segment) when deciding whether reflection token should be output
6.2 STATISTICAL RESULTS ON CHARCOUNT
To further confirm the model's excessive focus the input question, conduct statistical anal- ysis. We categorize all preceding tokens into three distinct categories: (1) Question, which refers to the tokens in the input question.
(2) Mid Results, which corresponds to tokens representing inter- mediate counting results, e.g. "3" or "three". (3) Others. For a token at position p, we can compute the average normalized score s6 it assigns to previous tokens of category c, and further aggregate them across groups G (Reflection, Other):
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a
1
1/p" p No i<p,iEc
S
c p
G
PEG
where N₂ the number of tokens that belongs to category e preceding to token p, and az is attention weight from token p to i. Dividing the by 1/p normalizes for positional imbalances, as it corresponds to the uniform attention baseline, enabling fair comparison across different token positions. The group-level score Sg quantifies the average extent to which tokens in group attend to category All calculations exclude the first three tokens of the sequences to exclude the attention sink phenomenon (Xiao et al. 2024
Reflection Ochar
Question
Mid Rasults
MN
Normalzen Attention Score
Others
115
Rabio (Retfaction
A374
0.6-
0.2
0.0
Resutss
(a)
Figure 4: (a) Group-level scores St with averaged attention scores from layers 21 to 30. Similar visualizations for other layers are in Appendix G and the trends are the same. (b) The ratio of Sh Reflection/ Other across all layers.
St
Figure 4(a) shows the model generally pays low attention to the question section during most of the reasoning process, while upon reflections it exhibits sharp increase more than four times its original level. Although attention to previous intermediate results also increases, the proportion of attention allocated to the question section still rises significantly. Figure 4(b) shows the layer-wist ratio group-level scores Sg for tokens in reflection steps versus other tokens. A sharp increase in attention to question tokens occurs in the middle-to-later layers, where the model engages in active reasoning (Zhao et al.) 2024 Wendler et al.) 2024). These attention patterns may reactivate internal bias during the model's decision whether initiate further reflection and lead overthinking.
7 MITIGATION TRIALS
Finally, we apply some existing techniques for mitigating overthinking and evaluate whether they can also reduce the influence of internal bias.
We reproduce two mainstream categories of approaches: (1) training-time reasoning optimiza- tion, exemplified by First-Correct Solutions with Reflection (denoted as FCS) (Chen et al. (025b), which focuses on constructing high-quality short reasoning data and refining the model through post-training via Supervised Fine-Tuning (SFT) or Direct Preference Optimization (DPO); and (2)
inference-time time reasoning interventions, including SEAL (Chen et al.] (025a)) which steers the rea- soning trajectory by intervening hidden states during decoding, and PROBE (Zhang et al.) (025a), which probes internal representations and determines an appropriate time to manully perform early stopping. We implement these methods RI RI-Distill-Qwen-Il
model (denoted Qwen-14B) and the 1000-sample subset of CharCount (zh) dataset; and additionally reproduce the best per- formed method FCS on AIME 2024. The implement details are in Appendix [1 The results are summarized in Table 4 where the symbols follow the same convention in Table 1 and include "Remove" entry corresponding the question-removal intervention ($ [5 reference.
On CharCount dataset, existing methods perform well in shortening reasoning trajectories, yet show minimal effectiveness mitigating internal bias, and in some cases even exacerbating

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Published as a conference paper at ICLR 2026
its influence, as evidenced by the compa- rable or even larger Ra. As a reference, question-removal can achieve an RA of only 15.2%. The training data for these methods drawn from the CharCount dataset itself, which may contribute to the observed performance improvement, No- tably, the simplest SFT method achieves the highest aceuracy and best RA among
Table Mitigation trials results in CharCount and FCSSFT results in AIME2024.
CharCount
Acc
Liow
Lnigh
Ra
Qwen-14B
73.4%
934.7
1229.1
31.5%
+ Remove
72.9%
727.3
837.8
15.2%
+ FCSpro
76.7%
555.3
812.8
46.3%
+ FCSSFT
78.9%
451.3
572.0
26.7%
existing methods. However, the more complex task AIME 2024, suffers sig-
+ SEAL
77.4%
581.3
805.1
38.5%
+ PROBE
73.1%
702.6
912.5
29.9%
nificant drop in accuracy, indicating that shortened reasoning chains may impair the model's reasoning ability for complex problems, which aligns with the findings Chen al. (2025b). These results sug- gest that existing methods achieve shorter reasoning traces only superficially, some- times even harming the model's reasoning capability. Crucially, the underlying reflection pattern driven by internal bias continues profoundly affect reasoning efficiency.
AIME2024
Acc
Ljow
Lhigh
Ra
Qwen-14BB
63.3%
8882.
12709.9
43.1%
+ Remove
66.7%
7252.6
8159.6
12.5%
+ FCSSFT
50.0%
7204.6
9718.1
34.9%
CONCLUSION AND DISCUSSION
We identify internal bias as a key reason for overthinking in reasoning models, and demonstrate this relevance across different experimental settings. Conterfactual bias intervention and bias injection results provide rigorous evidence the causal link between internal bias and overthinking. Further interpretability analyses reveal that excessive attention to the question during reflection likely reac- tivates the internal bias, leading to redundant reflections even after long reasoning chains. Finally, we find that existing overthinking mitigation methods may fail to eliminate the influence of internal bias, highlighting its resilience and the importance addressing this issue. Our findings call for shift in how we understand reasoning models and open new directions for building more efficient, self-aware, and adaptive reasoning systems.
Nevertheless, several aspects still merit deeper discussion, including the origins of internal bias, the dynamics of its influence, potential mitigation strategies, and the limitations of this work.
Origins of Internal Bias Internal biases may arise from multiple interacting factors rooted in training methods and data. For instance, reinforcement learning with verifiable rewards only may implicitly promote repeated self-checking, while inaccurate knowledge distributions can create con- flicts between the model's 's internal states and the correct reasoning path, triggering reflections, which has already received preliminary verification in the bias injection experiments in $5.2 Further analy- sis of knowledge storage mechanisms like FFN neurons may provide additional insights into biases.
Transition from Bias to Reasoned Output We are interested in when a model becomes "con- vinced" by the reasoning outputs and discards its internal bias. We hypothesize that this transition is gradual: as reasoning progresses, attention to the question becomes diluted, decreasing the influence of internal bias and allowing trust in the ongoing reasoning trajectory dominate. Supplementary results supporting this hypothesis are provided in Appendix K
Potential Solutions Since bias deeply rooted within the model, we argue that the most efficient approach is not to eliminate bias through training. but rather to decouple its influence during infer- ence. Inspired by the analysis in $ an attention- based early-exit mechanism may offer a promising direction. We explored this idea in Appendix
k and observed improvement on the CharCount (zh) dataset. However, developing more robust solutions remains future work.
Limitations
All analyses this paper are restricted to tasks with outputs that are explicit numer- ical values, ranges, or multiple-choice options. We acknowledge that for open-ended tasks, bias is yet well defined, as it may manifest in more complex forms.
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Published as a conference paper at ICLR 2026
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A SUPPLEMENTS FOR FIGURE
A.1 FULL ANSWERS
Below are the complete responses corresponding to the examples in Figure D The "<think>" and "</think>" tokens are added for clearer demonstration. And we mark thoughts in the reasoning process that are explicitly affected by the internal bias in bold font.
QUESTION:
How many letters are word strawberry?
DEEPSEEK-RE.
<think>
Okay, so need figure out how many times the letter "R" appears in the word "strawberry. Let me start writing down the word can look each letter one by one.
The word S-T-R-A- E-E-R-R- Y. Let me count each letter carefully.
Starting with the first letter: That's an S. not an R. Next is T. Still not an R. The third letter is R. Okay, that's one. Then A, W, B, E-tthos aren't R. Then comes another R. So that's two. Next is another R. Wait, let me check again. The word is strawberry. Let spell it out: S, T, R, A, W, B, E, R. R. Hmm, after there are Rs and then Y. after E, it's R. Y. So that would be Rs after Wait, me count again to be sure. Breaking it down:
total, the are at the third position, then the eighth and ninth positions?
1. S
(1)
A
5. W
R (2)
9. R (3)
10. Y
Wait, but that would make three Rs? But thought strawberry has Rs. Maybe I'm miscount- So after B, it's E,
ing. Let me check the spelling again. Strawberry is spelled AT-R-A-W-B-E-RAA then R, R, Y. So positions and are both Rs. So that's 'ss two Rs there, plus the one at position 3. So total of three Rs? But feel like that's not right. Let me verify by writing the word and underlining each R. ST R WEBERRY So the Rs are at the 3rd, 8th, and 9th letters. That's three Rs. Wait, but always thought strawberry has two Rs. Maybe I'm confusing it with another word. Let me check the correct spelling. Maybe I added an extra R by mistake. Let me look up the spelling of strawberry to confirm. [Pauses to recall] No, strawberry
So yes, Rs the end hefore the Y. So that's three Rs in total? Wait, no. I.ct me count again: (1), T (2). R (3), A (4), (5), (6), E (7). R (8), (9), Y (10). So positions 3, 8,8,5 9. That's three Rs. But think might be making a mistake here because when I say strawberry' out loud, it sounds like two Rs. Let me check another way. Maybe I'm breaking down the syllables wrong. Straw-ber-ry. Wait. the "ber" part is B-E-R. then "ry" is R-Y. So maybe in spelling, after B-E-R. there's another before So that's two in last part. total three? But I'm confused now. Let mc write it again:
ST R A W B E Y So letters: L.S, 2.T. 4.A, 5.W, 6.B, 7.E, 8.R, 9.R, 10.Y. Y. So indeed, three Rs. But I must be wrong because remember strawberry having two Rs. Wait, maybe the correct spelling is S-T-R-A-W--A E-R-Y, with only one after Let me confirm. No, no, think it's definitely spelled with two Rs at the end. Because the word comes from "straw" + "berry." and "berry" is B-É-R-R-Y. So "berry" has two Rs, right? Yes, berry is B-E-RA RE-R--- S-T-R-A-W- W-B-E-R-R- Y. Therefore, three Rs: eighth ninth letters). total of three Rs. Wait, but that contradicts my initial thought. Let check a dictionary or something. [Pretends to check a dictionary] Hmm, according to dictionary, strawberry spelled S-T-R-A- --A--------R-R R-Y. So yes, three Rs. Therefore, the answer should be three. But wait, when I say the word, it's pronounced with two Rs? Or maybe I'm just miscounting Let try pronouncing straw-ber-ry. The "ber" is R. and the "ry" is another R? No. actually, "berry" is pronounced with two Rs: B-E-R-R-Y in the spelling, it's two Rs, making total
R-Y. So strawberry is straw berry, so combining them would be "straw" (the third letter) and two in "berry" (the
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strawberry three. Wait, no. Wait, "straw" is S-T-R-A-W, which has one R. Then "berry" is B-E-R- R-Y, which has Rs. So combined, that's s three Rs. Therefore, the answer three. But I need to make sure not adding an extra Let me count again letter by letter:
S (1) nn
T (2)
(3)-1
(4)-1)
W (5) no
B (6) no
B. (7)
(
3
(10) -no
Yes, three Rs. So the answer is 3. But I'm second-guessing because maybe I'm including extra R. Wait, let me look up the word strawberry's spelling again mentally. Yes, it's S-T.R-A-WBBEE-RRR So three Rs. Therefore, the correct answer 3. </think>
To determine how many times the letter "R" appears in the word "strawberry," let's break it down step
by step:
Spell the word: S-T-R-A-W-----R-R--
Identify each letter's position:
1:
2: T
W
6:
7:
8:
10: Y
Count the occurrences of "R":
Position 3: R (1)
Position 8: (2)
Position 9: (3)
Final Answer: The word "strawberry" contains 3 letters "R".
QUESTION:
Let 2 be an integer. Call a positive integer n b-eautifidl has exactly two digits when expressed in base b, and these two digits sum to /n. For example, 81 is 13-eautiful because 81 63,x and 6+3 /81. Find the least integer b which there are more than b-cautiful integers.
DEEPSEEK-RI:
<thin
Okay. I need to find the least integer (where b is at least 2) such that there are more than ten b- eautiful numbers. A b-eautiful number is positive integer that has exactly two digits when written base and the sum of these two digits equals the square root n.
number is written in base as, say, "dl d0", then n d1 b+ d0. The two digits dI and d0 must satisfy dI d0 sqrt(n). Also, since it's two-digit number in base dl must be between and b-1 (since leading zeros aren't allowed), and d0 must be between 0 and b-1. So, the digits can't exceed b-1. 13-cautiful because
First, let me make sure understand the problem correctly.
Given that example, which is sqri(81) 1). That checks out.
13, written as (so d1=6, d0=3), and 6+3=9,
So. the task is to find the smallest base where there are more than ten such numbers. Let me formalize the problem. Let be two-digit number in base b, 1, .... b-1 do Therefore, for each base h, we can iterate over possible and (with from to b-1, from to b-1), compute = dI db + d0, check d1 + d0 sqrt(n). If yes, then is b-cautiful. We need to count how many such exist for each and find the smallest where the count exceeds 10.
dl * b d0, where dl É + d0 = sgrt(n).
.... b-1. Then, condition
So. the plan is: each base starting from upwards:
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For each possible two-digit number in base b (i.c,, dl from I b-1, d0 from to b- i. Compute dI 4b + d0.
Compute y = + d0.
ii.. Check if s2' = n. If yes, then n b-eautiful. b. Count how many such numbers exist for base b. the But doing this manually for each base would be tedious. Instead, maybe can find mathematical relationship between dI, d0, and to find possible solutions.
If the count more than 10, return
Given that n = d d0 and = dl + do = sqri(n). Therefore, s = dl d0. But s dl d0, so substituting s²2 d1 dd d0
But d0 = s d1, so substituting that into the equation:
s"2 d1 (s dl)
Simplify:
s"2 d1*(b- +
Therefore:
dl -'2
So:
s)/(b-)]]
Since dl must be positive integer (as it's digit in base b, from to b-1), and d1 d0 must be integer (as it's sum two integers). Therefore, (s"2 must divisible by (b- 1), and result must be an integer di where So. for each base b, possible values of must satisfy that (s"2 s) is divisible by (b 1), and the resulting dl and d0 must be valid digits in base b. Therefore, the key equation (s22-s x) mad (b- Which can be written as:
dl & b I, and = dl must satisfy 0 & d0 & b-
s(s 1) = 0 mod (b 1)
Therefore, for each b must divide s(s 1). So, b- I is divisor of s(s 1). are digits base b, so can be up to 1. and dO can up to
since = + dO, and dl b- Therefore. the maximum possible value of is (b 1) + (b 1) 2b The minimum possible is + 0= So ranges from to 2b 2.
But in reality, since s'2 is two-digit number in base b. which is least b (since the smallest two-digit number 10_b- bbbbbb b) and at most 1)*b+ (b 1)= b22 Therefore, n must satisfy b & sqrt(b) s & sqri(b'2 - 1) b- 1/(2b). Since an integer, s must be from ceil(sqr((b)) to b- 0-11.
x'2 & b22 Therefore, s must
Wait, but sqri(b"2 1) is less than h, so can be up to b- Therefore, ranges from ceil(sqrt(b)) to b I. But earlier. could be up 2b but in reality, because must be two-digit number base b, s can't be larger than If's is greater than or equal to b. then s'2 >= b²2, but n is at most b Therefore, must be less than Therefore, ranges from ccil(sqrt(b)) to b- 1. Wait, but if is less than b, then the maximum s b- 1. Therefore, is in [ceil(sqr((b)), 1- But then, if is from ccil(sqr((b)) b- then for each such s, can check if (s"2 is divisible by - 1).
since s'2 <= b"2
me check:
so, compute dI = (s s)/( 1). Then check if dl is between and dl is between 0 and b-
d0
Therefore, for each base b, the possible values are integers from ceil(sqrt(b)) up to b 1, such that (s'2 is divisible by (b 1). For each such compute dl and d0, check if they valid digits, and count that a b-cautiful number.
Therefore, the number of b-cautiful numbers equal to the number of integers in [ceil(sqrt(b)), b I] such that (s'2 2-2-s) s) mod (b- 1)= 0, and d1 (s s)/(b- - 1) is an integer hetween and b 1, and d0 =s- d1 is between ( and b 1. So, perhaps instead of iterating through all possible dI and d0 for each base which would he O(h"2) for each we can iterate through possible s values and check divisibility. This might more efficient, but even better, perhaps we can find a relationship between b and s.
Given that (s"2 s) must be divisible by (b 1), so (b 1) divides s(s 1). Therefore, (b- 1) must be consecutive integers, they coprime. Therefore, the divisors coprime, any
divisor of s(s 1). Since and s the products the divisors of and
of s(s
Therefore, since and
divisor s(s 1) can be written as dI d22 where divides and d2 divides s
But in our case, (b 1) is a divisor of s(s 1). Therefore, for each s, possible values of (b 1) are divisors of s(s 1). However, since from upwards. But since ranges up to b- and (b- 1) must least s/(something). Wait, perhaps this approach complicating things.
are looking for b >= 2, then (b 1) >= 1, so possible divisors
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Altematively, for each b, since (b- 1) divides s(s 1), and <= b- then is less than equal to (b- 1). So, is (cril()))(())). Therefore, since (b 1) divides s(s 1), and god(s, s 1)= 1, then 1) must be product a divisor of s and a divisor of s 1. So, (b- 1)= m, where divides and m divides s 1. Then, since s = and s- I m u for some integers I and Then, (b 1) m. So, perhaps this is getting too abstract.
11, and (b- divides s(s Let me note that and s-
coprime.
Alteratively, maybe it's easier fix b and find the possible 1) and then look for in (ceil(sqrt(b)), b 1J such that s(s 1) is divisible by (b 1). Then, for each such check if dl = (s - s)/(b 1) is an integer between and b and d0 = dl is between 0 and b-1. 1.
each base we can compute
But this requires checking for each h. Maybe instead, can parameterize and express in terms of and some divisor.
Let me try to rearrange the equation:
From s'2 $ ddd(b-1)
We can solve for b:
b= (s'2- s)/d1
Since must be an integer greater than di (since dI a digit in base b, so di < b). Also, di must divide 2-"2-3h s). Therefore, for cach d1 can be any divisor of s(s 1), and then b is determined as (s(s ()/dl) + 1. Then, we can check if dI is valid digit in base b (i.e., dl < b), and do =s dl is also valid digit (i.e., d0 >= and d0 b). since = d1 must be non-negative, we have
>= 0, so <= Also, since b, s <b But since b b= (s"2 s)/d1 + 1, so s- dI < <s - s)/11 + 1. Let's substitute into the inequality: s s-dl d1 (s s)/d1 +1
Multiply both sides by dl (assuming d1 which is since di digit from to b-1):
di(s- d1) < s
Expand left side:
dI <22 +
Bring all terms to left side:
di 2 -22 --
Combine like terms:
-dl22 + (s -1)dl - s'2 +
Multiply both sides by -1 (which reverses inequality):
(s-2-(s-1)0 + s2 -
Factor:
ds-2-(s-1)d1 + s(s 00
This quadratic in d1: d1 (s-1)d1 + s(s-1)
Let me compute discriminant:
D (s- (()2 40 11)=
+1) 4s(s =)= -ss +4s -3s
Hmm. discriminant -3522 +s
But since we have the quadratic expression d - -1)d1 s(s -1)> 0, and since the quadratic opens upwards (coefficient of d1 is 1), the expression is positive outside the roots. However, the roots are
given by dl = [[(s-1) (1) + sqrt(D)]/2. Since D is negative for large but let's check when D is positive. >0.
For to be positive: -3s"2
Multiply -1: 3s"2 -2s -1 <0.
Solve 3s"2 -1 <0.
Find roots of 38 -2s =0:
s= = sq +12)
sqn( #4/6
So #12 s=(2-4)/6=-1/3 So the quadratic is positive outside s=1 and s=-1/3. Therefore, 3s"2 -1 <0 for 1/3 <s <1. But since s is positive integer greater than or equal to 1, this is only possible for s=1. So, is positive only when s=1.
For -==1))) +2(1) +1= -322++++1=0 So D=0.
Therefore, except for s=1, D negative. Therefore, for >=2, the quadratic in d1 has real roots, which means quadratic expression is always positive. Therefore, inequality dl'2 (s -1)d1 s(s 1) >0 holds for all real numbers dl. But since dI is positive integer, this means that s dl < automatically satisfied? Wait, but the original inequality was s dl < b, which, after substitution and rearrangement, led the quadratic expression being positive. But since the quadratic always positive for s >=2, then dl < always true for s >=2. However, we must also ensure that d0 dl >=0. So, the main constraints are:
1. dl divides -1).
2. d1 <= (since d0 = dl >=0).
dl >=1 (since it's digit in base and digits start from for the first digit).
(s(s- -1)dd1) +1 must be greater than dl (since a digit base dl < b).
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So, given that, for each >=2, can consider all divisors d1 of s(s -1) such that <=d1 <=s, and check b (s(s -1)d11 +1 d1. Then, in such cases, d0 di must also less than But since
(s(s -I) +1, and d0 = dl, we can check if d0 <
s s-dI < (s(s-1)d1)) -Idd1) +1
Multiply both sides by dl:
di((s dI) s(s-))) +dl
Which is the same inequality as before, leading the quadratic expression which always positive for s >=2. Therefore, for >=2, as long as dI divides s(s -1) and <=dl <=s and b= (s(s -I)dd) +1 dl. then d0 will automatically be less than b. Therefore, the steps can be:
For each >=2:
a. Find all divisors dI of s(s -1) such that <=dl dl <=s.
b. For each such dI, compute b b= (s(s -1Vd1)
Check if b > dl. Since b= = (s(s -1dd1) +1, and dl <=s, then (s(s -1)dd1 >= (s(s -1)/s) = Therefore, b> (s -1) +1 =... Since dI and b >=s, so dl <=s and b >=s, then to have >d1, need that either b >s (5o dI <b) bs, then dl <s. But b (s(s -1) +1. If dl =s, then b= (s(s (1/s) +1= (s +l=s. So if dl=s, then b=s. But dl must be <b, because dI is digit in base b. Therefore, if d1 =s, then bs=, but d1 must be less than b, so d1 <b. Therefore, dI =s is only allowed if <b. But in this case, b=s, this case, when dl =s, b=s, but dl must be less than b, which would require <s, which is impossible. Therefore, cannot be s. Therefore, we must have di <s. So, for each s, dI must be a divisor of s(s -1) such that <=dl <s. Then, b = (s(s- I)dd)) +1. Then, since dI <s, s(s - -I)d1 >=s(s -I)((s - -))s,, so b b>=s+1. +1. Therefore, b >s, so d1 <s <b. Therefore, in this case, dl is valid digit (since digits go up b-1, and dl <b, dl <b). Then, d0 =s -dl must be between 0 and -1. Since d0 -dl <b (as shown earlier), and d0 =3 -dl >=0 because dl <=s. Therefore, as long as dl a divisor s(s -1), 1<=d1 <s, then b (s(s 11 I)dl)) +1, and n 522 is b-cautiful number.
<s is impossible. Therefore, dI cannot be equal to s. Therefore, in
Therefore, each pair (s, dI) where d1 is a proper divisor of s(s -1) (I<=d1 <s) gives base h b= (s(s -I)ddl) +1 and number 522 Therefore, for each the number of b-cautiful numbers in base b is equal to the number of divisors dl of s(s -1) with I<=ll <s. But each such divisor dl gives different base However.
same base can obtained from different s and di pairs. Therefore, to count the number of b-eautiful numbers for each base b, we need to consider how many pairs (s, dI) result in the same b.
Altemaatvely, for each base b, the number of b-cautiful numbers is equal to the number of such that there exists a where (s(s -I)d1) +1, I<=dl <s, and dl divides s(s -I).
This seems complicated. Maybe another approach: for each base b, find in [ceil(sqrt(b)), b -1] such that s(s -1) is divisible by (b -1), then compute dI = s(s -1/(bb -1) and check d1 is an integer hetween and b -1 and d0 =s -d1 is between 0 and b1 Therefore, perhaps it's possible the required range and check conditions. Let's try this approach.
straightforward
iterate
possible b and for each b. compute all
First, note that n = s'2 must be a two-digit number in base b. Therefore, n >=b and n <=b²2 -1. So, s'2 >= and s"2 <=b²2 Therefore, s ><cci(((qrt(b)) and s <=sqri(b"2 ( -1) Rbb -1((2)). Since s must be integer, <=b -1. Therefore, ranges from ceil(sqr((b) to b-1. For each iterate from ceil(sqr((b)) b-.. For each s, check if -1) is divisible by (b -1). If yes, compute =s(s -1)/(b- -1). Check if dl is integer, 1<=dl <=b -1, and do =s -dl is between and b-1. b-eautiful number.
all conditions are satisfied, then n =s'2
The number of such for each base b is the number of b-eautiful numbers. Therefore, solve the problem,
write program that for each starting from upwards: Compute the lower bound for ceil(sqrt(b)..
2. Iterate s from ceil(sqrt(b)) to b -1.
3. For each s, check if (1) mod (b -1) -1)==0. ==0.
4. If yes, compute di s(s -1)(hb -1).
Check if dI is integer (which it is if step passed), and I<=dl <=b-1. Compute
=s -dl, check O<=d00 <=b
7. If all conditions met, increment the count for base b.
8. After processing all s, if count >10, return b. But since this is a mental exercise, not programming one, I need to find a mathematical pattern propertics that allow me compute the number of solutions for each b.
Alternatioll, note that the equation s(s -1) = mod (b -1). Therefore, (b (1) divides s(s -1), Since s and s -1 are coprime, (b- -1) must divide either s or -1, but since they are coprime, (b -1) must divide the product, so (b -1) can be expressed as product of two coprime numbers, one dividing and the other dividing s -1. Therefore,
-1) m, where k divides and divides s -1, and ged(k, m)=1. Given that, since divides and m divides 11 and gcd(k, m)=1, By Chinese Remainder Theorem, there exists a unique solution modulo kmm for s. Therefore, for each
= mod and s = I mod m.
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factorization of (b-1) into k*m with ged(k, m)=1, there exists solutions =0 mod and = mod m. The number of solutions in the range (ceil(sqr((b)), b-1] would correspond the number of valid for each factorization.
But this might not directly help. Alternatively, consider that for a given b. the number of s in [crit()qr((b)). b -1] such that s(s -1) =0 mod (b -I). Since s(s -1) =0 mod (b -1), then either =0 mod (b-1) or =1 mod (b-1), (b -1) has factors that divide and s-1 respectively.
mod (b unless s =0, but >=ceil(sqrt(b)) >=2. So mod (b 1)). Since s <=b -1, the only possibility is s=1. But s >=cee(((qr((()), which is at least when b >=4. For b=2 or 3, ceil(sqr((2))=,, but can only be for b=2? Wait, no:
But
<=b can't be congruent =0 mod (b -1) is impossible. Similarly,
Wait for b=2, the possible range from ceil(sqn((2)=2 up -1=1. But >1, solutions. Therefore, for b=2, no b-cautiful numbers.
Similarly, for b=3: ceil(sqr((3))=2 s ranges from to (since b-1=2). So s=2. Check 2 mod 2. 2 divides 2, yes. Then dI = 12/2=1 Then d0=2 -1=1. Check digits: d1=1 (<=2), d0=1 (<=2). So n=1*3 +1=4. s=2, sqrt(4)=2. So yes, 4 3-eautiful. So for b=3, there number. But the question number of solutions increases.
find the minimal b with more than ten numbers. So perhaps for higher the
Alternativey, maybe we can find that the number of solutions for each base b is equal to the number of divisors of (b -1). Wait, but earlier, we saw that (b-1) divides s(s -1). However, since and coprime, (b -1) must be product
coprime numbers, dividing and the other dividing -1. So number solutions would correspond to number ways write =1)=kmm where k divides and m divides s -1, and and m are coprime. Therefore, for each divisor pair (k, m) of (b -1) with ged(k, m)=1, we can solve for = mod k and = mod The number of solutions in [cti)(sqr((b)), b would correspond the number of such pairs where the solution is in that interval.
But scems complex. Let's try specific examples.
Take h=13 (from the example). Then b -1=12. Factor pairs (k, m) of 12 where ged(k, m)=1:
1*12, 3*4, 4*3, 12*1. Since 12 can be factored into coprime pairs (1,12), (3,4), (4,3), (12,1), For each such pair:
(k=1, m=12):
Solve s = mod (always true), s = mod12. Since <=12 (b -1=12), s=1. But s must be >=ccil(sqn((13))44 So s=1 is invalid.
For (k=3,m=> m=4):
Solve 0 mod3. = mod4, Let's solve:
s=3a. Then 3a = mod4 =>
mod4 => mod4 (since 3*3=9=1 mod4). Therefore, a=4b
+3. Thus, s=3 +3)=12h +9. Since <=12, 2 the only solution is when b=0: s=9. Check if s=9 is in [4,12]. Yes. S=
For (k=4, m=3):
Solve 0 mod4, s mod3. s=4a. 4a =1 mod3 >> a mod3 (since mod3, 1 mod3 => a=1 mod3). Thus, a=3b +1. Therefore, s=4%(3b +1)=12b +4, For s <=12, when b=0: s=4.
Check s=4 in [4, Yes.
For (k=12, m=1): Solve ==0 mod12, s= modl (always true). s=12. Check if is [4,12]. Yes.
Therefore, for b=13, we have three solutions: s=4,9,12. Check each:
For s=4:
dl=(4#3/11=11/12-1, Then d0=4 -1=3, So digits and 3 in base13: 1#13 +3=16. s=4, sqrt(16)=4. So 16 is 13-cautiful.
Wait, but example given, 13-eautiful. Wait, perhaps I made mistake. Wait, when s=9; dl=(9"3/12-7//1266 Then d0-9 -6=3 So digits and 3: 6*13 +3=81. Which is the example. s=9,
sqrt(81)=9. Correct.
Fors=12.
So digits 11 and1. In base13, 3, 11 'B', B1_ 13=11 [1=143+1==144 s=12, sqrt(144=112. So also 13-eautiful. Therefore, b=13, there are three b-eautiful numbers: 16,11,,,,,14444
dl=(12#11/12-11. Then d0=12 -11=1.
But according to the problem statement, the example 81. So that's one of three. Therefore, in this ten b-cautiful numbers. So is much less than 10. Therefore, bases like 13 have 3 solutions. So, how can we get bases with more than ten solutions?
the number of solutions 3. But the problem asks for base with more than
Perhaps when b- -1 has many divisors, leading to multiple factor pairs (k, m) with ged(k, m)=1. The number of coprime factor pairs (k, m) of b is 2 2"(nmmer of distinct prime factors of b -1). Because each prime factor can go to k number coprime factor pairs 2'n.
For example, if b -1 is product of distinct primes, then the
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Therefore, if b has many distinct prime factors, then there many coprime factor pairs, Icading more solutions s. Therefore, the number of solutions is equal the number of coprime factor pairs (k, m) -1, where kmm =b -1, and the corresponding s is in (ceil()q))(b)), b b-1]. However, each coprime factor pair (k, m) gives a unique solution modulo mm. But since <=b -1, which is equal to k*m, so there is exactly one solution s in [1, k=m]. But we need to be in [ceil(sqri(b)), b-1]. Therefore, not all factor pairs will lead to in that interval.
Altematively, we can maximize the number of coprime factor pairs (k, m) of b -1, then we can maximize the number of solutions. Therefore, choosing b 11 to be number with many distinct prime factors. For example, if b-1 is product of the first few primes. then the number of coprime factor pairs would he 2 d'number_ae of of_primes, which increases exponentially.
But even then, each coprime factor pair may or may lead valid in the required interval. But let's consider that for each coprime factor pair (k, m) of b b1, we have unique solution s =0 mod k and s 1 mod m. Then, s can be calculated as:
s 0 mod k
s = mod
By Chinese Remainder Theorem, since and are coprime, there exists a unique solution modulo kmm =b-1. Therefore, the solution is s km + s0, where s0 is the unique solution modulo km. But since must be less than or equal to km (because b -1=km and s <=b -1=m)), the only solution is s=s0. Therefore, for each coprime factor pair (k, m) of b -1, there is exactly one s in [1, b-1] that satisfies the congruences. Therefore, the number of solutions is equal the number of coprime factor pairs (k, m) -1.
But each coprime factor pair is counted twice (k, m) and (m. k). except when k=m=1. Wait, no. we consider ordered pairs, then (k. m) and (m, k) are different if k =m. However, if we consider unordered pairs, then each pair is unique. But in our case, since divides and divides -1, the pairs (k, m) and (m, k) are distinct unless k=m=1.
But regardless, for each coprime factor pair (k, m) b-1, where and m are positive integers such that kmm =b- 11 and ged(k, m)=1, there is a unique solution in [1, b-1]. Therefore, the number of solutions is equal to the number of such coprime factor pairs. Since each coprime factor pair corresponds to unique s. and vice versa.
Therefore, the number b-eautiful numbers is equal the number of coprime factor pairs (k, m) of -I where the corresponding is in (cril(sqrt(b)). b-1]. But the number of coprime factor pairs (k, m) of b-1 2" (b -1), where a (n) is the number of distinct prime factors of n. Because for each distinct prime factor, we assign it to
Therefore, if b
distinct prime factors, there are 2"t coprime factor pairs.
However, each such factor pair (k, m) gives a unique s in [1, b11]. But we need s >=ceil(sqrt(h)). So, to find the number of solutions we need to count how many of these satisfy >=ceil((qrr((b)) Given that, perhaps for larger b. the majority of the solutions s would lie in the interval [ceil(sqr((b), b b-1j. the number of solutions would be approximately (b-1)...
But a (b -1) is maximized when -1 has many distinct prime factors. Therefore. to get more than ten solutions, we need 2" as (b- -1) >10. Therefore. (b (1) >=4, since 2'4=16>10.
Therefore, we need h -1 to have at lcast 4 distinct prime factors. Therefore, the smallest such b -1 would be the product of the first four primes: 2=315*7=210. Therefore, b=211. But this may be larger than necessary, because even some solutions are less than ceil(sqrt(b)), we might still have more than ten solutions where >=ceel(<qr((b)..
Alternatively, if b has many factors, even with fewer prime factors, but higher exponents. Wait, but exponents don't affect the number of coprime factor pairs, since coprime factor pairs determined by distinct prime factors. For example, if b -1= p then the only coprime factor pairs are (1, p'a), so only two factor pairs, but since they are not coprime unless p=1, which isn't. Wait, no. If prime power, then all factor pairs (k, m) must have k=p'i and m=p"j. but then ged(k, mn)=p'minii.j>> unless one of them is 1. Therefore, the only coprime factor pairs are (1, p'a and (p'a.1). Therefore, even for prime powers, there are only two coprime factor pairs, leading to two solutions But s=1 and s= p'a +12 Wait, no, would be congruent O mod1 and mad p'a, which is s=1 mod p'a. Since s<=p'a. then s=1. But s=1 less than ceil(sqrt(p's a +1)), which is at least 2. only s=1 a solution, but it's invalid. Similarly, s= p'a +1 is beyond the range. Therefore, for prime powers, there are no valid solutions. Hence, to get many solutions, we need b -1 to have multiple distinct prime factors. Therefore, the minimal b -1 would be the product of the first four primes: 2*3"5*7=210, leading to b=211. But maybe smaller number with four distinet prime factors exists. Wait, the product of the first four primes is 210, which is the smallest number with four distinct prime factors. Therefore, b=211. But let's check if b=211 actually has 16 solutions. However, s must be in [ceil(sqri(211)) 1)), 210]. 1455 so ceil(sqr((111)=
Therefore, ranges from 15 to210. However, for each coprime factor pair (k, of 210, which has 266 factor pairs, each leading to solution in [1,210]. However, some of these solutions might be less 15. Therefore, the actual number of solutions might be less 16 For example, consider factor pair (1,210): leads to s=1 mod210, so s=1, which is
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less 15. Similarly, (2,105); solve s= mod2, s=1 mod105. Find ==00aa +1. This must he even.
105a =0 mod2 =>105a E-1 mod2 =>a mod2. So a=2b +1. Then s=105~(2b +1)+1=210b
+106. s<=210, so b=0: s=106. Which is in [15,210]. Similarly, other factor pains may give in [15,210]. So, out of 16 factor pairs, how many lead to s>=15?
For each coprime factor pair (k, m) of 210:
(1,210): s=1 (invalid)
(2,105): s=106
3. (3,70): solve s=0 mod3, $=1 mod70. s=70a +1. 70a +1 +1 mod3 =>70 mod3 =>70=
mod3, so a=-1 mod3 =>a=3b -1. Therefore, (3b -1) +1=210b 700 +1=210b -69. For b=1:
==10 -69=141. For h=0: s=-69 (invalid). So s=141.
(5,42): solve s=0 mod5, s=1 mod42. s=42a +1. +1=0 mod5 =>42=-1 mod5 =>42=2 med5, so 2a=-1 mod5 =>2a=4 mod5 =>a= mod5. Therefore, a=5h s=42 +2) +1=210b +84 +1=210b +85. For b=0: s=85. b So s=85. S. (6,35): solve =0 mod6. ==1
m mod35.
s=35a +1. +1. 35a +1 =0 mod6 =>35a_1 mod6 =>35 mod6.
mod6, Sa mod6 =55=5 mod6
mod6
=>a=1
mod6. Thenefore, 4l. s=35# +1) +1=210b
+35 +1=210b +36. For b=0: F 200100000000......21222222223330 So s33.
solve = s0 mod7, =
s=30a
s=30a +1. 30a +1
mo =>30=2
=
so
For b=0: s=91. httphh..........r.....0000030313 solve |0100000000000001111222222222220 mod10, a |0000-100002222...11012222222110 For b-0: s=20 invalid. So 1=90 solve 8. htgpgh/000hnnnggnnaaaaaaaaaaaaa0 mod14, so ottp::/....aaaila.......11111111 b=0: s=-14 invalid. So s=16.
Similarly, the factor pairs reverse order (k, m)=mm))=)))))m..mmm1)))))))))=)) (70,3), (42,5), (35,6), (30,7), (21,10), (15,14), (210,1). check:
mod => mod2. So a=2b +1. s=105*(2b +1)=210b +005. For b=0: s=105. For b=1:210 +005=115>200 So s=105.
10. (105,2): solve = mod105, ==1 $ mod2. ==105.. 105a
(70.3): solve mod70, s=1 mod3. s=70a. 70a
mod3 =>70
mod3, so so
mod3, a=3b
+1. s370* +1)=210b +70. For b=0: s=70. For b=1:210 +70=200>210. +
So s=70
12. (42,5): solve ==0 mod42, s=1 mod5. s=42a. 42a mod5 =>42= mod5, so 2= mod5
=>a=3 mod5, a=5b +3. <5b +3)=210b +126 For b=0: s=126. For b=1:210
So
s=126.
13. (35,6): solve 0=0 mod35, s=1 mod6. s=35a. 35a mod6 =>35=-1 mod6, so -a mad6
=>a=-1 mod6. a=6b -1. <6b 1))-100b -35. For 15:00 For b=0: s=-35 invalid. So
s=175.
14. (30.7): solve = mod30. _ mod7 3=30a. 30a mod >>30 mod7, 2a mod7
= mod7. mod7, a=7b 5=30 +200 For b=0: s= 31 So
s=120.
15. 105
|| s0 mod10.
a=10b +1. 11.b+++1.221110000 (+1
16. (15,14): s0lve +21 For b=0: s=21, 12220
s=
mod144 s=15a. 15a= 1
s mod14.
a=14b +1. +=1+ +15. b=0: s=15. For 1::20001555
s=15.
17. (210,1): solve ==0 mod210, s= modl. s=210a. s<=210, so s=210. Check if s=210 is in
[15,210]. Yes.
So. compiling all solutions for b=211:
From the above:
1416,141,85,366 153366.9...0.111111177..2...1...
Wait, let's count them:
1. (2,105):
(3,70):s=141
(5,42):s=85
4. (6.35):s=36
5. (7.30):s=91
6. (10,21):s=190
(14.15):s=196
(105,2):s=105
9. (70,3):s=70
10. (42.5):==126
11. (35.6)s=175
(30,7)s=120
13. (21,10):=21
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14. (15.14):==15
15. :2=00 But 1=,1,,5,,0,,8, 105, 120, 17,, Wait,
120,,,0,11111. 1i2.120,,2,,14, 14,211111177,, 196,210. Total of 15 solutions. However, some of these values are less than ceil(sqr((111)221455 which is 15. So s=15 is okay (since (il(sqr((211)=1555 but s=21,36, etc., above 15. Wait, s=15 is the lower bound. So ranges from15 to210. Therefore, the solutions 3=15,21.36.00,85,,, 18,,9,,005.106 12,11,,6,,,,1,,1111 Total 15 solutions. So for b=211, there are 15 b-cautiful numbers, which is more than ten. Therefore, b-211 is a candidate.
But perhaps there is smaller b with more than ten solutions. Because 211 1 is quite large. Let's check smaller numbers with multiple prime factors. For example, take =1=60=22=*$55 It has three distinct prime factors. Therefore, the number of coprime factor pairs is 2*3=8. Let's check for b=6l. Compute ranges from c to60. For each coprime facctor pair (k, m) of60:
(1,60):s=1 (invalid)
2. (3,20): solve ==0 mod3, s=1 mod20. s=20a +1. 20a +1 modi =120 = mod3 =>20= +21. For b=0:
mod3, so 21 mod3 =>2a mod3 =>> mod3. a=3b +1.
s=21. For b=1:60 +21=81>60. So s=21
mod15. s=15a +l. 15a +1 1 mod4 => 15 mod4, so 31 mod4 33 mou4 =>> mod4. 4=4b (00= +16. For b=0:
3. (4,15): solve = mod4,
s=16. For b=1:60 +16=666666 So
4. (5,12): solve = mod5, = mod12. 5=12a +l. +1 m mod5 >>
mod5, so 2a mod5 4 => mod5. a=5b +2. +2)+) +25 For b=0:
s=25. For b=1:60 +25=85>60. $=5
(12,5): solve ==0 mod12, x=1 mod5. s=12a. 12a=1 mod5 =>12=2 mod5, so 2==1 mod5
=>a=3 mod5. a=5h s= 1 +36. For b=0: s=36. For b=1:60 +36=96260, So s=36.
6. (15,4): solve ==0 mod15, s=1 mod4, s=15a. 15a=1 mod4 =>15=3 mod4, 3a mod4
=>a =3 mod4. a=4b +3. 5=15 +3)=60b +45. For b=0: s=45. For b=1:60 +45-105>0. So s=45.
(20,3): solve ==0 mod20, s mod3 s=20a. 20a mod3 =>20=2 mod3, so 2a mod3
=>a=2 mod3. a=3b +2. ==20 +2)=60b +40. For b=0: s=40, For b=1:60 +40=100>60. So s=40.
8. (60,1): 5 mod60, ==1 modl. s=60. s=60.
Now, reverse factor pairs:
(60.1): s=60.
10. (20,3): s=40.
11. (15,4): s=45.
12. (12.5): s=36.
13. (5,12): s=25.
(4.15): s=16.
15. (3,20): s=21.
16. (1,60): s=l.
But the unique solutions (ccil(sqr(61)==)::
Let's check which of these are >=8 That's solutions. For b=61, there are b-eautiful numbers. Which is less than ten. So not enough. Another example, take b b-1=120=2734345 Three distinct primes, so 2'3=8 coprime factor pairs. But let's check for b=121. s ranges from ceil(sqr((22))=11 to120.
All of them are >=16. So 16,6,,,,5,,3
Factor pairs:
1. (1,120) invalid.
2. (3,40): s mod33 x=1 mod40. s=40a +1. 00 =040= mod3, so = mod3.
a=3b (=10 -39. For h=1: 8=81. For h=0: s= invalid. So s=81.
mod5. so - mod5 =>4a mod5 =>a=1 mod5. a=5b +1. (5b4 +1) +1=120b +25. For b=0: s=25. For b=1:120
(5.24): solve m0 s mod24. s=24a +1. 24a
+25=145>120, So s=25
- 15 mod8 = 15 4. (8,15): solve == mod8, ==1 mod15. s=15a +1. 15a +1 mo
mod8, so -1 mod8 7 mod8 =>a mod8. a=8b +1. 3= + 16. For b=0:
s=16. For b=1:120 +16=136>120. s=66
5. (15,8): solve = mod15, == mod8 s=15a. 15a=1 mod8 =>15=7 mod8 so 7a=1 mod8
=>a=7 mod8. ab +7. s=15+(8b +7)=120b +105. For b=0: 105 For 1 So
1015
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6. (24,5): solve s=0 mod24, s=1 med5. s=24a. 24a=1 I mod5 =>24=4 mod5, 4a mod5
=>a=4 mod5. a=5b +4. (5b +4)=120b +96. For b=0: s=96. For b=1:120 +96=216> So $=96.
7. (40.3): 0=0 mod40, s=1 mod3. s=40a. 40a mod3 =>40= mod3, so a mod3, a=3b
+1. s=40% +1)=120b +40. For b=0: s=40. For b=1:120 +40=160>1200 So s=40.
(120,1): s=120.
Reverse pairs:
Same as above. The unique solutions are 10,,,,5,40,,8,,,,,100
Check which are >=11: All of them are >=16 except s=16 itself. So total of7 solutions. Still less than ten.
Therefore, even with three distinct prime factors, only get 7 solutions.
Another example: take b -= Three distinct primes, factor pairs. For h=31.
8 ranges from ceil(sqr((11)= to30.
Factor pairs:
1. (1.30)ss=1 invalid.
2. (2,15): solve ==0 mod2, = mod15, s=15a +1. 15a +1 mod2 =>15a=11 mod2 =>15== mod2 =>a mod2. a=2b +1. s=15 +1) +1=30b +16. For b=0: s=16. For b=1:30 +16=46>30.
So s=16.
3. (3,10): solve ==0 mod3, == mod10. 5=10 +1. 10a +1 0 mod3 => 10 -1 mod3 >>1
mod3, s0 mod3. a=3b -1. 1=0= -1) +1=30b -9. For b=1: s=21. For b=0: s=9 invalid. So s=21.
(5.6): solve ==0 mod 1
== +1. 6a 6a mod5 = mod5 =>6=1 mod5, so
a mod5, a=5b -5
For b=1: s=25, For b=0: 5= invalid. So s=25.
5. (6.5): solve = mod6, - mod5. 5= 6a mod5 =>6= mod5, a mod5. a=5b +1.
5=6*(bb +6. For b=0: =6 For b=1:30 +6=36>30. +6-3006663636333 So s=
(10.3): solve m
s=10a. 10a mod3 =>10= mod3, so a= mod3, a=3b 100.
+1. x310 +1)=300 For h=0: s=10. For b0:30
7. (15.2): =0 1 mod2. s=15a. 15a mod2 =>15a mod2 =>a mod2. a=2b
+1. s=15 +1)=30b For b=0: s=15. For +15:::3055:53300.. So s=15
8. (30,1): solve mod30. s mod1. $=30.
So solutions 1=6.100.1516221,25..3. Check which are >=6 (ceil(sqt((11))=)) 156,,0011,,1,,,,,,5,.2 1,,2,500 Total solutions. Still less than ten.
are valid. So Hmm. So maybe b needs four distinct prime factors. Let's check b=211 gives 15 solutions, which more than ten. Is there smaller number with four distinct prime factors? The smallest number with four distinct prime factors 27315 So b=211 the minimal such
Wait, but let's check another number with four distinct prime factors but a smaller product. For example, 243=5#7=210. The next one would be 2=3*5#11=333 which is larger. Therefore, 210 is the smallest number with four distinct prime factors. Therefore, b=211 is the minimal base where -1=210 has distinct prime factors, leading range. Therefore, b=211 the answer.
coprime factor pairs and solutions the valid
But to confirm, let's check for b=211, we have 15 solutions as previously enumerated. But the problem states "more than ten", so 15 is acceptable. Is there smaller b with, say, b -1 210/2=105, which is 3*5"7, three distinct primes. Wait, b=106, with b b-1=105. Let's check for b=106. s ranges from ceil(sqrt(106) 106)= to105.
For b=106, b b-1=105=3=5*7, Number of coprime factor pairs=2"3=8.
Factor pairs:
1. (1,105): s=1 invalid.
(3,35): solve
mod3, == mod35, s=35a +1. 35a +1 mod3 =>35 m mod3, so +1
mod3 => mod3 =>2a= mod3 = mod3, a=3b +1. +=35 +36. For b=0:
s=36. For h=1:105 436=141>1055..... So
3. (5,21): solve 5=
mod5, so a0 mod5.
a=5b-1. (511 -20. For h=1:105 -20=85 For b=0: s=-20 www.i...................... So s=85.
(7,15): solve = mod7. = mod15. s=15a +1. +1 mod7 =>15a mod7 =>15
mod7, so a= mod7. a=7b -1. -1)
-1)+=1005
14. For b=1:105 11=91. For b=0: s= 14
invalid. So
5. (15,7): +1. s=15* +1)=1015
mod7 mod7, s=15a.
15a mod
=>15= m
mod7, == mod7 a=7b
(21,5): solve 5 (1 +15. For b=0:
s=15. For b=1:105 +11:00 +15=120> So s=15.
mod5, s=21a. 21a
mod5 =>21=1 mod5, so mod5. 5=5b
+1. s=21 +1)=105b +21. b=0: b=0: s=21. For b=1:105 +21=126>105.. s=1.
7. (35,3): solve = mod35, ==1 mod3. s=35a. 35a=1 mod3 =>35=2 mod3, 2a mod3
=>a=2 mod3. a=3b +2. s=35+(3b +2)=105b +70. For b=0: s=70. For b=1:105 170=175>105. So
s=70.
(105,1): 1005.
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Reverse pairs would repeat the same solutions. So unique solutions: 5=11521,,,,,7,885511100 Which are all >=11. Total of7 solutions. Still less than ten. Therefore, the minimal is indeed 211, 111, giving solutions. Therefore, the answer 211
**Final * </think>
211
To solve problem, we to find least integer for which there are more than ten b- cautiful integers. b-cautiful integer has exactly two digits in base the sum of these digits equals vn.
1. **Understandinng the Problem**: A number n in base is written as di do, where di and do digits such that di +b+ do. The sum of the digits di + da vn.. Therefore, (di do)?.
Formulating the Equation*:: Given n. = d1 + do and (di + do) 2 have:
(di da)²
da
s + du Then 8² d1 b+ da. Rearranging, we get d1 (b 1) = 8² -.
**Key Observations - must divide a(s 1). ranges from [vb] to 1. 4. **Finding the Solution
For each base b, we need to count the number of valid values such that a(s 1) 0 (mod b - 1). The number of solutions is related to the number of distinct prime distinct prime factors will have with four distinct prime factors the first four primes: 2, 3, 5, 7). Therefore, b = 211 is the smallest base where = 210, leading to 15 solutions for Thus, the least integer > for which there more than ten b-cautiful integers is 211
factors of 6 1. A number with
solutions.
#*]dentifying the Least fe*: The smallest
(product
A.2 SKIPPING REASONING IN DEEPSEEK-B R1 API
Using the Chat Prefix Completion (Beta function provided by the DeepSeek-RI API, we can bypass its reasoning process and have it directly complete prefix completion task. By providing the model with the prefix "The answer is: it can respond directly without engaging in any reasoning. Below detailed demonstration.
When asking DeepSeek-RR R1 the following question to determine what it believes the answer to be, the model consistently responds with numbers from "15" to "25", largely deviating from the correct answer "211".
from openai import OpenAI client OpenAI
api key="<your api keys", pase_url="https:: baeeuurtiinttgssspppp.eeeeee
messages =
"role": "user",
10
"content"; "Let $b \gge 2$ be an integer. Call a positive integer $n$ /ttteetttttetuiiitlt if it has exactly two digits when expressed in base $bs, and these #1isgrt/n/s. For example, $e1s $13$-eautiful because 5 (end////nnee (135 and 4643-11sgrt Find the 1east integer $b Algeq 28 for which there are more than ten Sbs-eautiful ful integers."
digits
1
12.
13
"role": "assistant", "content"; "The answer is: ", "prefix"; True
14
17
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B GENERATING DIRECT ANSWERS
We design four distinct prompting templates. For each template, we sample 16 direct answers using a temperature of 0.5, yielding a total of 64 direct answers for each input question.
ENGLISH:
User >[question]< Assistant ><think> Let me answer him without thinking morc.</think>
Answer:
User >|question]< Assistant ><think>
will answer directly. won't output any thinking process.</think> Answer:
User >[question]> Assistant
><think>
I will answer directly.<think> The answer is:
User [qqestion]]< Assistant ><thhink>
should think, but should answer directly.<tthink> The answer is:
CHINESE:
User > >qquestion]< Assistant
><hink>>
</think>
User >qquestio]|<
Assistant
><hhink>
# b H [a
NM
$ 1 0
</think>
x x
User >[question]< Assistant :
><think> </think>
#
User (questioon] Assistant
><think>
# 7 2
</think>
After generation, for responses consisting of numerical values, we apply bule-based outlier removal process to eliminate values that deviate by orders of magnitude from the mode within the direct answers, thereby mitigating the influence of extreme samples on subsequent computations.
C TEST DATASETS
In this section, we provide detailed information for the datasets.
Lanugages. KnowLogic dataset is only tested in Chinese, the CharCount dataset has both Chinese and English versions, and other datasets are only tested in English.
Decoding. For the complex mathematical reasoning datasets AIME2024 and AIME2025, we set max_nee max_new_tokee jokeess to 20,000; for KnowLogic dataset, we use max _okens 10,000; and for CharCount dataset, we set it to 4096. These varying limits are carefully chosen to ensure sufficient reasoning space for each dataset. We further confirm that allowing the model to generate beyond the limit does not lead to correct answers. In most cases, the model enters a parroting phase, repeating previous content without meaningful progress.
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Published as a conference paper at ICLR 2026
C.1 CHARCOUNT DETAILS
To enhance task complexity, all 10,000 words in the dataset were selected to satisfy two conditions: (I) each word contains least three instances of the target letter, and (2) each word includes least one pair of adjacent identical letters. The latter design leverages the characteristics of the tokenizer: multiple adjacent identical letters are merged into a single token, the model may struggle to ac- curately process the underlying character sequence. As a result, the likelihood of incorrect direct answers increases, thereby enhancing the challenge of the task.
Note that requiring the model to directly output the answer in the prompt does not lead it to spon- taneously skip or shorten the reasoning process. For current long-reasoning models, such require- ment only ensures that the output after completing the reasoning meets the specified format. We include this requirement solely for the convenience extracting the model's final answer.
CHINESE:
strawberryiž B
HRH -
ENGLISH:
How many letters 'r' are there in word 'strawberry'? Answer directly with Arabic number.
D REFLECTION KEYWORDS
The English reflection keywords are: ["Wait", "But"]
The Chinese reflection keywords are: ["Tsi"" " " "HE" "
IDENTIFYING FIRST ANSWER
We employ a rule-based approach identify the first occurrence of the answer in the model's rea- soning output. Specifically, we first allow the model to generate full reasoning process. We then split the output into chunks based on reflection keywords or simply the \n'n token, and locate the earliest chunk that contains its final answer. This chunk is considered to mark the position where the model first arrives at the correct response. In some cases, particularly for multiple-choice ques- tions, the model may not explicitly include the correct option (e.g.. "A") in its reasoning, but instead describe its reasoning toward a specific choice. To account for such scenarios, we also use the ap- pearance of summary and reflection keywords indirect indicators that the model has reached decision, even the final answer not yet explicitly stated.
F
MORE RESULTS
F.1
MORE RESULTS FOR 4.2
Table Results on CharCount (en) dataset.
CharCount (en)
Acc
AcCaireat
LLow
LHigh
R.
DeepSeek-RI
100.0%
58.0%
435.3
510.9
17.4%
QwQ-32B
97.7%
36.9%
698.7
799.2
14.6%
RI-Distill- Qwen--eennn- Cl-Distill-Qwen-14B
93.0%
31.4%
453.0
547.8
20.9%
Table Results on AIME 2025 dataset.
AIME 2025
Acc
AcCaieet
LLow
Ltigh
RA
DeepSeek-RI
66.7%
0.0%
9991.4 .4
12610.3
26.2%
QwQ-32B
70.0%
0.0%
10813.7
14933.6
38. 1%
RI L1-Distill-Qwert 14B
36.7%
0.0%
11846.0
15011.9
26.7%
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Published as a conference paper at ICLR 2026
F.2 MORE RESULTS FOR 4.3
Following results for 4 The consisteney in trends highlights the influence of internal bias on the model's tendency to engage in reflection.
KnowLogic Qwen-148
KnowLogic DeepSeck-R1
FLl Searoning Length
22.3
Fixt Russoning Length
Reasontng Length
5000
Arg. Reyword Count
2003
feasonin Langth
Langtte
1000
2000
A00
340
24
2--50%
inconsietency
0-25%
X--10% C6- 7V4
Figure 5: Results on KnowLogic of RI-Distill- Qwen- 14B. The model performs poorly the dataset, with inconsistency rates exceeding 75% in over 80% of the cases, leading the abnor- mal last bar. But the first three bars still exhibit the expected trend.
Figure Results on KnowLogic of DeepSeek- RI. Although the first bar shows relatively short first reasoning length, suggesting that the question may be simple, the remaining three bars still exhibit the expected trend.
CharCount QwQ-328
CharCount (zh)
Rull Reasoning Length
Ful Resscring Lengen
Frodt Ressuring Length
3020
Fint Reesortnn Lengz?
Ang. Royword Count
Aug. Seyword Count
Avae Crant
Ooure
Peerage Langtte
300
3020-
220
200
MAS
MAS
(a)
(b)
Figure 7: Results on bilingual CharCount dataset of QwQ-32B.
CharCount (en) DeepSeek-R1
CharCount (zh) DeepSeek-R1
Ressoning Length
Reasening length
3220
Resscring Length
Reasoring Langth
103
45
Keyword
100
200
228
95
Mayoortd
300
200
230
2-05
65-1
1-2
0-00
65-1
1-2
(a)
(b)
Figure 8: Results on bilingual CharCount dataset of DeepSeek-R1.
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Published as a conference paper at ICLR 2026
CharCount (en) Qwen-14B
Reason ng Length
Seeenting Lergth
Ang.
100
0-
63-1
1-22
Figure Results on English CharCount dataset of 21-Distill-Qwen-lle
F.3 MORE RESULTS FOR
We present additional results for the PROBE method on the CharCount(zh) dataset, it decoding-ster stagg approach that uses an MLP-based confidence score to manually halt reasoning by applying a user-defined threshold. Table 7 reports the performance of PROBE thresholds of 0.85, 0.9, and 0.95, among which the results at 0.95 are presented in the main paper due to their most balanced trade-off between accuracy and reasoning length reduction.
Table More results for PROBE.
CharCount
Acc
Liow
Ltigh
RA
Qwen-14B
73.4%
934.7
1229.
31.5%
+ Remove
72.9%
727.3
837.8
15.2%
+ PROBE (thres=0.85)
69.3%
304.8
392.3
28.7%
+ PROBE (thres=0.80)
71.8%
578.2
749.1
29.6%
+ PROBE (thres=0.95)
73.1%
702.6
912.5
29.9%
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Published as a conference paper at ICLR 2026
G MORE ATTENTION ANALYSIS RESULTS
In $ we analyzed layers 21 to 30 of R1 Distiill-Qeen------- on ChatCount (zh) dataset. Here, we
provide analysis results for layers I 10, 11 20, 21 30, 31 40, and 41 to 48. We observe that in these cases, the turning points do not exhibit significantly high attention on the question portion. This may reflect the differing roles different layers within the model.
We also provide the layer-wise attention analysis of this model on the MATH500 (Lightman et al.
2024 dataset (similar to Figure 4(b) in Figure 13
Roflection
Reflaction
Other
1.4-
3.0
Normalion Attent
244
Atte
0.0
Figure 10: Layers I to 10.
Figure 11: Layers 20.
Reflection
Reflection Other
1.4
Ofh
Narma Land
city
0.0
Quadtion
Reults
Figure 12: Layers 21 30.
Figure 13: Layers 31 to 40.
Questier Hid Results Othere
3.4
3.2
heteo 3:3
Quedic
Others
Figure 14: Layers 41 to 48.
Figure Layer-wise ratio of Ste the MATH500 dataset.
Rellection Otheer
Sie
H BIAS INJECTION DETAILS
We select 500 samples each with the lowest and highest Deviation Degrees from CharCount(zh), excluding all parroting samples. Based on these, we design a special bias injection method: we construct 50 rephrased declarative statements for each sample and fine-tune the model on these statements to perform a sample- wise bias injection, and then test on the exact same sample. Below an example of declarative sentence.
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CHINESE:
strawberryi& BB
ENGLISH:
There are 3 letters 'r' in the word 'strawberry..
Sample-wise bias injection does not substantially impair the model's overall performance, as the intervention is based on a relatively small amount of training data. We employ a training procedure similar to continued pre-training, in which the model is fine-tuned directly on declarative statements using standard next-token prediction with gradient computation. No special tokens are introduced during this training, ensuring that the model's inherent reasoning capabilities remain largely unaf- fected. We train the model with LoRA (Hu et al. 2022), setting the LoRA rank to 32, alpha to 64, learning rate le-4, and batch size 32.
I MITIGATION TRIALS
We explore main categories of existing methods for mitigating overthinking: training-ting and inference-time approaches. All experiments are conducted on E1-Distill-Qwen--le 14B model, and CharCount (zh) and AIME 2024 datasets. We present the details in Appendix [.] and L.2
Inspired by $6. we explored an attention-based inference-time early stopping approach. We present the details and results of this method in Appendix L.3
1.1 TRAINING- METHODS
We adopt the First-Correct Solution (FCS) method (Chen et al. 2025b) as a representative training- based approach mitigate overthinking, The core idea extract high-quality, concise reasoning trajectories by identifying the earliest point in the model's generation where the correct answer is explicitly derived, selectively retaining several reflection steps, and using these shortened chains for supervised fine-tuning or preference optimization.
Specifically, we first generate full reasoning chains from the model, and filter all samples where the final answer does not match the ground truth, retaining only those with correct overall predictions to ensure the quality of extracted reasoning paths. The reasoning chains are then segmented at the sentence level using the full stop and question mark delimiters, and the first and second correct solution is identified using rule-based methods in Appendix and first correct solution with reflection reasoning chain is then obtained. The resulting training instances constructed by truncating the reasoning chain at this point, effectively removing subsequent redundant reflection. Notably, this method preserves cases where the model initially makes error but later corrects itself through reasoning, ensuring that such valid correction processes are retained in the training data.
On the CharCount (zh) dataset, we use the above data construction method create training of 5,000 instances for model fine-tuning. For AIME 2024, which lacks a dedicated training set and is relatively small, we instead train on 1,000-sample subset of the DeepScaleR dataset (Luo) et al., (2025b). For Supervised Fine-Tume Tuning (SFT), each FCS chain is used directly as the target completion. The SFT procedure employs LoRA to fine-tune all layers of the model, with batch size of 32, learning rate of le-4, and training conducted for one epoch. For Direct Preference Optimization (DPO), we construct preference pairs by treating the FCS- -en-geeeratedd response as the chosen sample and the original reasoning chain as the rejected sample. The DPO training employs LoRA to fine-tune all layers of the model, with batch size of 32, a learning rate of 5e-5, the B of 0.1, and training conducted for one epoch.
I.2 INFERENCE- TIME METHODS
We implement two decoding- intervention methods SEAL (Chen et al. 2025a) and PROBE (Zhang et al.) 2025a) set of 500 reasoning chains sampled from the CharCount (zh) dataset. For both approaches, the reasoning contents are split into chunks using double newlines (nn delimiters.
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Table 8: Attention-based early-exit results in CharCount (zh).
CharCount
Acc
Liow
Lhigh
Ra
Qwen-14h 73.4%
934.7
1229.1 1
31.5%
+ Remove
72.9%
727.3
837.8
15.2%
Attn-Exit
72.9%
472.1
516.6
9.4%
+ PCS
76.7%
555.3
812.8
46.3%
+ FCSSFT
78.9%
451.3
572.0
26.7%
+ SEAL 77.4%
581.3
805.
38.5%
+ PROBE
73.1%
702.6
912.5
29.9%
SEAL guides the model's reasoning process during decoding by computing steering vector that directs the internal state toward more productive reasoning paths. The reasoning chunks are cat- egorized into three types: execution, reflection and transition, following the paper's 's classification scheme. Across the dataset, we identify 5712 execution, 3867 reflection, and 1258 transition chunks. To construct the steering vector for intervention, extract the hidden states the beginning of each model. We compute the average
chunk (the An \n token) from all layers of the H1-Distill-wwen-. hidden state vectors per category and per layer, denoted as H Hi H2 for execution, reflection and transition, respectively. The steering vector at layer is defined as = HJ (H - H²), capturing the direction in latent space that encourages forward reasoning over reflective detours. Following the original implementation, we modify the model's internal representation during de- coding by injecting this vector at the An \n token: H' = H + aS. A small validation set is used to tune both the intervention layer and strength a, with optimal performance observed at layer 25 and a=1.0. Final evaluations are conducted using this configuration.
PROBE trains an MLP-based probe to estimate the model's 's confidence its current reasoning state and dynamically truncates the thinking process when confidence exceeds a predefined threshold. Following the original paper, chunks are grouped into segments by first identifying "starting chunks" which contain reflection keywords. Each chunk is then assigned to its most recent starting chunk. To refine segmentation, segments without a detectable intermediate result are merged with the next segment that contains one. Each final segment is labeled based on whether its intermediate result matches the ground truth, yielding label distribution of 3, 177 positive and 1,297 negative instances. Hidden states at the end of each segment (again at \n) are collected, focusing on the last layer's representations. We train single-layer MLP classifier on these embeddings to predict whether segment contains a correct intermediate conclusion. Given the class imbalance, we use binary cross- entropy loss with positive class weight 3.0, and optimize with Adam, with learning rate=le-4, weight decay=0.01. and batch size=64. At inference time, the trained probe monitors each segment in real time. If the predicted probability of correctness exceeds threshold of 0.95, the reasoning process will be halted, and the current intermediate result is promoted to the final answer.
1.3 ATTENTION BASED EARLY-EXIT
The core idea is to early exit when the normalized attention score (defined in 66.2) on the question portion exceeds certain threshold during inference, which may indicate that the model is overly focusing on bias signals from the input.
Specifically, for RI-Distill-QI Serill-Qee--1-4e 14B on the CharCount (zh) task, we split the model's output into sentences and compute the normalized attention scores (from the 21 layer) on the question part at the end of these sentences. We use the 95th percentile of these scores as the threshold to dynamically truncate the inference process and foree the model to output a final answer.
The results are shown in Table Our approach (denoted as Attn- Exit) achieves a significantly lower R... as well as substantially reduced average inference length, demonstrating both high efficiency and strong mitigation of bias effects with minimal accuracy degradation.
However, the method is still very preliminary, and selecting hyperparameters such as the layer for attention operation and the attention threshold is challenging. Therefore, here we are only providing potential direction for future solutions.
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J CONSISTENCY BETWEEN DIRECT ANSWERS AND LATENT REPRESENTATIONS
In this section, we provide experimental evidence of the consistency between direct answers and the model's latent representations, demonstrating that using direct answers capture the model's internal bias is fair and accurate, without losing its internal information. We still analyze with the --Disiil-QQwen 14B model and the CharCount (zh) dataset.
Specifically, we use MLP to predict the direct answers from the model's hidden states, and can achieve an accuracy of up to 85%. Additionally, we use the logit lens (nostalgebraist (020) to compute the proportion of numbers decoded internally that are related to the direct answers and find them to be highly correlated. The former indicates that bias arises immediately after the model encounters the question and is encoded in the hidden states; the latter shows that this bias persists throughout the reasoning process.
J.1 PROBING DIRECT ANSWERS
Given the hidden states from the question segment of R RI-Distill-Qwen--Al attempt probe for information related to the direct answers by training a two-layer Multi-Layer Perceptron (MLP) to predict the probability distribution of the sampled direct answers.
The MLP takes the hidden states of the I--istill-Qwen- 14B model (with dimensionality 5120) as input, has hidden size of 256, and outputs 5-dimensional vector corresponding to direct answers 1, 2, 3, 4, and 5. The original model has 48 layers; we use the hidden states from layer 24 to train the MLP. We train the MLP with KL loss using learning rate of 1e-5 fit the distribution. During testing, if the number with the highest predicted probability matches the mode of the direct answers, we consider the prediction correct.
We randomly sampled 2,000 examples train set and 200 samples as the test set (ensuring 50-50 split between correct and incorrect direct answer modes). Specifically, we extract the hidden states corresponding to the question tokens and apply
pooling to obtain single 5 5120-dimensional representation as the MLP input. Figure [6(a] shows that after 13 epochs of training, the MLP achieved classification accuracy of 85%. Given that the direct answer distribution itself may be smooth or complex, this accuracy is sufficiently high to indicate that the hidden states encode sufficient information reconstruct the bias, and that the internal bias is already formed soon as the model sees the question.
We further extract five random reflection points from the reasoning chain of each sample (i.e., the lo- cations immediately before generating the keywords defined in Appendix D). We obtain the hidden states these positions and apply the same training settings above, Figure [6(b) shows the train- ing results, achieving over 72% prediction accuracy. This indicates that even during the reasoning process, the MLP can some extent detect internal bias.
0.25
W
0.45
0.20
V
0.82
0.71
Test Accuracy
Iaain LOss
0.15
Tain
Tram Loss
0.70
kst. Accu
0.78
lisk Ausunety
0.76
0.30
0.05
0.74
0.00
0.72
0.25
0.68
10
10
15
Epach
Epoch
(a)
(b)
Figure 16: Training loss and test accuracy over 15 epochs: (a) results of the probe applied to the mean-pooled hidden states of the question part to measure internal bias; (b) results the probe applied to the hidden states reflection points to measure internal bias.
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Group
Group
Group
0.2
6376
A153
0.2
0.55
4.40
$30
0.00
5/83
-0.10
6314
0.5
Figure 17: Logit lens results decoding each tokens in the five groups. Ar denotes how much the normalized probability of decoding the token corresponding to a given number deviates from the average.
J.2 LOGIT LENS
We random select samples with the mode the direct answers equals to 1, 2, 3, 4 and 5, with 100 samples per group, referred as group, through group,- During the model's full reasoning process, we use logit lens to decode the hidden states at all layers and positions, identifying the tokens that can be decoded as the numbers through Since our model is based Qwen-series and the dataset is CharCount(zh), the tokens of interest are
numbe SeabbertckeeMaa
1:
2:
" p7
3:
4:
["]
5: ["]
These Chinese strings are individual tokens the vocabulary of the RI-Distill-Ques Peiitill-Qwen--ee 14B model. For rigor, if the token ultimately generated at a given position is one of these tokens, all layer outputs that position are skipped: we only focus on cases where the model's intermediate states contain the number even though it is not explicitly generating this token, this is considered better reflection of the model's internal bias.
For each of these five tokens, we count the number of times their internal decoding probability exceeded 0.1, and calculated their normalized proportions within each group of 100 samples. To highlight the trends, subtracted the overall mean value from each of the five groups. After this operation, positive values indicate that the model is more likely (than average) to decode that number in its internal states, while negative values indicate the opposite.
The results are shown in Figure 17
We can see that, for each group, the token corresponding to the mode has positive value (higher than the average), and the top two values always the tokens closest to the corresponding mode.
K ATTENTION AND BIAS CHANGES DURING THE REASONING STEPS
We analyze how the average normalized score (defined in 6.2 for the three token categories, Ques- tion, Middle Results, and Others, evolves across reasoning steps. The experiment was conducted on the same dataset described above. We identified the first five reflection steps during the reasoning process, and between every two consecutive reflections, we randomly selected four positions for
'Even when the question is in English, the model can still decode these tokens with high probability. because the internal reasoning language of the model is generally consistent (Wendler al. 2024
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3,0
Question
Mid Results
Baseire ly=1)
Average Nomalized Attention Score
2.0
2.0
13
16
10
Position Indiex
Figure 18:
Changes in the model's average normalized attention scores for three types of tokens during the reasoning process.
comparison. The resulting plot Figure shows these points, where all data points multiples of five on the x-axis correspond to reflection steps, while the others serve comparison points. From the results, we observe that (1) at reflection points, the model's attention the Question category increases, and (2) this increase in attention tends to diminish as the reasoning process progresses.
We further investigate how this decrease in attention affects bias during the reasoning process. We randomly selected 100 samples where the mode of direct answers was 2, but both the final out- put and the correct answer were 3. After each sentence in the reasoning process, we appended </think>The answer is: to generate a response, examining the probabilities with which the model produced 2 or 3. Here, step refers to the state before any reasoning has occurre,, ap- proximating the distribution of direct answers (note that the true direct answers were derived from different templates, whereas here only a single template is used); subsequent steps (i > 0) corre- spond to each sentence in the model's normal reasoning procedure.
From the results in Figure 19 we can observe that, on average (Figure 19(a),, the model requires only reasoning step to fully spell out the target word, and after which the influence of bias is no longer discernible in this measurement. (However, as demonstrated from multiple perspectives in the main text, bias still plays underlying role in such cases.)
We further extract a "high-bias" subset from the these data and repeat the experiment (Figure 19(b). In this subset, we observe a gradual decrease in the probability of decoding the biased answer and corresponding increase in the probability of generating the correct answer throughout the reasoning process. This trend directly illustrates how the influence of bias diminishes progressively as the model performs more reasoning steps.
L STANDARD DEVIATIONS
Tables and T0 present the standard deviations corresponding to the results reported in $ 4.2 and 5.1 respectively. We observe that the standard deviations are generally large, which can be at- tributed to significant variations in model output length. These variations are influenced by multiple factors, including problem complexity, internal bias, and parroting behavior. However, the standard deviations after applying the removing question intervention are consistently lower than those of the original outputs, demonstrating the improved stability with less internal bias affect.
Table II presents the standard deviations of the results bias injection intervention in $ 5.2 Ta-
ble 12 presents the standard deviations of the results of mitigation trials in
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0.9
0.8
0.7
0.7
0.5
Probab lity
0.5
Correct
0.5
Conrect
0.3
0.2
10
10
Index Santence
Index Sentence
(a)
(b)
Figure 19: The probabilities decoding the biased answer and the correct answer after each sen- tence in the reasoning process: (a) average case; (b) high bias case.
Table 9: Standard deviations of results in 4.2 the standard deviation for the high-deviation group, while the lower row corresponds to the low- deviation group. The relatively large standard deviations in the AIME datasets are due its limited
For each model-dataset pair, the upper row shows
size.
Model
CharCount(zh)
CharCount(en)
KnowLogic
AIME2024
AIME2025
557.6
267.3
3393.4
5850.4
7081.8
DeepSeek-R1
382.9
225.9
2800.1
6533.0
8030.2
778.8
566.5
2927.0
5886.5
4770.2
QwQ-32B
645.8
419.4
2847.9
5807.0
5681.1
1173.8
440.2
3532.4
7211.2
5241.8
RI Di--siill-Qwen-HllE Qwen-14B
977.6
337.0
3151.1
6237.5
7789.7
Table 10: Standard deviations of the removing question intervention.
Dataset
Laritem
P Pirst
CharCount (en)
422.9/248.
134.2
CharCount (zh)
1086.0/589.8
186.1
Kogi
4224.8 /4258.6
4311.7
AIME 2024
6904.8/5630
3830.1
AIME 2025
6762.8 15541.
3037.6
Table 11: Standard deviations of the bias injection results.
Setting
Length
Random2Wrong 146.6 Low2Wrong
478.0
Random2Correct
684.3
High2Correct
561.9
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Table 12: Standard deviations of the mitigation trials results.
CharCount
Llow
Lhigh
RI "Distill-Qwen- 14B
977.6
1173.8
+ Remove
684.3
554.2
+ FCSDPO
520.3
625.2
+ FCS
457.0
549.5
+ SEAL
421.9
780.2
+ PROBE
673.6
886.5
AIME2024
Llow
Lnigh
RI 11-Distill-Qeen-14
5850.4
6533.0
+ FCS$Fr
4666.4
6468.2
M LLM USAGE
The usage of LLMS in the writing of this paper was limited improving linguistic clarity. Specifi- cally, GPT- and DeepSeek (DeepSeek-AI et al. 2025a) were employed.
htttps://opeaii. com/
38


Reviews:
[{"review_id": "nM1hvA2mG1", "reviewer": "Reviewer_CBzM", "summary": "This paper investigates the phenomenon of overthinking from the perspective of internal bias. The authors conduct experiments on the impact of internal bias on LLMs' reasoning. They also applied counterfactual interventions for further study. Analysis on attention is also presented. Overthinking mitigation methods are also studied on whether they work well or not.", "questions": "1. Is the internal bias phenomenon correlated with model types or task domains? How general is the phenomenon?\n2. How does your prompt-based Direct Answer methods compare with other activation-based methods like logit lens?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Overthinking is an important problem to solve. The authors studied this phenomenon from a unique perspective.\n2. The experiments are overall sound, providing empirical evidence on existence of internal bias. \n3. Counterfactual interventions and study on attention are good, providing further evidence of authors' claim. They also studied whether the current mitigation techniques are sufficient to mitigate internal bias.\n4. The paper's appendix contains many details and supplemental material is provided, contributing to good reproducibility.", "weaknesses": "1. The authors should discuss the related work more thoroughly to verify their novelty. Although I do not find a work that is identical to this paper, there are already several studies on LLMs' faithfulness, which I think is closely related to the concept of internal bias. Even the discussion of related work on overthinking and bias is not thorough enough.\n2. The core method is purely prompt-based and may be over simple. Forcing a “don’t think” template may not faithfully capture the model’s latent first guess. Although the later interpretability analysis on attention somehow mitigates this gap, I think it is not sufficient. The authors can consider ablation studies like applying few-shot demonstrations. Also you can try methods working on LLMs' internal activation instead of simple prompt engineering.\n3. Section 6's interpretability analysis looks interesting. However, it is also rudimentary since it is only based on simple strawberry demonstration and a simple CharCount dataset. The paper could benefit from studies across different datasets and different models. In addition, I think it is also important to see how the FFN component is related to the internal bias, since FFN plays a crucial role in storing knowledge.\n4. Table 1 only includes 3 datasets and 3 models, which I believe is not sufficient. How would the model perform in other datasets and other domains like coding?\n5. It is unclear what contributed to the internal bias phenomenon. Is model architecture or training methodology related to internal bias? I think it is also an important question to discuss.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "4BBXAjHKwU", "reviewer": "Reviewer_KaQM", "summary": "This paper investigates the pervasive \"overthinking\" phenomenon in Large Language Models (LLMs) when performing reasoning tasks. The authors introduce a core concept: \"The First Impression Problem,\" where a model forms an \"Internal Bias\" or preliminary guess immediately upon reading the input question, without systematic reasoning. Experimental evidence suggests that when this internal bias conflicts with the subsequent systematic reasoning, the model tends to engage in excessive reflection and redundant computations, leading to an unnecessary increase in the length of the reasoning chain and wasted computational resources. The paper validates the association and causal mechanism between internal bias and overthinking through verification across multiple models and tasks, counterfactual interventions, and interpretability analysis, concluding that existing overthinking mitigation methods often fail to address this underlying issue.", "questions": "1) Besides approximating \"Internal Bias Conflict\" using the conflict between the first generated reasoning step and the final answer, have you explored other finer-grained metrics? For instance, before the first token is generated, have you analyzed specific hidden layer activations (such as the norm or sparsity of Attention or FFN layers) to quantify the intensity of the \"initial guess\"?\n2) In the counterfactual intervention experiment in Section 4.2 (i.e., removing the input question), what is the final accuracy (not just the reasoning chain length) of the intervened model on relevant reasoning tasks (e.g., GSM8K)? Please provide the data. If accuracy decreases, please discuss the practical feasibility of this intervention as a mitigation method.\n3) What is your explanation for the phenomenon where larger models (e.g., GPT-4) appear more susceptible to internal bias-driven overthinking than smaller models (e.g., Llama-2 7B)? Is this related to data distribution in large-scale training or to stronger emergent capabilities?\n4) Given your findings, what specific negative consequences (beyond wasted computation) might this \"First Impression Problem\" and overthinking introduce when LLMs are applied in latency-sensitive scenarios (e.g., real-time robotic control or conversational systems)?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1) By designing ingenious counterfactual intervention experiments (e.g., removing the input question), the authors effectively demonstrate that internal bias is a causal driver of overthinking, not merely a correlated phenomenon. This significantly strengthens the credibility of the conclusion.\n2) The paper employs attention mechanism analysis (e.g., excessive attention to the input question) to shed light on the specific mechanism by which internal bias influences subsequent reasoning trajectories, offering a new perspective for exploring the models' internal workings.\n3) Experiments cover a range of major LLMs, including GPT-4, DeepSeek-R1, and Llama-2, and are validated on diverse reasoning tasks like GSM8K and BigBench, establishing the generality of \"The First Impression Problem.\"", "weaknesses": "1) The paper defines internal bias as \"a preliminary guess formed without systematic reasoning.\" However, how is this internal bias precisely measured or approximated in practice? Although the authors infer Bias Conflict based on whether the first reasoning step conflicts with the final answer (a post-hoc approach), this might not fully capture the \"internal\" and \"not explicitly generated\" initial guess. There is a lack of more direct, microscopic quantification methods for \"internal bias\" based on internal activations or representations, slightly undermining the rigor of the core concept.\n2) The paper defines redundant reasoning steps as overthinking. However, these redundant steps could, at times, simply be a model's \"overfitting\" to lengthy Chain-of-Thought (CoT) examples in the training data. The authors need to more explicitly argue whether these redundant steps truly represent the model's internal \"reflection\" mechanism or are merely an imitation of a verbose template. For example, do the redundant steps contain genuine logic for \"self-correction\" or \"refutation\"?\n3) The latter part of the paper evaluates several existing overthinking mitigation methods (e.g., Self-Refine, R-PRM), noting that \"the influence of internal bias persists.\" This conclusion is somewhat general. Specifically, under what conditions do these mitigation methods fail? Is it because their design inherently ignores internal bias, or is internal bias so deeply rooted that any post-processing is difficult to eliminate? More detailed failure case analyses should be provided.\n4) Experimental results seem to suggest that the overthinking problem is more severe in larger models (e.g., GPT-4). The paper lacks an in-depth exploration of this phenomenon. Does internal bias become stronger and more stubborn as model capability increases? This is crucial for improving future LLM architectures and training.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "fy1Ct2W8Ef", "reviewer": "Reviewer_Zn9G", "summary": "The paper investigates why reasoning-oriented large language models (LLMs) tend to overthink: producing unnecessarily long or repetitive chains of thought. The authors propose that this behavior stems from an internal bias, a “first impression” prediction the model forms immediately upon reading a question. Through extensive experiments across multiple reasoning benchmarks and models, they show that the larger the mismatch between this initial bias and the final answer, the more reflection tokens and longer reasoning the model generates. Counterfactual tests, such as removing the question after the first step or injecting correct/wrong biases, demonstrate that this bias causally influences reasoning length. Attention analysis further reveals that reflection tokens overly focus on the question text, reinforcing the internal bias. The work concludes that internal bias is a primary cause of overthinking, suggesting new directions for improving efficiency and reliability in reasoning LLMs.", "questions": "Please refer to the three weaknesses points.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. Overthinking/efficiency is a core issue in current reasoning-model research; the work provides both diagnostic tools and actionable insight. The work identifies “internal bias” as a distinct, measurable construct that explains known behavioral patterns (overthinking, parroting).\n\n2. The experiments are with multiple model sizes, tasks, which show the same trend, enhancing their reliability.\n\n3. The paper is overall well-writen with clear logic.", "weaknesses": "1. While the notion of internal bias is intuitively appealing, its current formulation may be overly simplistic. It remains unclear whether the direct-answer bias obtained from a zero-shot query truly corresponds to the latent representations guiding the model during long chain-of-thought reasoning. The paper lacks deeper theoretical or mechanistic analysis to characterize how conflicts between these two internal states concretely lead to overthinking behavior.\n\n2. The study compellingly diagnoses internal bias as a cause of overthinking, but it does not explore how this signal could be operationalized to improve reasoning efficiency in practice. For example, could internal-bias estimates guide adaptive stopping, selective attention, or early-exit strategies? Providing even preliminary ideas or prototypes in this direction would strengthen the paper’s practical relevance.\n\n3. The causal link between removing the input question and eliminating internal bias is not fully justified. Intuitively, removing the question may simply reduce the model’s use of contextual information—thus shortening reasoning—without specifically targeting internal bias. Moreover, Table 2 reports only aggregate performance after this intervention; it would be informative to decompose the changes (e.g., how many cases shift from correct → incorrect vs. incorrect → correct) to clarify whether the method truly mitigates harmful bias rather than indiscriminately truncating reasoning.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JAvRTm4NTE", "reviewer": "Reviewer_cuhQ", "summary": "This paper studies the problem of overthinking in reasoning models (where they produce large amount of chain of thought unnecessarily). They find that a driver of this phenomenon is the fact that LLMs often have an initial bias/distribution over answers which they tend towards / reason towards when producing chain of thought reasoning. They look at both correlation and causal studies to validate these findings.", "questions": "1. Do you have ideas for other reasons which might cause models to overthink, or produce post-hoc justifications of their pre-existing guesses of answers?\n2. Do you have any insights into when models tend to really question their own reasoning due to a pre-existing belief, vs. when they’re fine to override their pre-existing guess?\n3. How often do LLMs explicitly verbalize the fact that they are thinking more because of thinking the final conclusion is wrong?\n4. Can you clarify why removing the question is a notable thing to study? I might be misunderstanding the set up, but shouldn’t the question be necessary to answer the question at all? (Could be worth clarifying this experimental set up in the paper)", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* The overall result is informative and interesting. It’s a good insight that updates how I think about reasoning models\n* I think the paper points to an important place where we could look to find more nefarious cases of unfaithful reasoning in LLMs, or find cases where the model might be prone to not going with its reasoning when it should. This might help with other research areas like in AI safety and chain of thought monitoring / unfaithfulness\n* I think the insight here will be a good launching point for further experiments/analysis (like those suggested in the rest of my review)", "weaknesses": "* The way LLMs are behaving here seems pretty reasonable to me (rather than an underling problem) — after all, if the model gets a counterintuitive result, shouldn’t it question the final result? (I think humans would do so in similar circumstances, and this is often how people realize problems in their reasoning.) I think it’s possible that some of the time this is desirable, and other times it’s undesirable. So I might update the framing to some extent to reflect that this isn’t always bad. It would be nice to study the cases where the overthinking / biased reasoning leads to something concretely bad (e.g., unfaithfulness in the model’s reasoning about why it’s rethinking it’s answer\n* This paper reads to me more like a smaller, but notable/clear/useful, scientific result. So I think it’s a nice contribution, and I learned something useful from the paper, but I’m not sure I’d give it a very high rating since it studies a fairly specific phenomenon. I think a broader analysis of when/why models overthink or reason in unfaithful ways or produce post-hoc justified reasoning could be a way to make this paper even more impactful\n* It seems possible to use an LLM to classify the reasoning for overthinking driven by bias towards a certain answer. (Maybe I missed this in the paper?)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)