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mlfoundations-dev
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math_100000_samples

instruction stringlengths 22 15.1k	completion stringlengths 2 49.3k	__original_row_idx int64 0 100k	conversations listlengths 2 2
If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be?	If I were to choose a specific book for a mathematician at the beginning of their career, I would recommend "How to Prove It: A Structured Approach" by Daniel J. Velleman. This book provides a solid foundation in mathematical reasoning and proof techniques, which are essential skills in the field. It emphasizes the imp...	57	[ { "from": "human", "value": "If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be?\n" }, { "from": "gpt", "value": "If I were to choose a specific book for a mathematician at the beginning of their caree...
I am solving the following inequality, please look at it and tell me whether am I correct or not. This is an example in Howard Anton's book and I solved it on my own as given below, but the book has solved it differently! I want to confirm that my solution is also valid.	I'd be happy to help you with your inequality! Please provide the inequality that you solved, along with your solution and the method used in the book. Once I have that information, I can compare the two approaches and confirm whether your solution is valid.	585	[ { "from": "human", "value": "I am solving the following inequality, please look at it and tell me whether am I correct or not. This is an example in Howard Anton's book and I solved it on my own as given below, but the book has solved it differently! I want to confirm that my solution is also valid.\n\n" ...
mathfactor is one I listen to. Does anyone else have a recommendation?	If you enjoy MathFactor, you might also like the following math-related podcasts: 1. Numberphile - This podcast features discussions about interesting numbers and mathematical concepts with engaging guests. 2. Math Encounters - A podcast that showcases mathematical ideas in an accessible way, often featuring g...	1	[ { "from": "human", "value": "mathfactor is one I listen to. Does anyone else have a recommendation?\n" }, { "from": "gpt", "value": "If you enjoy MathFactor, you might also like the following math-related podcasts:\n\n1. Numberphile - This podcast features discussions about interesting numb...
What's the meaning of the double turnstile symbol in logic or mathematical notation? : $\models$	The double turnstile symbol, denoted as \(\models\), is used in logic and mathematical notation to signify semantic entailment or truth in a model. It indicates that a certain statement or set of statements (often referred to as a theory or a set of formulas) logically entails another statement. In more formal terms, ...	101	[ { "from": "human", "value": "What's the meaning of the double turnstile symbol in logic or mathematical notation? :\n\n$\\models$\n\n" }, { "from": "gpt", "value": "The double turnstile symbol, denoted as \\(\\models\\), is used in logic and mathematical notation to signify semantic entailment o...
I'm looking for the name of a shape that is like a capsule, but where each circle can have different radii. The shape could be described using two circles (two centers and two radii). Something like this (this is a 2D shape, not a lame attempt at 3D):	The shape you are describing sounds like a "lens" or more specifically, a "curvilinear quadrilateral" or "rounded rectangle" if you prefer, where two circles with different radii are connected by straight edges. In the context of a 2D shape formed by two circles with different radii, it could also be referred to as a "...	297	[ { "from": "human", "value": "I'm looking for the name of a shape that is like a capsule, but where each circle can have different radii. The shape could be described using two circles (two centers and two radii). Something like this (this is a 2D shape, not a lame attempt at 3D):\n\n" }, { "from": "...
I'm looking for an online or software calculator that can show me the history of items I typed in, much like an expensive Ti calculator. Can you recommend any?	Certainly! If you're looking for a calculator that keeps a history of your calculations similar to a TI calculator, there are a few options you might consider: 1. Desmos: This is a powerful online graphing calculator that keeps a history of your calculations. You can create and save different projects, making it e...	9	[ { "from": "human", "value": "I'm looking for an online or software calculator that can show me the history of items I typed in, much like an expensive Ti calculator. \nCan you recommend any?\n" }, { "from": "gpt", "value": "Certainly! If you're looking for a calculator that keeps a history of yo...
This is problem taken from Problem 4. I couldn't find the solution anywhere and I am curious to see a solution for this problem, as I can at least comprehend the question and it seems that the mechanism for the solution involved will be somewhat understandable.	It seems like you are referencing a specific problem (Problem 4) but did not provide the details of that problem. If you could share the problem statement or describe the topic or concept involved, I would be more than happy to help you understand it and guide you through the solution!	880	[ { "from": "human", "value": "This is problem taken from Problem 4.\nI couldn't find the solution anywhere and I am curious to see a solution for this problem, as I can at least comprehend the question and it seems that the mechanism for the solution involved will be somewhat understandable.\n" }, { ...
Please suggest a suitable approach for this problem.	Of course! However, I need more details about the specific problem you're facing in order to suggest a suitable approach. Could you please provide more context or describe the problem you're working on?	1,332	[ { "from": "human", "value": "Please suggest a suitable approach for this problem.\n" }, { "from": "gpt", "value": "Of course! However, I need more details about the specific problem you're facing in order to suggest a suitable approach. Could you please provide more context or describe the probl...
I can't seem to solve this one. I've tried Dirichlet's test but to no avail. Any help is greatly appreciated.	I'd be happy to help! However, I need more information about the specific problem you are trying to solve. Could you please provide the details of the series or integral you're working on? With that information, I can assist you better!	1,623	[ { "from": "human", "value": "I can't seem to solve this one. I've tried Dirichlet's test but to no avail.\nAny help is greatly appreciated.\n" }, { "from": "gpt", "value": "I'd be happy to help! However, I need more information about the specific problem you are trying to solve. Could you please...
I apologise for the really basic question. This didn't really fit on any other StackExchange website so the Maths one was the closest one where I could ask. Really Basic Question- What are $4$ sided shapes called again? Like how triangles are $3$ sided shapes, octagons are $8$ sided shapes, ... What are the $4$ sided ...	No need to apologize! A four-sided shape is called a quadrilateral. The term "quadrilateral" comes from the Latin words "quadri-" meaning four, and "latus" meaning side. There are many specific types of quadrilaterals, such as squares, rectangles, rhombuses, and trapezoids, but they all fall under the category of q...	1,284	[ { "from": "human", "value": "I apologise for the really basic question. This didn't really fit on any other StackExchange website so the Maths one was the closest one where I could ask.\nReally Basic Question- What are $4$ sided shapes called again? \nLike how triangles are $3$ sided shapes, octagons are $8...
It seems like functions that are continuous always seem to be differentiable, to me. I can't imagine one that is not. Are there any examples of functions that are continuous, yet not differentiable? The other way around seems a bit simpler -- a differentiable function is obviously always going to be continuous. But ...	It's a common intuition that continuous functions should be differentiable, but there are indeed continuous functions that are not differentiable. A classic example of this is the absolute value function: \[ f(x) = \|x\| \] This function is continuous everywhere, but it is not differentiable at \( x = 0 \). At that poi...	36	[ { "from": "human", "value": "It seems like functions that are continuous always seem to be differentiable, to me. I can't imagine one that is not. Are there any examples of functions that are continuous, yet not differentiable?\nThe other way around seems a bit simpler -- a differentiable function is obvi...
Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number? It seems as though formerly $0$ was considered in the set of natural numbers, but now it seems more common to see definitions saying that the natural numbers are precisely ...	The classification of zero as a natural number is a topic that has sparked discussion and variation in definitions within the mathematical community. There are primarily two conventions regarding the natural numbers: 1. Including Zero: In this convention, the set of natural numbers is defined to include zero. This...	60	[ { "from": "human", "value": "Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number?\nIt seems as though formerly $0$ was considered in the set of natural numbers, but now it seems more common to see definitions sayi...
I covered hyperbolic trigonometric functions in a recent maths course. However I was never presented with any reasons as to why (or even if) they are useful. Is there any good examples of their uses outside academia?	Hyperbolic trigonometric functions indeed have several practical applications outside of academia. Here are a few examples that illustrate their usefulness in various fields: 1. Physics: Hyperbolic functions describe the shape of hanging cables or chains, known as catenaries. When a flexible chain or rope is suspe...	31	[ { "from": "human", "value": "I covered hyperbolic trigonometric functions in a recent maths course. However I was never presented with any reasons as to why (or even if) they are useful.\nIs there any good examples of their uses outside academia?\n" }, { "from": "gpt", "value": "Hyperbolic trigo...
The knight's tour is a sequence of 64 squares on a chess board, where each square is visted once, and each subsequent square can be reached from the previous by a knight's move. Tours can be cyclic, if the last square is a knight's move away from the first, and acyclic otherwise. There are several symmetries among kni...	The enumeration of knight's tours on a chessboard is a well-studied problem in combinatorial game theory and graph theory. While the exact count of knight's tours that are distinct under all symmetries (including rotation and reflection) is complex, there have been notable findings. As of the latest research, the numb...	148	[ { "from": "human", "value": "The knight's tour is a sequence of 64 squares on a chess board, where each square is visted once, and each subsequent square can be reached from the previous by a knight's move. Tours can be cyclic, if the last square is a knight's move away from the first, and acyclic otherwis...
I just got my statistics test back and I am totally confused about one of the questions! A study was done that took a simple random sample of 40 people and measured whether the subjects were right-handed or left-handed, as well as their ages. The study showed that the proportion of left-handed people and the...	Your understanding of correlation and causation is generally correct—correlation does not imply causation. However, the way you phrased your conclusion is problematic because it implies a specific directional relationship (i.e., that people become right-handed as they grow older), which is not supported by the informat...	89	[ { "from": "human", "value": "I just got my statistics test back and I am totally confused about one of the questions! \n\nA study was done that took a simple random sample of 40 people and measured whether\n the subjects were right-handed or\n left-handed, as well as their ages.\n The study showed that ...
Is there a collective term for the $XY, YZ$ and $ZX$ planes in $3D$ co-ordinate geometry? I was thinking "principal planes" but I'm not sure where I heard that.	Yes, the terms "principal planes" or "coordinate planes" can be used to refer to the \(XY\), \(YZ\), and \(ZX\) planes in 3D coordinate geometry. These planes are defined by the coordinate axes: - The \(XY\) plane is defined by \(z = 0\). - The \(YZ\) plane is defined by \(x = 0\). - The \(ZX\) plane is defined by \(y...	344	[ { "from": "human", "value": "Is there a collective term for the $XY, YZ$ and $ZX$ planes in $3D$ co-ordinate geometry? I was thinking \"principal planes\" but I'm not sure where I heard that.\n" }, { "from": "gpt", "value": "Yes, the terms \"principal planes\" or \"coordinate planes\" can be use...
What is the proper definition of a Countable Set?	A countable set is defined as a set that has the same cardinality as the set of natural numbers, which means it can either be finite or infinite but can be put into a one-to-one correspondence with the natural numbers \( \mathbb{N} = \{1, 2, 3, \ldots\} \). In more formal terms: 1. Finite Countable Set: A finite ...	704	[ { "from": "human", "value": "What is the proper definition of a Countable Set?\n" }, { "from": "gpt", "value": "A countable set is defined as a set that has the same cardinality as the set of natural numbers, which means it can either be finite or infinite but can be put into a one-to-one corres...
In Intro Number Theory a key lemma is that if $a$ and $b$ are relatively prime integers, then there exist integers $x$ and $y$ such that $ax+by=1$. In a more advanced course instead you would use the theorem that the integers are a PID, i.e. that all ideals are principal. Then the old lemma can be used to prove that ...	Yes, it is indeed possible for a ring to have all finitely generated ideals as principal while also having some ideals that are not finitely generated. To illustrate this, consider the ring of integers \(\mathbb{Z}\), which is a principal ideal domain (PID). In \(\mathbb{Z}\), every ideal is generated by a single ele...	113	[ { "from": "human", "value": "In Intro Number Theory a key lemma is that if $a$ and $b$ are relatively prime integers, then there exist integers $x$ and $y$ such that $ax+by=1$. In a more advanced course instead you would use the theorem that the integers are a PID, i.e. that all ideals are principal. Then...
I need $n$ cakes for a party. I go to the cake shop and there are $k$ different kinds of cake. For variety, I'd like to get at least one of each cake. How many ways can I do this?	To determine how many ways you can select \( n \) cakes from \( k \) different kinds of cake with the requirement that you get at least one of each kind, we can utilize the "stars and bars" combinatorial method. First, since we need at least one cake of each kind, we will initially allocate one cake of each kind. This...	167	[ { "from": "human", "value": "I need $n$ cakes for a party. I go to the cake shop and there are $k$ different kinds of cake. For variety, I'd like to get at least one of each cake. \nHow many ways can I do this?\n" }, { "from": "gpt", "value": "To determine how many ways you can select \\( n \\) ...
When given an algrebraic expression of the form $$ \ 6A + 7{A^2}\ $$ and asked to simplify, would the correct answer be a) $$ \ A(6 + 7A)\ $$ or b) That this cannot be simplified further	To simplify the expression \( 6A + 7A^2 \), we can factor out the common term, which is \( A \). Starting with the expression: \[ 6A + 7A^2 \] We can factor out \( A \): \[ A(6 + 7A) \] So, the correct answer is: a) \( A(6 + 7A) \) Therefore, the expression can indeed be simplified further.	1,119	[ { "from": "human", "value": "When given an algrebraic expression of the form\n$$ \\ 6A + 7{A^2}\\ $$\nand asked to simplify, would the correct answer be\na)\n$$ \\ A(6 + 7A)\\ $$\nor\nb)\nThat this cannot be simplified further\n" }, { "from": "gpt", "value": "To simplify the expression \\( 6A + ...
Is a prime number in the decimal system still a prime when converted to a different base?	Yes, a prime number remains prime when expressed in a different base, but the representation of the number changes, not its intrinsic properties. A prime number is defined as a natural number greater than 1 that has no positive divisors other than 1 and itself. This definition is independent of the numeral system use...	941	[ { "from": "human", "value": "Is a prime number in the decimal system still a prime when converted to a different base?\n" }, { "from": "gpt", "value": "Yes, a prime number remains prime when expressed in a different base, but the representation of the number changes, not its intrinsic properties...
I have a square that's $10\mathrm{m} \times 10\mathrm{m}$. I want to cut it in half so that I have a square with half the area. But if I cut it from top to bottom or left to right, I don't get a square, I get a rectangle! I know the area of the small square is supposed to be $50\mathrm{m}^{2}$, so I can use my calcul...	To find the side length of a square with an area of \(50 \, \text{m}^2\) without using a calculator, we can solve the problem step by step using some basic math principles. 1. Understanding the Area of a Square: The area \(A\) of a square is given by the formula: \[ A = s^2 \] where \(s\) is the le...	35	[ { "from": "human", "value": "I have a square that's $10\\mathrm{m} \\times 10\\mathrm{m}$. I want to cut it in half so that I have a square with half the area. But if I cut it from top to bottom or left to right, I don't get a square, I get a rectangle! \nI know the area of the small square is supposed to ...
The scatter graph at the Wikipedia article seems to suggest so. Has anyone attacked this before? Is there a known proof?	It seems like you're referring to a specific concept or hypothesis related to a scatter graph that appears in a Wikipedia article. However, without additional context about which topic or concept you're discussing, it's difficult to provide a direct answer. In general, many mathematical or statistical hypotheses repre...	1,865	[ { "from": "human", "value": "The scatter graph at the Wikipedia article seems to suggest so. Has anyone attacked this before? Is there a known proof?\n" }, { "from": "gpt", "value": "It seems like you're referring to a specific concept or hypothesis related to a scatter graph that appears in a W...
I have to divide $2860$ by $3186$. The question gives only $2$ minutes and that division is only half part of question. Now I can't possibly make that division in or less than $2$ minutes by applying traditional methods, which I can't apply on that division anyways. So anyone can perform below division using faster tec...	To quickly estimate the division of \( 2860 \) by \( 3186 \), we can first simplify the fraction \( \frac{2860}{3186} \). First, let's find approximate values for both numbers: 1. Calculate \( \frac{2860}{3186} \): - Both numbers are fairly close to each other. We can round them to simpler numbers for estimation. ...	351	[ { "from": "human", "value": "I have to divide $2860$ by $3186$. The question gives only $2$ minutes and that division is only half part of question. Now I can't possibly make that division in or less than $2$ minutes by applying traditional methods, which I can't apply on that division anyways.\nSo anyone c...
A quick question-Something that I learned that I forgot throughout the years. If a question asked you for a multiple of 9 and 12, which would be less than 40, would "36" be an answer, as both 9 and 12 can multiply to get 36, yet it fits under the criteria for being under 40?	Yes, you are correct! The multiples of 9 are: 9, 18, 27, 36, 45, ... The multiples of 12 are: 12, 24, 36, 48, ... The least common multiple (LCM) of 9 and 12 is 36. Since 36 is indeed a multiple of both 9 and 12 and is less than 40, it fits the criteria perfectly. So, 36 is a valid answer.	1,546	[ { "from": "human", "value": "A quick question-Something that I learned that I forgot throughout the years. \nIf a question asked you for a multiple of 9 and 12, which would be less than 40, would \"36\" be an answer, as both 9 and 12 can multiply to get 36, yet it fits under the criteria for being under 40?...
This is such a basic question I'm sure, but I've been trying to find a robust solution to it for a while. Lets say I have the numbers 0-100 in series. As the number increases, I want another number say 1000, to decrease. so that when you got to 100, it was at 0 0 = 1000 50 = 500 100 = 0 What is this kind of series call...	What you're describing is a linear relationship between two variables, where one variable decreases as the other increases in a linear manner. Specifically, in your case, you want the values to be inversely proportional: when one variable increases from 0 to 100, the other variable decreases from 1000 to 0. The mathem...	931	[ { "from": "human", "value": "This is such a basic question I'm sure, but I've been trying to find a robust solution to it for a while.\nLets say I have the numbers 0-100 in series.\nAs the number increases, I want another number say 1000, to decrease. so that when you got to 100, it was at 0\n0 = 1000\n50 =...
Suppose $A$ is some algebraic structure, and $x$ and $y$ are two elements of the underlying set. Is there any more concise way of stating, "there exists an automorphism in $A$ which maps $x$ to $y$"?. It seems like the obvious thing to say is "$x$ and $y$ are symmetric in $A$", but I've never come across this phrasin...	The concept you're describing, where there exists an automorphism in the algebraic structure \( A \) that maps \( x \) to \( y \), can indeed be succinctly expressed. The term you might be looking for is "conjugate". Specifically, you can say that "elements \( x \) and \( y \) are conjugate in \( A \)" if such an autom...	620	[ { "from": "human", "value": "Suppose $A$ is some algebraic structure, and $x$ and $y$ are two elements of the underlying set. Is there any more concise way of stating, \"there exists an automorphism in $A$ which maps $x$ to $y$\"?. It seems like the obvious thing to say is \"$x$ and $y$ are symmetric in $...
I forget for a while, we don't need the compactness condition here right?	It seems like you're referring to a concept from topology or functional analysis, possibly in the context of a theorem or a property of certain spaces or sets. The compactness condition is often a crucial assumption in various mathematical results, such as the Arzelà-Ascoli theorem in analysis or certain theorems in to...	1,180	[ { "from": "human", "value": "I forget for a while, we don't need the compactness condition here right?\n" }, { "from": "gpt", "value": "It seems like you're referring to a concept from topology or functional analysis, possibly in the context of a theorem or a property of certain spaces or sets. ...
There is a theorem of Riemann to that effect. How to prove it? Note: This was asked by Kenny in the beta for "calculus".	It seems like you are referencing the Riemann Hypothesis or possibly theorems related to Riemann surfaces or Riemann sums in calculus, but it's not entirely clear from your question. Let's clarify a few possibilities: 1. Riemann Hypothesis: This is a conjecture in number theory about the distribution of the zeros ...	204	[ { "from": "human", "value": "There is a theorem of Riemann to that effect. How to prove it?\nNote: This was asked by Kenny in the beta for \"calculus\".\n" }, { "from": "gpt", "value": "It seems like you are referencing the Riemann Hypothesis or possibly theorems related to Riemann surfaces or ...
A map $s : \mathbb{N} \to X$ is a computable sequence in $(X,\nu_X)$ when there exists a computable map $f : \mathbb{N} \to \mathbb{N}$ such that $s(n) = \nu_X(f(n))$ for all $n \in \mathrm{dom}(\nu_X)$. My best guess would be, "A map s taking N onto X is a computable sequence in the ??? (X, nu??) when there exists a c...	You can read the statement aloud in a way that maintains clarity while also conveying all the relevant elements. Here’s a suggested phrasing: "A map \( s \) from the natural numbers to \( X \) is a computable sequence in the space \( (X, \nu_X) \) when there exists a computable function \( f \) from the natural number...	1,625	[ { "from": "human", "value": "A map $s : \\mathbb{N} \\to X$ is a computable sequence in $(X,\\nu_X)$ when there exists a computable map $f : \\mathbb{N} \\to \\mathbb{N}$ such that $s(n) = \\nu_X(f(n))$ for all $n \\in \\mathrm{dom}(\\nu_X)$.\nMy best guess would be, \"A map s taking N onto X is a computabl...
You've spent your whole life in the hyperbolic plane. It's second nature to you that the area of a triangle depends only on its angles, and it seems absurd to suggest that it could ever be otherwise. But recently a good friend named Euclid has raised doubts about the fifth postulate of Poincaré's Elements. This postula...	To model Euclidean geometry within the hyperbolic plane, you can use a creative approach. One intuitive way to accomplish this is by constructing what we'll call the "Conformal Projection Model" of Euclidean geometry in the hyperbolic plane. ### The Conformal Projection Model 1. Choose a Hyperbolic Model: Start ...	314	[ { "from": "human", "value": "You've spent your whole life in the hyperbolic plane. It's second nature to you that the area of a triangle depends only on its angles, and it seems absurd to suggest that it could ever be otherwise.\nBut recently a good friend named Euclid has raised doubts about the fifth post...
I'm very interested in Computer Science (computational complexity, etc.). I've already finished a University course in the subject (using Sipser's "Introduction to the Theory of Computation"). I know the basics, i.e. Turing Machines, Computability (Halting problem and related reductions), Complexity classes (time and s...	It's great to hear that you've built a solid foundation in computability and complexity theory! Here are some recommendations for more advanced texts that delve deeper into various aspects of computational complexity: 1. "Computational Complexity: A Modern Approach" by Sanjeev Arora and Boaz Barak - This is an exc...	74	[ { "from": "human", "value": "I'm very interested in Computer Science (computational complexity, etc.). I've already finished a University course in the subject (using Sipser's \"Introduction to the Theory of Computation\").\nI know the basics, i.e. Turing Machines, Computability (Halting problem and related...
Please suggest suitable approach for this problem	I'd be happy to help! However, I need a bit more context about the specific problem you're referring to. Could you provide more details about the issue or situation you're facing? The more information you provide, the better I can tailor my suggestions to your needs.	2,486	[ { "from": "human", "value": "Please suggest suitable approach for this problem\n" }, { "from": "gpt", "value": "I'd be happy to help! However, I need a bit more context about the specific problem you're referring to. Could you provide more details about the issue or situation you're facing? The ...
Why are differential equations called differential equations?	Differential equations are called so because they involve derivatives, which are a type of differential operation. In mathematics, derivatives measure how a function changes as its input changes. A differential equation is an equation that relates a function with its derivatives, thus capturing the relationship between...	1,114	[ { "from": "human", "value": "Why are differential equations called differential equations?\n" }, { "from": "gpt", "value": "Differential equations are called so because they involve derivatives, which are a type of differential operation. In mathematics, derivatives measure how a function change...
Lifted from Mathoverflow: I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best. Then what might be the 2nd best? It can be a book, preprint, online lecture note, webpage, etc. One suggestion per answer please. Also, please include an explanation of why you like the book, or what makes...	One highly regarded second choice for a comprehensive resource in algebraic geometry is "Algebraic Geometry" by Robin Hartshorne's own advisor, David Mumford. This book is unique and valuable for several reasons: 1. Geometric Intuition: Mumford's approach emphasizes geometric intuition alongside formalism. His...	243	[ { "from": "human", "value": "Lifted from Mathoverflow:\nI think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best.\nThen what might be the 2nd best? It can be a book, preprint, online lecture note, webpage, etc.\nOne suggestion per answer please. Also, please include an explana...
So you have $2$ dice and you want to get at least a $1$ or a $5$ (on the dice not added). How do you go about calculating the answer for this question. This question comes from the game farkle.	To solve the problem of finding the probability of rolling at least one \(1\) or one \(5\) with two dice, we can use the complementary counting method. Here are the steps: 1. Calculate the Total Possible Outcomes: Each die has 6 faces, so with 2 dice, the total number of outcomes is: \[ 6 \times 6 = 36. ...	70	[ { "from": "human", "value": "So you have $2$ dice and you want to get at least a $1$ or a $5$ (on the dice not added). How do you go about calculating the answer for this question.\nThis question comes from the game farkle.\n" }, { "from": "gpt", "value": "To solve the problem of finding the pr...
Follow up to Say I have triangles as shown: Its easy to see that $ABC$ is adjacent to $ACJ: ABC$ and $ACJ$ share $2$ vertices ($A$ and $C$). However $FEC$ and $ACJ$ are also adjacent, but only one vertex is shared. How can I find if two triangles are adjacent when they differ in size?	In the context of geometry, two triangles are considered adjacent if they share at least one vertex or one edge. The adjacency can be evaluated using the following criteria: 1. Shared Vertices: If two triangles share at least one vertex, they are adjacent. This is the case you mentioned with triangles \( ABC \) an...	968	[ { "from": "human", "value": "Follow up to\nSay I have triangles as shown:\n\nIts easy to see that $ABC$ is adjacent to $ACJ: ABC$ and $ACJ$ share $2$ vertices ($A$ and $C$).\nHowever $FEC$ and $ACJ$ are also adjacent, but only one vertex is shared.\nHow can I find if two triangles are adjacent when they di...
This is a problem from "Introduction to Mathematics - Algebra and Number Systems" (specifically, exercise set 2 #9), which is one of my math texts. Please note that this isn't homework, but I would still appreciate hints rather than a complete answer. The problem reads as follows: If 3p2 = q2, where $p,q \in \mathbb{Z...	You're on the right track in showing that \( 3 \) divides \( q \). To show that \( 3 \) also divides \( p \), you can proceed with a proof by contradiction. Here's a hint: 1. Assume for contradiction that \( 3 \) does not divide \( p \). This means that \( p \) can be expressed in the form \( p = 3k + r \) where \( ...	671	[ { "from": "human", "value": "This is a problem from \"Introduction to Mathematics - Algebra and Number Systems\" (specifically, exercise set 2 #9), which is one of my math texts. Please note that this isn't homework, but I would still appreciate hints rather than a complete answer.\nThe problem reads as fol...
You know that a couple has two children. You go to the couple's house and one of their children, a young boy, opens the door. What is the probability that the couple's other child is a girl? If you list all possibilities for the sexes of two children, $BB, BG, GB, GG$, you see that $2$ of the $3$ pairs that have B (for...	This question is a classic problem in probability that revolves around conditional probability. Let's analyze it step by step. First, we list all possible combinations of two children, assuming they can either be a boy (B) or a girl (G). The possible combinations are: 1. BB (both are boys) 2. BG (one boy and one girl...	641	[ { "from": "human", "value": "You know that a couple has two children. You go to the couple's house and one of their children, a young boy, opens the door. What is the probability that the couple's other child is a girl?\nIf you list all possibilities for the sexes of two children, $BB, BG, GB, GG$, you see ...
The Sum of 2 sets of Measure zero might well be very large for example, the sum of $x$-axis and $y$-axis, is nothing but the whole plane. Similarly one can ask this question about Cantor sets: If $C$ is the cantor set, then what is the measure of $C+C$?	To find the measure of \( C + C \) where \( C \) is the Cantor set, we begin by examining the nature of the Cantor set and how its sum behaves. 1. Understanding the Cantor Set: The Cantor set \( C \) is a subset of the interval \([0, 1]\) that is constructed by iteratively removing the middle thirds of segments. I...	568	[ { "from": "human", "value": "The Sum of 2 sets of Measure zero might well be very large for example, the sum of $x$-axis and $y$-axis, is nothing but the whole plane. Similarly one can ask this question about Cantor sets:\nIf $C$ is the cantor set, then what is the measure of $C+C$?\n" }, { "from": ...