Search found 602 matches

by Cauchy
Fri Aug 19, 2016 2:00 am UTC
Forum: Logic Puzzles
Topic: Factorial puzzle
Replies: 6
Views: 3872

Re: Factorial puzzle

What is the point of having m and n? The Euler totient function is multiplicative , so unless you plan to have solutions with m and n sharing a factor, we can just talk about mn*phi(mn). Really, though, we should be splitting m and n up across their prime factors. So we have m = p_1^{m_1}p_2^{m_2}.....
by Cauchy
Sun Aug 14, 2016 8:22 am UTC
Forum: Mathematics
Topic: Misunderstanding basic math concepts, help please?
Replies: 338
Views: 80534

Re: Misunderstanding basic math concepts, help please?

What is the complement of an axiomatic system? Does it have the same symbols? The same well-formed formulae? The same rules of inference? The same axioms? You use the term like you expect us to know what it means, but no one else talks about the complement of a theory. Te complement of a set, I know...
by Cauchy
Fri Aug 12, 2016 2:27 pm UTC
Forum: Logic Puzzles
Topic: Sharing secret information publicly
Replies: 27
Views: 7213

Re: Sharing secret information publicly

Once I found the best and elegant solution I will post it. Just letting you know, it's already in the thread. Tirear posted most of it, and I put the finishing touches on it. The Fano plane solution (which you have yet to accurately write down) is pretty, and a natural jump when you're talking abou...
by Cauchy
Fri Aug 12, 2016 3:48 am UTC
Forum: Logic Puzzles
Topic: Connect all points to form a single tree
Replies: 12
Views: 5883

Re: Connect all points to form a single tree

Solution:

Spoiler:
deltoidal-trihexagonal-tree-medium-01 solution.png
by Cauchy
Thu Aug 11, 2016 6:46 pm UTC
Forum: Logic Puzzles
Topic: Sharing secret information publicly
Replies: 27
Views: 7213

Re: Sharing secret information publicly

The solution that Tirear posted has been proved correct, so you don't have to worry about that. It's quite elegant. Goahead, your solution can't work. Once A says something, B will never learn any more information, since he already knows the information he's about to say, and that's everything else ...
by Cauchy
Wed Aug 10, 2016 4:18 pm UTC
Forum: Mathematics
Topic: Formula for the Volume of a n-cone
Replies: 20
Views: 6179

Re: Formula for the Volume of a n-cone

Euclid proved the volume of the cone. Are you are suggesting that he knew calculus? I'm suggesting that he did what we would now put under the umbrella of calculus, whether or not he "knew calculus" or knew about calculus. The method of exhaustion is seen as a precursor to the methods of ...
by Cauchy
Wed Aug 10, 2016 3:44 am UTC
Forum: Serious Business
Topic: Problem about free will
Replies: 126
Views: 28889

Re: Problem about free will

There's no flaw in Adam's logic, so far as I can tell, though he's using the term "free will" a little weirdly (which you've explained in subsequent posts, JS). I question what kind of doctor would "tell" a patient what medicine to take as opposed to recommending a medicine, but ...
by Cauchy
Tue Aug 09, 2016 3:55 pm UTC
Forum: Logic Puzzles
Topic: Sharing secret information publicly
Replies: 27
Views: 7213

Re: Sharing secret information publicly

The order being required knowledge doesn't make the problem any harder. If you have a solution in which the order is not required, just append the relative values of your three cards in your statement, e.g. "low, high, middle". Anyone who knows your three cards will be able to determine th...
by Cauchy
Sun Aug 07, 2016 11:28 pm UTC
Forum: Mathematics
Topic: Formula for the Volume of a n-cone
Replies: 20
Views: 6179

Re: Formula for the Volume of a n-cone

Although it might be possible to phrase what I described in terms of Riemann sums, I didn't, and I don't think it is a very productive way of approaching most geometric problems - a circle is not even the graph of a function, and it's entirely possible for the base of a cone to be stranger (think s...
by Cauchy
Tue Aug 02, 2016 6:48 pm UTC
Forum: Mathematics
Topic: Misunderstanding basic math concepts, help please?
Replies: 338
Views: 80534

Re: Misunderstanding basic math concepts, help please?

It's What the Tortoise Said to Achilles . I might grant you that the empty statement (which says nothing about anything) is vacuously true, but otherwise you're going to fall into this trap. You have to bootstrap somewhere if you want to get meaning, and bootstrapping is inherently this risky proces...
by Cauchy
Tue Aug 02, 2016 6:08 pm UTC
Forum: Logic Puzzles
Topic: Unblock puzzle with pipes
Replies: 6
Views: 3623

Re: Unblock puzzle with pipes

You move a given edge row or column into place, then don't touch it again and work on a smaller grid, working on smaller and smaller grids until you're down to a 2x2 square. The one wrinkle is that due to parity considerations, it's possible that you'll be unable to solve naively if you have identic...
by Cauchy
Tue Aug 02, 2016 6:04 pm UTC
Forum: Mathematics
Topic: Formula for the Volume of a n-cone
Replies: 20
Views: 6179

Re: Formula for the Volume of a n-cone

Both of the things you described here are things I would call "calculus". Really? To me, "calculus" has always meant "a symbolic method of calculation" (some variation of this definition is called for if you want to know why "propositional calculus", "la...
by Cauchy
Sun Jul 31, 2016 5:40 pm UTC
Forum: Logic Puzzles
Topic: Unblock puzzle with pipes
Replies: 6
Views: 3623

Re: Unblock puzzle with pipes

The description and screenshots make the game look like you figure out how the pipes are supposed to go, and then you slide them into place. Both of those halves sound well-worn and not particularly interesting. The sliding part in particular is just the 15-puzzle, whose solution methods are pretty ...
by Cauchy
Sun Jul 31, 2016 9:52 am UTC
Forum: Logic Puzzles
Topic: Unblock puzzle with pipes
Replies: 6
Views: 3623

Re: Unblock puzzle with pipes

What makes this not just solving a tile-swapping puzzle to find a valid tile setup, followed by a 15 puzzle to get the blocks in the right place?
by Cauchy
Sun Jul 31, 2016 9:29 am UTC
Forum: Mathematics
Topic: Formula for the Volume of a n-cone
Replies: 20
Views: 6179

Re: Formula for the Volume of a n-cone

There is no need for calculus. Just approximate the base of the cone with a bunch of little quadrilaterals (or hypercubes, in higher dimensions). In the two dimensional case, this is the same as breaking up a triangle's base into a bunch of smaller bases, cutting the whole triangle into a bunch of ...
by Cauchy
Sat Jul 30, 2016 7:00 am UTC
Forum: Mathematics
Topic: Formula for the Volume of a n-cone
Replies: 20
Views: 6179

Re: Formula for the Volume of a n-cone

I have thought about it a lot and I still cannot figure out if the pattern I noticed works for higher dimensional figures. It does, but it probably won't be intuitive why until you take calculus. If you want to try to bend your mind: An isosceles right triangle is half of a square. That is to say: ...
by Cauchy
Thu Jul 28, 2016 9:14 pm UTC
Forum: Mathematics
Topic: Misunderstanding basic math concepts, help please?
Replies: 338
Views: 80534

Re: Misunderstanding basic math concepts, help please?

I realize I didn't say this in my last post, but with regards to the Socratic argument part of your treatise, Treatid, you said: "A given system (language, theorem, set of well formed formulae, set of axioms) consists of a set of symbols and a set of rules that specify how we manipulate and int...
by Cauchy
Thu Jul 28, 2016 6:31 pm UTC
Forum: Mathematics
Topic: Goahead52's Math Posts
Replies: 148
Views: 19386

Re: Theorem and consequences

The difference between x^k and (x+1)^k is on the order of kx^{k-1} (by the derivative), and the difference between y^{k+1} and (y+1)^{k+1} is on the order of (k+1)y^k. The smallest k+1-st power number above x^k is larger than (x^{k/(k+1)})^{k+1}, so the difference between it and the next k+1-st numb...
by Cauchy
Wed Jul 27, 2016 10:34 am UTC
Forum: Mathematics
Topic: Misunderstanding basic math concepts, help please?
Replies: 338
Views: 80534

Re: Misunderstanding basic math concepts, help please?

Fundamentally, Treatid, if you have different definitions and axioms than us, you aren't talking about axiomatic mathematics anymore, because that's *we're* talking about, and you're talking about something different. We disagree with your definitions and axioms in the sense that they don't do what ...
by Cauchy
Tue Jul 26, 2016 3:40 pm UTC
Forum: Mathematics
Topic: Misunderstanding basic math concepts, help please?
Replies: 338
Views: 80534

Re: Misunderstanding basic math concepts, help please?

I'm not going to respond to Treatid's treatise in full. Heck, I'm not even going to read most of Treatid's treatise. His "Socratic argument" is rather insulting. But based on what I can glean from some parts I tried to read: 1. The difference between a language and a theory is the axioms. ...
by Cauchy
Mon Jul 25, 2016 12:49 am UTC
Forum: Mathematics
Topic: Dataset of linear systems
Replies: 21
Views: 5034

Re: Dataset of linear systems

Well that would generate matrices of maximal rank with probability 1, right? Maybe TC wants to also have examples with more linear dependencies.
by Cauchy
Fri Jul 22, 2016 6:05 am UTC
Forum: Mathematics
Topic: The liar paradox as a basis for kantian metaphysics
Replies: 11
Views: 3809

Re: The liar paradox as a basis for kantian metaphysics

I wasn't sure if type theory was exactly the right term. Thanks! Really, the problem I have with both the definitions you provide is that "false" ends up being "all things that aren't true" rather than "the negations of all true things". When you change the definition o...
by Cauchy
Thu Jul 21, 2016 6:05 pm UTC
Forum: Mathematics
Topic: The liar paradox as a basis for kantian metaphysics
Replies: 11
Views: 3809

Re: The liar paradox as a basis for kantian metaphysics

According to conventional logic which doesn't treat truth as a set, that would state the liar paradox is false. Well that's not true. After all, if the liar paradox were false, then the sentence itself would be true. I'm not exactly sure why truth should be assumed just because Arthur Prior said so...
by Cauchy
Wed Jul 20, 2016 4:36 pm UTC
Forum: Mathematics
Topic: The liar paradox as a basis for kantian metaphysics
Replies: 11
Views: 3809

Re: The liar paradox as a basis for kantian metaphysics

Let us assume ∃x{x≡~(x∈T)}. This obviously results in ~x∈T & x∈T. This is the liar paradox. Therefore, ~∃x{x≡~(x∈T)}. What I have just stated summarizes Arthur Prior's solution to the liar paradox. Truth is automatically assumed when the statement is presented, such that "this sentence is ...
by Cauchy
Fri Jul 15, 2016 7:10 am UTC
Forum: Logic Puzzles
Topic: Monty Hall-ish Problem
Replies: 4
Views: 3522

Re: Monty Hall-ish Problem

I don't think this problem is tractable, since the other player has agency. The lack of agency is what makes the Monty Hall Problem tractable, and its presence is what makes Let's Make a Deal interesting. Anyway, switching yields a 52% chance of winning (26 black cards out of 50 cards not visible to...
by Cauchy
Fri Jul 15, 2016 5:09 am UTC
Forum: Mathematics
Topic: Trisecting an angle
Replies: 63
Views: 12024

Re: Trisecting an angle

So we know all the constructable angles, and triple any constructable angle is a (constructable) trisectable angle. Cool. But as people pointed out, 3 pi/7 is a non-constructable trisectable angle. So an interesting question arises: what are all the trisectable angles?
by Cauchy
Tue Jul 12, 2016 4:16 am UTC
Forum: Mathematics
Topic: Trisecting an angle
Replies: 63
Views: 12024

Re: Trisecting an angle

Being able to trisect an angle θ is equivalent to being able to construct an angle of θ/3 given an angle of θ. If you can construct θ/3 without having θ, then more power to you, and you can certainly construct θ/3 when given θ. That is to say, constructing an angle of 120 degrees from scratch is an ...
by Cauchy
Mon Jun 27, 2016 9:01 am UTC
Forum: Logic Puzzles
Topic: Two puzzles for the price of 1!
Replies: 17
Views: 6359

Re: Two puzzles for the price of 1!

ConMan wrote:It's a series of puns (sore/saw, whole/hole, hoarse/horse).


What dialect do you speak such that sore and saw are homophones? I've never, ever heard that. They don't even have the same vowel for me, let alone the same post-vowel sound (which is "none" for saw).
by Cauchy
Fri Jun 24, 2016 11:49 pm UTC
Forum: Logic Puzzles
Topic: Seeking Advise from Puzzle Masters
Replies: 20
Views: 5621

Re: Seeking Advise from Puzzle Masters

Here's one option with 5 clues, all of which appear necessary for a unique solution ABCD BAED - 25 BACD - 50 DDAA - 0 CCCC - 25 ABEA - 50 None of the clues individually give away much information, and this can be logically worked through in a couple minutes. Edit: looking back, the first or fourth ...
by Cauchy
Thu Jun 23, 2016 3:50 pm UTC
Forum: Logic Puzzles
Topic: Logic Puzzle - Scavenger Hunt
Replies: 9
Views: 5650

Re: Logic Puzzle - Scavenger Hunt

I think the obvious first step is to calculate the frequency distribution; that is, figure out how many times each letter occurs. Once you have those numbers (in order from greatest to fewest), something about subtraction? I'm too lazy to actually tabulate.
by Cauchy
Wed Jun 22, 2016 11:15 am UTC
Forum: Logic Puzzles
Topic: Two puzzles for the price of 1!
Replies: 17
Views: 6359

Re: Two puzzles for the price of 1!

1: Your solution doesn't follow. They could all be dead from a traffic accident...or, they could not. Some subset of them could be going to the party. Or not. There's not enough information in the description to say. This is just an https://xkcd.com/169/ problem. And even if we're going...
by Cauchy
Mon Jun 20, 2016 12:50 am UTC
Forum: Mathematics
Topic: Misunderstanding basic math concepts, help please?
Replies: 338
Views: 80534

Re: Misunderstanding basic math concepts, help please?

My deepest thanks to everyone in this thread and previous ones. Proof: The premises for this proof are the axioms/rules of propositional logic. That is, we assume that all the rules of propositional logic are true. The proof is intended to show that the axioms/rules of propositional logic are incon...
by Cauchy
Mon Jun 13, 2016 9:42 am UTC
Forum: Mathematics
Topic: Completions, Cauchy sequences, and countability
Replies: 7
Views: 3870

Re: Completions, Cauchy sequences, and countability

I don't know how to write such a proof that mirrors Cantor's argument, since non-equivalence is a tail condition, and hence you can't pick only one sequence member per equivalence class you're trying to avoid. And while I can pretty easily construct a sequence that isn't any of the equivalence class...
by Cauchy
Thu Jun 02, 2016 10:51 am UTC
Forum: Mathematics
Topic: Math: Fleeting Thoughts
Replies: 434
Views: 161586

Re: Math: Fleeting Thoughts

And every non-discrete, nontrivial topological space also has subsets which are neither closed nor open. I guess it depends on what you mean by non-trivial, but this doesn't seem right. Take the space on two points, A and B, whose open sets are {}, {A}, and {A, B}. More generally, take the open set...
by Cauchy
Thu Jun 02, 2016 9:12 am UTC
Forum: Mathematics
Topic: Misunderstanding basic math concepts, help please?
Replies: 338
Views: 80534

Re: Misunderstanding basic math concepts, help please?

⋅ A system X is specified such that, in principle, all well formed propositions within and relating to X resolve to either true or false. Any conceivable alternative state(s) than 'true and false' are explicitly rejected (rename the labels - but there is no semi-true). As you note, being ...
by Cauchy
Tue May 31, 2016 10:28 am UTC
Forum: Mathematics
Topic: Misunderstanding basic math concepts, help please?
Replies: 338
Views: 80534

Re: Misunderstanding basic math concepts, help please?

I'll take #8. 1. A ⊃ (B & C) [Prem] 2. ~B [Prem] 3. (B & C) ⊃ B [Taut. #4] 4. A ⊃ B [1, 3, Hyp. Syll.] 5. ~A [4, 2, M.T.] ∎ I'd also nitpick both of the previous solutions. For #14, ~L ⊃ U doesn't directly follow from L V U by Disj. Syll.. (After all, Disj. Syll. need...
by Cauchy
Mon May 30, 2016 1:35 am UTC
Forum: Mathematics
Topic: Misunderstanding basic math concepts, help please?
Replies: 338
Views: 80534

Re: Misunderstanding basic math concepts, help please?

Personally, if a deduction has logical flaws, I might consider pointing out those logical flaws so that the deducer can learn from their errors. People have done that constantly with you. That's what Twistar was doing when he "graded" your response to the worksheet question. You seemed......
by Cauchy
Sun May 29, 2016 8:25 am UTC
Forum: Logic Puzzles
Topic: Cipher Puzzle
Replies: 2
Views: 3114

Re: Cipher Puzzle

Spoiler:
WE ARE ALL IN THE GUTTER, BUT SOME OF US ARE LOOKING AT THE STARS
by Cauchy
Sun May 29, 2016 8:16 am UTC
Forum: Mathematics
Topic: Misunderstanding basic math concepts, help please?
Replies: 338
Views: 80534

Re: Misunderstanding basic math concepts, help please?

The whole point and purpose of The Law of Excluded Middle is that the middle is explicitly excluded. There is no middle ground between 'specified' and 'not-specified'. "specified imperfectly" has been deliberately removed from the table as a possibility. For the thousandth time, the Law o...
by Cauchy
Sat May 28, 2016 11:38 am UTC
Forum: Mathematics
Topic: Find the number to divide with another one to obtain zero remainder.
Replies: 7
Views: 3088

Re: Find the number to divide with another one to obtain zero remainder.

If we're going to admit non-integer solutions, why bother with factoring at all? We're looking for x such that 29.7/x = n, where n is an integer. Then, 29.7 = xn, so x = 29.7/n, where again n is an integer. This completely classifies the possible divisors: for each integer n, 29.7/n is one solution ...

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