Mathematics

NaturalReasoning

NaturalReasoning: 2.8M challenging reasoning questions backtranslated from pretraining corpora, spanning mathematics, physics, computer science, and economics/social sciences. Each question carries a reference answer and model-generated responses (Llama-3.3-70B-Instruct).

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Response matrix

Every model, scored item by item.

Each row is an AI model and each column an item, ordered so the strongest models and easiest items gather toward one corner. 1 subjects × 143,048 items, 100% of cells evaluated.

NaturalReasoning response matrix: AI models (rows) against items (columns) — Correct (1)Incorrect (0)Unobserved
Scale: 1 = correct · 0 = incorrect

Sample items

What the questions look like — and how subjects answer.

A spread of items across the difficulty range, each shown with a few subjects' actual answers.

Item 1100% solve rateanswer: A compiler translates a high-level programming language into machine code for a specific processor architecture, and the operating system provides system calls and libraries that the compiler can use to generate machine code. The implications of using a cross-compiler versus a native compiler are that a cross-compiler can generate machine code for a different processor architecture, while a native compiler generates machine code for the same processor architecture as the compiler.

Design a compiler that can translate a high-level programming language into machine code for a specific processor architecture. Discuss the role of the operating system in this process and how it interacts with the compiler. What are the implications of using a cross-compiler versus a native compiler? Provide a detailed example of how a program written in a high-level language would be compiled and executed on a machine with a different operating system.

How subjects answered

Llama-3.3-70B-Instruct correct
Designing a Compiler for High-Level Programming Languages

Overview

A compiler is a software tool that translates high-level programming languages into machine code for a specific processor architecture. The compilation process involves several stages, including lexical analysis, syntax analysis, semantic analysis, intermediate code generation, optimization, and code generation.

Role of the Operating System

The operating system plays a crucial role in the compilation process by providing the necessary resources and services for the compiler to function. These resources include:

Memory management: The operating system …

Item 2100% solve rate

Design a correlational study to investigate the relationship between the use of positive reinforcement techniques in counseling and the reduction of anxiety in group therapy settings.

How subjects answered

Llama-3.3-70B-Instruct correct
Here's a potential design for a correlational study investigating the relationship between the use of positive reinforcement techniques in counseling and the reduction of anxiety in group therapy settings:

Title: Exploring the Relationship between Positive Reinforcement Techniques and Anxiety Reduction in Group Therapy Settings

Research Question: Is there a significant correlation between the use of positive reinforcement techniques in counseling and the reduction of anxiety in group therapy settings?

Hypothesis: It is hypothesized that there will be a negative correlation between the use of positive reinforcement techniques and anxiety levels in group therapy settings, …

Item 3100% solve rateanswer: $\\boxed{f(x_1,\\dots,x_n)=0}$ represents a hypersurface in $\mathbb{R}^n$.

Consider a smooth function $f(x,y) = 0$ representing a curve in $\mathbb{R}^2$ . Explain how the Newton algorithm can be generalized to solve this equation, including the role of the Jacobian matrix in this context. Provide a step-by-step derivation of the Newton-Raphson method for finding a point $(x, y)$ that satisfies $f(x, y) = 0$ , given an initial guess $(x_0, y_0)$ .

How subjects answered

Llama-3.3-70B-Instruct correct
Step 1: Introduction to the Newton-Raphson Method for Multivariable Functions

The Newton-Raphson method is a powerful technique for finding the roots of a real-valued function. For a multivariable function $f(x,y) = 0$ , the goal is to find a point $(x, y)$ that satisfies this equation. The method starts with an initial guess $(x_0, y_0)$ and iteratively improves this guess until it converges to a root.

Step 2: Linear Approximation Using the Jacobian Matrix

The key to the Newton-Raphson method is the linear approximation of $f(x,y)$ around the current estimate $(x_n, y_n)$ . This approximation is given by the Taylor series expansion up to the first order: [f(x_n + \Delta x, y_n + …

Item 4100% solve rateanswer: $\\frac{2GMr}{(a^2+r^2)^{3/2}}$ towards the center of mass.

Compute the magnitude and direction of the gravitational field at a point P on the perpendicular bisector of the line joining two objects of equal mass separated by a distance of 2a.

How subjects answered

Llama-3.3-70B-Instruct correct
Consider two objects each of mass m, separated by a distance 2a. Let P be a point on the perpendicular bisector of the line joining the objects, and let r be the distance of P from each object. The magnitude of the gravitational field at P due to each object is: E = Gm / r2 Since the objects have equal mass, the magnitude of the gravitational field at P due to each object is the same. The direction of the gravitational field at P due to each object is along the line joining the object and point P. Since point P is on the perpendicular bisector of the line joining the objects, the directions of the gravitational fields at P due to each object are equal and opposite. Therefore, the net …

Item 5100% solve rateanswer: The stationary distribution alone does not directly inform the hitting time without additional context.

Consider a biased random walk on the positive integers with a reflecting boundary at the origin, where a +1 step has probability $p$ , a -1 step has probability $q$ , and staying in place has probability $r = 1 - (p+q)$ , with $p \leq q$ . The stationary distribution is given by $P(X=n) = \left(\frac{p}{q}\right)^n \left(1-\frac{p}{q}\right)$ . Initialize the walk at position $0 \leq i < M$ and consider the average hitting time at position $M$ . Discuss how the sum $S = \sum_{n=M}^{\infty} \left(\frac{p}{q}\right)^n \left(1-\frac{p}{q}\right)$ relates to the average hitting time $H(M)$ , and derive an expression for $H(M)$ if possible.

How subjects answered

Llama-3.3-70B-Instruct correct
Step 1: Understand the given problem and the stationary distribution

The problem describes a biased random walk on the positive integers with a reflecting boundary at the origin. The probabilities for a +1 step, a -1 step, and staying in place are given as $p$ , $q$ , and $r = 1 - (p+q)$ respectively, with the condition that $p \leq q$ . The stationary distribution is provided as $P(X=n) = \left(\frac{p}{q}\right)^n \left(1-\frac{p}{q}\right)$ . We need to discuss how the sum $S = \sum_{n=M}^{\infty} \left(\frac{p}{q}\right)^n \left(1-\frac{p}{q}\right)$ relates to the average hitting time $H(M)$ at position $M$ .

Step 2: Analyze the sum $S$ and its relation to the stationary …

Item 6100% solve rate

Consider a turbulent flow through a rifled pipe. Analyze how the rifling affects the friction between the fluid and the pipe wall, considering the Coriolis effect and the potential increase in surface area due to rifling. Discuss the conditions under which rifling could reduce friction and improve fluid flow, referencing relevant experimental studies and theoretical principles.

How subjects answered

Llama-3.3-70B-Instruct correct
Rifling a pipe involves creating a spiral groove on the inner surface of the pipe, which can significantly impact the flow characteristics of fluids, particularly in turbulent regimes. The effects of rifling on friction between the fluid and the pipe wall are multifaceted, involving the Coriolis effect, increased surface area, and alterations in flow patterns.

Theoretical Principles

Coriolis Effect: In a rifled pipe, the Coriolis force acts on the fluid due to its rotation as it moves through the spiral groove. This force can potentially organize the turbulent flow into more coherent structures, reducing the mixing and interaction with the wall, which might decrease the wall …

Item 7100% solve rateanswer: The relative error for $256.786$ represented as $2.568 \cdot 10^2$ is $0.00545\%$, and for $256750000$ represented as $2.568 \cdot 10^8$ is $0.0194\%$. Therefore, the final answer is: $\\boxed{0.0194\%}$.

Given a five-place floating-point register, discuss how the relative error in representing numbers can vary, using the examples of $256.786$ and $256750000$ . Explain why the order of relative error might not be the same for every number, considering the representation of numbers in decimal or binary form and the impact of the leading digit on relative error.

How subjects answered

Llama-3.3-70B-Instruct correct
Step 1: Understand the representation of numbers in a five-place floating-point register

In a five-place floating-point register, numbers are represented with a leading digit (which can be any non-zero digit) followed by four more digits that represent the fractional part or the digits after the leading digit. This means the first digit (most significant digit) is always non-zero, and the rest are the mantissa.

Step 2: Analyze the representation of $256.786$

For $256.786$ , when represented in a five-place floating-point format, it would be something like $2.5679 \times 10^2$ , assuming the format allows for a power of ten to be specified (though the exact method of representing the …

Item 8100% solve rateanswer: -50t^2 + 200t - 100

A new set of automatic sliding doors at the entrance to a supermarket is being designed. The doors will consist of a pair of 100cm wide glass panels which are programmed to slide open in opposite directions when a sensor is triggered. The panels are identical except for the direction in which they move. For the purposes of this question, you will consider the motion of the right-hand panel. Let y be the position, measured in centimeters, of the left edge of the panel relative to its position when closed. The position is modeled as a function of time, t, in seconds. (i) It is proposed that the doors should be programmed to open in two stages with the motion of the glass panel described by the following two equations: stage position time period 1 50t 0 < t < 1 2 y= -50t^2 + 200t - 100 1 < t < 2 (a) For each stage of motion, plot the function on your calculator, using an appropriate window for the given time interval. Hence sketch the position-time graph over the range 0 to 2 seconds, adding appropriate labeling and indicating each stage of motion clearly. (b) For each stage of motion, plot the rate of change of the function. Hence sketch the velocity-time graph over the range 0 to 2 seconds, labeling it clearly. (c) Copy and complete Table 2, by finding the position, velocity, and acceleration of the panel at the times listed. Table 2 Stage time/ position/ Velocity/ Acceleration seconds cm cm s^-1 cm s^-2 1 0 1 0.5 2 1.5 2 2 (ii) After some discussion, the managers of the supermarket decide that the sliding doors should be programmed differently. They would like the motion of the panels to be described by an equation of the form y = At^2 + Bt + C for 0 < t < 2. The following features are also required: at time t = 0, the position should be y = 0, and the parabola described by the equation should have a vertex at (t, y) = (1, 100). (a) Find the values of A, B, and C that satisfy these requirements.

How subjects answered

Llama-3.3-70B-Instruct correct
Step 1: Understand the given problem and identify the requirements for the motion of the glass panel in the first scenario.

The problem describes the motion of a glass panel in two stages, with given equations for each stage: y = 50t for 0 < t < 1, and y = -50t^2 + 200t - 100 for 1 < t < 2.

Step 2: Plot the functions for each stage of motion on a calculator and sketch the position-time graph over the range 0 to 2 seconds.

To plot these functions, we use a graphing calculator. For the first stage (0 < t < 1), we plot y = 50t. For the second stage (1 < t < 2), we plot y = -50t^2 + 200t - 100. The position-time graph shows a linear increase from 0 to 50 cm over the first second, …

Item 9100% solve rateanswer: The potential 'red flags' in DOMO Electronics' control environment include: 1. Lack of a formal mechanism for top management or the Board of Directors to receive confidential information from employees. 2. The audit committee's lack of independence from management. 3. The highly aggressive target of 40% revenue increase, which creates pressure on management to meet targets. 4. The lack of a method for evaluating work quality and performance reviews. 5. The large changes in systems, processes, and programs, which creates a high level of inherent risk for error. These 'red flags' could impact the company's financial statements by increasing the risk of material misstatements due to fraud or error. The audit steps to address these concerns include: 1. Evaluating the company's internal controls and identifying areas for improvement. 2. Conducting interviews with employees to assess their understanding of the code of conduct and the reporting mechanisms for confidential information. 3. Reviewing the audit committee's composition and ensuring that it is independent from management. 4. Analyzing the company's revenue growth targets and assessing the pressure on management to meet these targets. 5. Evaluating the company's systems, processes, and programs to identify areas of high inherent risk and recommending controls to mitigate these risks.

DOMO Electronics, a publicly traded company, has a written code of conduct that all employees are required to understand and follow. However, the company lacks a formal mechanism for top management or the Board of Directors to receive confidential information from employees. The company's audit committee consists of several financially savvy individuals who are also top managers in the company. The company has a highly aggressive target of 40% revenue increase, which creates pressure on management to meet targets. Identify the potential 'red flags' in DOMO Electronics' control environment and explain how they could impact the company's financial statements. Provide a detailed analysis of the potential risks and consequences of these 'red flags' and recommend audit steps to address these concerns.

How subjects answered

Llama-3.3-70B-Instruct correct
The control environment at DOMO Electronics presents several 'red flags' that could potentially impact the company's financial statements. These 'red flags' are:

Lack of a Formal Mechanism for Confidential Reporting: The absence of a formal mechanism for employees to report concerns or unethical behavior confidentially to top management or the Board of Directors can lead to undetected fraud or misconduct. This lack of transparency and accountability can result in management overriding controls or engaging in unethical practices without fear of detection.

Audit Committee Composition: The audit committee, comprising financially savvy individuals who are also top managers, may …

Item 10100% solve rateanswer: The reaction shows an increase in entropy.

Does the reaction KCLO3(s) ---> 2KCL(s) + 3O2(g) show an increase or decrease in entropy? Explain your answer using the concept of degrees of freedom and the properties of solids and gases.

How subjects answered

Llama-3.3-70B-Instruct correct
Step 1: Understanding the concept of entropy

Entropy is a measure of disorder or randomness in a system. It can also be thought of in terms of the number of available microstates for a system, which relates to the degrees of freedom of the particles in the system.

Step 2: Applying the concept to the reaction

The given reaction is KCLO3(s) ---> 2KCL(s) + 3O2(g). This reaction involves the decomposition of a solid (potassium chlorate) into two solids (potassium chloride) and a gas (oxygen).

Step 3: Analyzing the change in degrees of freedom

In solids, the particles (atoms, molecules, or ions) have restricted movement, meaning they have fewer degrees of freedom compared to gases. In …

Item 11100% solve rateanswer: Electromagnetic waves do not require a medium in the classical sense, but rather propagate through the electromagnetic field, which permeates spacetime.

Is it correct to say that electromagnetic waves do not require a medium, and if so, how do they propagate through empty spacetime?

How subjects answered

Llama-3.3-70B-Instruct correct
A fundamental question in physics!

Yes, it is correct to say that electromagnetic waves do not require a medium to propagate. This is a key characteristic that distinguishes them from mechanical waves, such as sound waves or water waves, which require a physical medium (like air, water, or a solid object) to transmit energy.

Electromagnetic waves, including light, radio waves, X-rays, and others, can propagate through the vacuum of empty spacetime. This seems counterintuitive, as we're used to thinking of waves as requiring a medium to transmit energy. However, electromagnetic waves are a unique case, and their propagation can be explained by the principles of electromagnetism and the …

Item 12100% solve rate

Design an experiment to investigate the effect of pH on the activity of trypsin. Predict the results of the experiment, explain the predicted outcomes, and discuss potential sources of error in the method. How could the experiment be modified to determine the optimum pH for trypsin activity more accurately?

How subjects answered

Llama-3.3-70B-Instruct correct
Step 1: Introduction to the Experiment

The experiment aims to investigate how the activity of trypsin, a digestive enzyme, is affected by different levels of pH. Trypsin is known for breaking down proteins into smaller peptides or amino acids, and its activity can be influenced by the pH of its environment.

Step 2: Experimental Design

To conduct this experiment, several solutions with different pH levels (e.g., pH 2, 4, 6, 7, 8, 10) will be prepared. A substrate for trypsin, such as casein or a synthetic peptide, will be added to each solution. The same amount of trypsin will then be added to each solution, and the mixtures will be incubated at a constant temperature (e.g., 37°C) for …

Subjects

The models, agents, and reward models evaluated.

1 subjects, ranked by mean response (accuracy) across this benchmark's items.

1Llama-3.3-70B-Instruct1

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