Mathematics

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1500 questions
88
votes
10 answers

100 Soldiers riddle

One of my friends found this riddle. There are 100 soldiers. 85 lose a left leg, 80 lose a right leg, 75 lose a left arm, 70 lose a right arm. What is the minimum number of soldiers losing all 4 limbs? We can't seem to agree on a way to approach…
Pieter van Niekerk
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88
votes
3 answers

Complexity class of comparison of power towers

Consider the following decision problem: given two lists of positive integers $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_m$ the task is to decide if $a_1^{a_2^{\cdot^{\cdot^{\cdot^{a_n}}}}} < b_1^{b_2^{\cdot^{\cdot^{\cdot^{b_m}}}}}$. Is this…
Vladimir Reshetnikov
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87
votes
14 answers

Stirling's formula: proof?

Suppose we want to show that $$ n! \sim \sqrt{2 \pi} n^{n+(1/2)}e^{-n}$$ Instead we could show that $$\lim_{n \to \infty} \frac{n!}{n^{n+(1/2)}e^{-n}} = C$$ where $C$ is a constant. Maybe $C = \sqrt{2 \pi}$ . What is a good way of doing this? Could…
James
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87
votes
5 answers

How to use the Extended Euclidean Algorithm manually?

I've only found a recursive algorithm of the extended Euclidean algorithm. I'd like to know how to use it by hand. Any idea?
Andrew
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87
votes
4 answers

Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$

Using $\text{n}^{\text{th}}$ root of unity $$\large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$ Prove that $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
Ali_ilA
  • 973
87
votes
8 answers

What is integration by parts, really?

Integration by parts comes up a lot - for instance, it appears in the definition of a weak derivative / distributional derivative, or as a tool that one can use to turn information about higher derivatives of a function into information about an…
Elle Najt
  • 21,722
87
votes
5 answers

Proving that the tensor product is right exact

Let $A\stackrel{\alpha}{\rightarrow}B\stackrel{\beta}{\rightarrow}C\rightarrow 0$ a exact sequence of left $R$-modules and $M$ a left $R$-module ($R$ any ring). I am trying to prove that the induced sequence $$A\otimes_R M\xrightarrow{\alpha\otimes…
Klaus
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87
votes
6 answers

Can $\det(A + B)$ be expressed in terms of $\det(A)$, $\det(B)$, $n$?

Let $A$ and $B$ be $n \times n$ matrices. Can $\det(A + B)$ be expressed in terms of $\det(A)$ , $\det(B)$ , $n$ ?
Sidharth Ghoshal
  • 18,528
87
votes
11 answers

Logic problem: Identifying poisoned wines out of a sample, minimizing test subjects with constraints

I just got out from my Math and Logic class with my friend. During the lecture, a well-known math/logic puzzle was presented: The King has $1000$ wines, $1$ of which is poisoned. He needs to identify the poisoned wine as soon as possible, and with…
Justin L.
  • 15,298
87
votes
9 answers

What is the proof that the total number of subsets of a set is $2^n$?

What is the proof that given a set of $n$ elements there are $2^n$ possible subsets (including the empty-set and the original set).
Celeritas
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87
votes
6 answers

A new pythagorean proof

I am writing to you today because I believe I may have developed a new visual proof for the Pythagorean theorem. The proof is based on a geometric dissection method, equating the area of a larger composite square to the sum of its smaller…
kingyoon
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87
votes
0 answers

Does the $32$-inator exist?

Background It is common popular-math knowledge that as we extend the real numbers to complex numbers, quaternions, octonions, sedenions, $32$ -nions, etc. using the Cayley-Dickson construction , we lose algebraic properties at each step such as…
pregunton
  • 6,440
87
votes
7 answers

Is an automorphism of the field of real numbers the identity map?

Is an automorphism of the field of real numbers $\mathbb{R}$ the identity map? If yes, how can we prove it? Remark An automorphism of $\mathbb{R}$ may not be continuous.
Makoto Kato
  • 44,710
87
votes
1 answer

Does every Cauchy sequence converge to *something*, just possibly in a different space?

Question. If I attempt to prove that space $X$ is complete by pursuing the strategy, “Assume $x_n \rightarrow x$ ; the space $X$ is complete if $x \in X$ ,” then why is that wrong? Context. I know the definition of Cauchy sequences and convergent…
1Teaches2Learn
  • 2,589
87
votes
6 answers

Any rectangular shape on a calculator numpad when divided by 11 gives an integer. Why?

I have come across this fact a very long time ago, but I still can't tell why is this happening. Given the standard calculators numpad: 7 8 9 4 5 6 1 2 3 if you dial any rectangular shape, going only in right angles and each shape consisting of 4…
noncom
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