nyuuzyou/wikis huggingface dataset:
Locally Convex Space is Hausdorff iff induces Hausdorff Topology

== Theorem ==
Let $\struct {X, \mathcal P}$ be a locally convex space.
Let $\tau$ be the standard topology on $\struct {X, \mathcal P}$.
Then $\struct {X, \mathcal P}$ is Hausdorff $\struct {X, \tau}$ is a Hausdorff topological space. 
== Proof ==
=== Sufficient Condition ===
Suppose that $\struct {X, \tau}$ is a Hausdorff space.
Let $x \in X$ have $x \ne {\mathbf 0}_X$.
If there exists no such $x$, then we have the claim by vacuous truth.
Otherwise, since $\struct {X, \tau}$ is Hausdorff, there exists $U, V \in \tau$ with $x \in U$ and ${\mathbf 0}_X \in V$ and $U \cap V = \O$. 
Then $x \not \in V$ and ${\mathbf 0}_X \not \in V$.
From Open Sets in Standard Topology of Locally Convex Space, there exists $\epsilon > 0$ and seminorms $p_1, p_2, \ldots, p_n$ such that: 
:$\set {z \in X : \map {p_k} z < \epsilon, \text { for all } k \in \set {1, 2, \ldots, n} } \subseteq V$
Then we have, since $x \not \in V$:
:$x \not \in \set {z \in X : \map {p_k} z < \epsilon, \text { for all } k \in \set {1, 2, \ldots, n} }$
Then there exists $k \in \set {1, 2, \ldots, n}$ such that: 
:$\map {p_k} x \ge \epsilon > 0$ 
In particular: 
:$\map {p_k} x \ne 0$
with $p_k \in \mathcal P$.
So $\struct {X, \mathcal P}$ is Hausdorff.
=== Necessary Condition ===
Suppose that $\struct {X, \mathcal P}$ is Hausdorff.
Let $x, y \in X$ have $x \ne y$.
We aim to find $U, V \in \tau$ such that $x \in U$ and $y \in V$ with $U \cap V = \O$.
Since $x \ne y$, we have $y - x \ne {\mathbf 0}_X$.
So from the definition of a Hausdorff locally convex space, there exists $p \in \mathcal P$ such that: 
:$\map p {y - x} \ne 0$
That is: 
:$\map p {y - x} > 0$
since seminorms map onto the non-negative real numbers. 
Let: 
:$\epsilon = \map p {y - x}$
Then consider: 
:$U = \set {z \in X : \map p {z - x} < \epsilon/2}$
and:
:$V = \set {z \in X : \map p {z - y} < \epsilon/2}$
From Open Sets in Standard Topology of Locally Convex Space, we have that $U, V \in \tau$. 
From Seminorm Maps Zero Vector to Zero, we have: 
:$\map p { {\mathbf 0}_X} = 0 < \epsilon/2$
So we have $x \in U$ and $y \in V$.
We now show that $U \cap V = \O$.
Suppose that $U \cap V \ne \O$.
Let $z \in U \cap V$.
Then: 
:$\map p {z - x} < \epsilon/2$
and:
:$\map p {z - y} < \epsilon/2$
Then, we have: 
a contradiction.
So there exists no such $z \in U \cap V$, so $U \cap V = \O$.
Since $x, y \in X$ were arbitrary, we have that $\struct {X, \tau}$ is Hausdorff.
Category:Locally Convex Spaces
Category:Hausdorff Spaces
Locally Convex Space is Topological Vector Space

== Theorem ==
Let $\GF \in \set {\R, \C}$. 
Let $\struct {X, \PP}$ be a locally convex space over $\GF$ equipped with the standard topology $\tau$. 
Then $\struct {X, \tau}$ is a topological vector space. 
 
=== Corollary ===
== Proof ==
From Vector Addition on Locally Convex Space is Continuous, vector addition on $X$ is continuous. 
From Scalar Multiplication on Locally Convex Space is Continuous, scalar multiplication on $X$ is continuous. 
So $\struct {X, \tau}$ is a topological vector space.
Category:Locally Convex Spaces
Category:Topological Vector Spaces
Category:Locally Convex Space is Topological Vector Space
Locally Convex Space is Topological Vector Space/Corollary

== Theorem ==
Let $\GF \in \set {\R, \C}$. 
Let $\struct {X, \PP}$ be a Hausdorff locally convex space over $\GF$ equipped with the standard topology $\tau$. 
Then $\struct {X, \tau}$ is a Hausdorff topological vector space. 
 
== Proof ==
From Locally Convex Space is Topological Vector Space, $\struct {X, \tau}$ is a topological vector space.
From Locally Convex Space is Hausdorff iff induces Hausdorff Topology, $\struct {X, \tau}$ is a Hausdorff space.
So $\struct {X, \tau}$ is a Hausdorff topological vector space.
Category:Hausdorff Topological Vector Spaces
Category:Locally Convex Space is Topological Vector Space
Locally Euclidean Space has Countable Local Basis Homeomorphic to Open Balls

== Theorem ==
Let $M$ be a locally Euclidean space of some dimension $d$.
Let $m \in M$.
Then: 
:there exists a countable local basis $\family{U_n}_{n \in \N}$ of $m$ where each $U_n$ is the homeomorphic image of an open ball of $\R^d$
== Proof ==
By definition of a locally Euclidean space:
:there exists an open neighbourhood $U$ of $m$ which is homeomorphic to an open subset $V$ of Euclidean space $\R^d$.
Let $\phi: U \to V$ be a homeomorphism.
By definition of the Euclidean space $\R^d$ the topology on $\R^d$ is the topology induced by the metric:
:$\ds \map {d_2} {x, y} := \paren {\sum_{i \mathop = 1}^n \paren {x_i - y_i}^2}^{1 / 2}$
where $x = \tuple {x_1, x_2, \ldots, x_d}, y = \tuple {y_1, y_2, \ldots, y_d} \in \R^d$.
By definition of the induced topology:
:$\exists \epsilon > 0 : \map {B_\epsilon} {\map \phi m} \subseteq V$
where $\map {B_\epsilon} {\map \phi m}$ is the open ball of radius $\epsilon$ and center $\map \phi m$
Consider the set of open balls:
:$\BB_m = \set{\map {B_{\epsilon / n}} {\map \phi m} : n \in \N_{>0}}$
From Sequence of Reciprocals is Null Sequence:
:the sequence $\sequence{\dfrac \epsilon n}_{n \in \N_{>0}}$ is a null sequence
From Null Sequence induces Local Basis in Metric Space:
:$\BB_m$ is a countable local basis of $\map \phi m$ in $\R^d$
From Local Basis of Open Subspace iff Local Basis:
:$\BB_m$ is a countable local basis of $\map \phi m$ in $V$
For each $n \in \N_{>0}$, let:
:$U_n = \phi^{-1} \sqbrk {\map {B_{\epsilon / n}} {\map \phi m}}$
Consider the set:
:$\BB'_m = \set{U_n : n \in \N_{>0}}$
From Inverse of Homeomorphism is Homeomorphism:
:$\phi^{-1} : U \to V$ is a homeomorphism
From Homeomorphic Image of Local Basis is Local Basis:
:$\BB'_m$ is a local basis of $m$ in $U$
From Local Basis of Open Subspace iff Local Basis:
:$\BB'_m$ is a local basis of $m$ in $M$
From Restriction of Homeomorphism is Homeomorphism:
:$\forall n \in \N_{>0}: U_n = \phi^{-1} \sqbrk {\map {B_{\epsilon / n}} {\map \phi m}}$ is homeomorphic to the open ball $\map {B_{\epsilon / n}} {\map \phi m}$
The result follows.
Category:Locally Euclidean Spaces
Locally Euclidean Space has Countable Neighborhood Basis Homeomorphic to Closed Balls

== Theorem ==
Let $M$ be a locally Euclidean space of some dimension $d$.
Let $m \in M$.
Then: 
:there exists a countable neighborhood basis $\family{N_n}_{n \in \N}$ of $m$ where each $N_n$ is the homeomorphic image of a closed ball of $\R^d$
== Proof ==
By definition of a locally Euclidean space:
:there exists an open neighbourhood $U$ of $m$ which is homeomorphic to an open subset $V$ of Euclidean space $\R^d$.
Let $\phi: U \to V$ be a homeomorphism.
By definition of the Euclidean space $\R^d$ the topology on $\R^d$ is the topology induced by the metric:
:$\ds \map {d_2} {x, y} := \paren {\sum_{i \mathop = 1}^n \paren {x_i - y_i}^2}^{1 / 2}$
where $x = \tuple {x_1, x_2, \ldots, x_d}, y = \tuple {y_1, y_2, \ldots, y_d} \in \R^d$.
By definition of the induced topology:
:$\exists \epsilon > 0 : \map {B_\epsilon} {\map \phi m} \subseteq V$
where $\map {B_\epsilon} {\map \phi m}$ is the open ball of radius $\epsilon$ and center $\map \phi m$
Consider the set of closed balls:
:$\NN_m = \set{\map {B^-_{\epsilon / 2 n}} {\map \phi m} : n \in \N_{>0}}$
From Sequence of Reciprocals is Null Sequence:
:the sequence $\sequence{\dfrac \epsilon {2 n}}_{n \in \N_{>0}}$ is a null sequence
From Null Sequence induces Neighborhood Basis of Closed Sets in Metric Space:
:$\NN_m$ is a countable neighborhood basis of $\map \phi m$ in $\R^d$
From Neighborhood Basis of Open Subspace iff Neighborhood Basis:
:$\NN_m$ is a countable local basis of $\map \phi m$ in $V$
For each $n \in \N_{>0}$, let:
:$U_n = \phi^{-1} \sqbrk {\map {B_{\epsilon / n}} {\map \phi m}}$
Consider the set:
:$\NN'_m = \set{U_n : n \in \N_{>0}}$
From Inverse of Homeomorphism is Homeomorphism:
:$\phi^{-1} : U \to V$ is a homeomorphism
From Homeomorphic Image of Neighborhood Basis is Neighborhood Basis and Continuity Defined from Closed Sets:
:$\NN'_m$ is a Neighborhood basis of $m$ in $U$
From Neighborhood Basis of Open Subspace iff Neighborhood Basis:
:$\NN'_m$ is a local basis of $m$ in $M$
From Restriction of Homeomorphism is Homeomorphism:
:$\forall n \in \N_{>0}: N_n = \phi^{-1} \sqbrk {\map {B_{\epsilon / n}^-} {\map \phi m}}$ is homeomorphic to the closed ball $\map {B_{\epsilon / n}^-} {\map \phi m}$
The result follows.
Category:Locally Euclidean Spaces
Locally Euclidean Space is First-Countable

== Theorem ==
Let $M$ be a locally Euclidean space of some dimension $d$.
Then $M$ is first-countable.
== Proof ==
Follows immediately from Locally Euclidean Space has Countable Local Basis Homeomorphic to Open Balls.
Category:Locally Euclidean Spaces
Category:First-Countable Spaces
Locally Euclidean Space is Locally Compact/Proof 1

== Theorem ==
== Proof ==
Let $m \in M$ be arbitrary.
By definition of locally Euclidean space, there exists an open neighborhood $U$ of $m$, homeomorphic to an open subset of $\R^d$.
By the definition of an open set, there is some open ball:
:$B = \map {B_\delta} {\map \phi m} = \set {x \in \R^d: \size {x - \map \phi m} < \delta}$
of radius $\delta$ containing $\map \phi m$, contained in $U$.
By Closure of Open Ball in Metric Space and Topological Closure is Closed, the set:
:$C = \set {x \in \R^d: \size {x - \map \phi m} \le \dfrac \delta 2}$
is closed, and $C \subseteq B \subseteq U$.
Moreover, $C$ is trivially bounded, hence compact by the Heine-Borel Theorem.
Now if $\phi$ is a homeomorphism $U \to \R^d$, then by definition $\phi^{-1}$ is continuous.
Therefore by Continuous Image of Compact Space is Compact, $\phi^{-1} \sqbrk C \subseteq M$ is compact.
Furthermore $m \in \phi^{-1} \sqbrk C$ because $\map \phi m \in C$.
Thus every point of $M$ has a compact neighborhood.
Category:Locally Euclidean Space is Locally Compact
Locally Euclidean Space is Locally Compact/Proof 2

== Theorem ==
== Proof ==
Let $m \in M$ be arbitrary.
From Locally Euclidean Space has Countable Neighborhood Basis Homeomorphic to Closed Balls:
:there exists a countable neighborhood basis $\family{K_n}_{n \in \N}$ of $m$ where each $N_n$ is the homeomorphic image of a closed ball of $\R^d$
From Closed Ball in Euclidean Space is Compact:
:$\forall n \in \N: K_n$ is the homeomorphic image of a compact subspace
From Continuous Image of Compact Space is Compact:
:$\forall n \in \N: K_n$ is a compact subspace
Hence:
:there exists a countable neighborhood basis $\family{K_n}_{n \in \N}$ of $m$ consisting of compact subspaces
It follows that $M$ is locally compact by definition.
Category:Locally Euclidean Space is Locally Compact
Locally Euclidean Space is Locally Path-Connected

== Theorem ==
Let $M$ be a locally Euclidean space of some dimension $d$.
Then $M$ is locally path-connected.
== Proof ==
Let $m \in M$ be arbitrary.
From Locally Euclidean Space has Countable Local Basis Homeomorphic to Open Balls:
:there exists a local basis $\family{U_n}_{n \in \N}$ of $m$ where each $U_n$ is the homeomorphic image of an open ball of $\R^d$.
For all $n \in \N$, let:
:$U_n = \phi_n \sqbrk {B_n}$
where $B_n$ is an open ball of $\R^d$ and $\phi_n: B_n \to U_n$ is a homeomorphism.
From Open Ball in Normed Vector Space is Path-Connected:
:$B_n$ is path-connected.
From Continuous Image of Path-Connected Set is Path-Connected: Metric Space:
:$U_n$ is path-connected.
Hence $m$ has a local basis of path-connected sets. 
Since $m$ was arbitrary, it follows that $M$ is locally path-connected by definition.
Category:Locally Euclidean Spaces
Category:Path-Connected Spaces
Locally Euclidean Subspace of Euclidean Space is Manifold

== Theorem ==
Let $\R^n$ be an Euclidean space for $n \in \N$.
Let $M = \struct{H, \tau_H}$ be a subspace of $\R^n$, where $H \subseteq \R^n$.
Let $M$ be locally Euclidean of dimension $d$.
Then $M$ is a $d$-manifold.
== Proof ==
From Metric Space is Hausdorff, $\R^n$ is a Hausdorff space.
From Subspace of Hausdorff Space is Hausdorff, $M$ is a Hausdorff space.
From Euclidean Space is Second-Countable, $\R^n$ is second-countable.
From Second-Countability is Hereditary, $M$ is second-countable.
Therefore, $M$ is a $d$-manifold.
== Sources ==
* : $\S 3$: New Spaces From Old: Subspaces. Topological Embeddings
Category:Topological Manifolds
Locally Finite Connected Graph is Countable

== Theorem ==
Let $G = \struct {V, E}$ be a graph which is connected and locally finite.
Then $G$ has countably many vertices and countably many edges.
== Proof ==
We first show that $V$ is countable.
If $V$ is finite, then it is surely countable.
Suppose instead that $V$ is infinite.
Choose an arbitrary vertex $q \in V$.
Recursively define a sequence $\sequence {S_n}$:
:Let $S_0 = \set q$.
:Let $S_{n + 1}$ be the set of all vertices that are adjacent to some element of $S_n$ but not adjacent to any element of $S_k$ for $k < n$.
:That is, $S_n$ is the set of vertices whose shortest path(s) to $q$ have $n$ edges.
Since $G$ is connected, $V = \ds \bigcup_{n \mathop \in \N} S_n$.
Furthermore, $S_n$ is finite and non-empty for each $n$.
But by Countable Union of Finite Sets is Countable, $\ds \bigcup_{n \mathop \in \N} S_n$ is countable, so $V$ is countable.
By Cartesian Product of Countable Sets is Countable, $V \times V$ is countable.
Let $E' = \set {\tuple {a, b}: \set {a, b} \in E}$.
By Subset of Countable Set is Countable, $E'$ is countable.
Let $g: E' \to E$ be defined by $\map g {a, b} = \set {a, b}$.
Then $g$ is surjective.
Thus by Surjection from Natural Numbers iff Countable and Composite of Surjections is Surjection, $E$ is countable as well.
Category:Graph Theory
Locally Finite Set of Subsets is Sigma-Locally Finite Set of Subsets

== Theorem ==
Let $T = \struct {S, \tau}$ be a topological space.
Let $\AA$ be a locally finite set of subsets.
Then:
:$\AA$ is a $\sigma$-locally finite set of subsets
== Proof ==
For each $n \in \N$, let
:$\AA_n = \AA$.
Then:
:$\AA = \ds \bigcup_{n \in \N} \AA_n$
The result follows.
Category:Locally Finite Sets of Subsets
Category:Sigma-Locally Finite Sets of Subsets
Locally Integrable (f(x+ct) + f(x-ct))/2 is Weak Solution to Wave Equation

== Theorem ==
Consider the wave equation:
:$\dfrac {\partial^2 u} {\partial t^2} - c^2 \dfrac {\partial^2 u} {\partial x^2} = 0$
with the initial conditions:
:$\map u {x, 0} = \map f x$
:$\map {\dfrac {\partial u} {\partial t}} {x, 0} = 0$
and $c \in \R$.
Then it has a weak solution of the form:
:$\map u {x, t} := \dfrac {\map f {x + ct} + \map f {x - ct}} 2$ 
where $f \in \map {L^1_{loc} } \R$ is a locally integrable function.
== Proof ==
Let $\map u {x, t} = \map f {x + ct}$ be a locally integrable function.
We have that a locally integrable function defines a distribution.
Let $T_u \in \map {\DD'} {\R^2}$ be a distribution associated with $u$.
Let $\phi \in \map \DD {\R^2}$ be a test function.
Then:
Therefore, $\ds \map u {x,t} := \frac {\map f {x + ct} + \map f {x - ct}} 2$ is a weak solution to the wave equation.
== Sources ==
* : Chapter $\S 6.3$: A glimpse of distribution theory. Weak solutions
Category:Examples of Weak Solutions
Category:Wave Equation
Locally Integrable Function defines Distribution

== Theorem ==
Let $f \in \map {L^1_{loc}} {\R^d}$ be a locally integrable function.
Let $\map \DD {\R^d}$ be the test function space.
Let $T_f : \map \DD {\R^d} \to \C$ be a mapping.
Then $T_f$ is a distribution.
== Proof ==
=== Existence ===
Let $\phi \in \map \DD {\R^d}$ be a test function.
Let $T_f$ be defined as
:$\ds T_f = \int_{\R^d} \map f {\mathbf x} \map \phi {\mathbf x} \rd \mathbf x$
By definition, $\phi$ has the compact support.
Together with the properties of $f$ we have that $T_f$ is bounded with respect to any compact range of integration.
=== Linearity ===
By Linear Combination of Integrals we have that $T_f$ is a linear mapping.
=== Continuity ===
By Convergent Sequence Minus Limit, we can shift the sequence to set its limit to zero.
Let $\mathbf 0 : \R^d \to 0$ be the zero mapping.
Let $\phi_n$ converge to $\mathbf 0$ in $\map \DD {\R^d}$:
:$\phi_n \stackrel \DD {\longrightarrow} {\mathbf 0}$
Let $K \subseteq \R^d$ be the compact support of all $\phi_n$.
By definition, $T_f$ is a distribution.
== Sources ==
* : Chapter $\S 6.1$: A glimpse of distribution theory. Test functions, distributions, and examples
Category:Examples of Distributions
Locally Integrable f(x+ct) is Weak Solution to Transport Equation

== Theorem ==
Consider the transport equation:
:$\dfrac {\partial u} {\partial t} - c \dfrac {\partial u} {\partial x} = 0$
with the initial condition:
:$\map u {x, 0} = \map f x$
where $c \in \R$.
Then it has a weak solution of the form:
:$\map u {x, t} := \map f {x + ct}$ 
where $f \in \map {L^1_{loc} } \R$ is a locally integrable function.
== Proof ==
Let $\map u {x, t} = \map f {x + ct}$ be a locally integrable function.
We have that a locally integrable function defines a distribution.
Let $T_u \in \map {\DD'} {\R^2}$ be a distribution associated with $u$.
Let $\phi \in \map \DD {\R^2}$ be a test function.
Then:
 
Introduce real variables $\xi, \eta$ such that:
:$\xi = x + ct$
:$\eta = t$
Then:
:$x = \xi - c \eta$
:$t = \eta$
Let the expressions above define the mapping $\Psi$ between these variables:
:$\tuple {\xi, \eta} \stackrel {\Psi} {\mapsto} \tuple {x, t}$
Suppose $F : \R^2 \to \R$ is a real function.
By Change of Variables Theorem (Multivariable Calculus):
where $\circ$ denotes the composition of mappings, and $\mathbf J_\Psi$ is the Jacobian matrix:
Hence:
:$\map \det { {\mathbf J_\Psi} } = 1$.
Suppose:
:$\map F {x, t} = \map f {x + ct} \paren {c \map {\dfrac {\partial \phi}{\partial x} } {x, t} - \map {\dfrac {\partial \phi}{\partial t} } {x, t} }$
Then:
Hence:
Therefore, $\map u {x, t} := \map f {x + c t}$ is a weak solution to the transport equation.
== Sources ==
* : Chapter $\S 6.3$: A glimpse of distribution theory. Weak solutions
Category:Examples of Weak Solutions
Locally Path-Connected Space is Locally Connected

== Theorem ==
Let $T = \struct {S, \tau}$ be a topological space which is locally path-connected.
Then $T$ is also locally connected.
== Proof ==
Let $x \in S$ be any point of $T$.
Let $\BB$ be a local basis of path-connected sets for $x$.
From Path-Connected Space is Connected, $\BB$ is a local basis of connected sets.
Thus, $T$ is locally connected by definition.
== Sources ==
* : Part $\text I$: Basic Definitions: Section $4$: Connectedness
Category:Locally Path-Connected Spaces
Category:Locally Connected Spaces
Locally Path-Connected Space is not necessarily Locally Arc-Connected

== Theorem ==
Let $T = \struct {S, \tau}$ be a topological space which is locally path-connected.
Then it is not necessarily the case that $T$ is also a locally arc-connected space.
== Proof ==
Let $T$ be the Either-Or topological space.
From Either-Or Topology is Locally Path-Connected, $T$ is a locally path-connected space.
From Either-Or Topology is not Locally Arc-Connected, $T$ is not a locally arc-connected space.
Hence the result.
== Sources ==
* : Part $\text I$: Basic Definitions: Section $4$: Connectedness
Category:Locally Path-Connected Spaces
Category:Locally Arc-Connected Spaces
Locally Path-Connected Space is not necessarily Path-Connected

== Theorem ==
Let $T = \struct {S, \tau}$ be a topological space which is locally path-connected.
Then it is not necessarily the case that $T$ is also path-connected.
== Proof ==
Let $T := \struct {\R, \tau_d}$ be the real number line $\R$ under the usual (Euclidean) topology $\tau_d$.
Let $a, b, c \in \R$ where $a < b < c$.
Let $A$ be the union of the two open intervals:
:$A := \openint a b \cup \openint b c$
Let $T' := \struct {A, \tau_A}$ be the subspace composed of $A$ with the subspace topology induced on $\tau_d$ by $A$.
From Union of Adjacent Open Intervals is Locally Path-Connected, $T'$ is a locally path-connected space.
From Union of Adjacent Open Intervals is not Path-Connected, $T$ is not a path-connected space.
Hence the result.
== Sources ==
* : Part $\text I$: Basic Definitions: Section $4$: Connectedness
Category:Locally Path-Connected Spaces
Category:Path-Connected Spaces
Locker Problem

== Problem ==
There is a school with $100$ students, and correspondingly $100$ lockers, all of which start off closed.
The first student opens every locker.
The second student closes every other locker, starting with the second.
The third student changes the state of every third locker, starting with the third.
That is, if the locker is open, he or she closes it, and if it is closed, he or she opens it.
This continues similarly until all $100$ students have passed along the lockers.
After the $100$th student has finished, which lockers are open and which are closed?
== Solution ==
The square numbered lockers ($1, 4, 9, \ldots$) are open, and all others are closed.
== Proof 1 ==
== Proof 2 ==
Category:Number Theory
Category:Square Numbers
Category:Named Theorems
Category:Classic Problems
Category:Locker Problem
Locker Problem/Proof 1

== Problem ==
== Proof ==
Lockers switch state when a student whose number is a factor of the locker's number makes a pass.
Because they start closed, only those lockers whose state has been switched an odd number of times will finish open.
By definition, any factor of a number $n$ has another (not necessarily distinct) matching factor such that the product of the two is $n$.
In most cases, these pairs of factor have distinct elements.
However, if the number in question is a square, then one of its pairs of factors is identical.
Indeed, both elements of the pair are the number's square root.
Thus:
:square numbers will have their state switched an odd number of times, and will end up open
:all other numbers will have their state switched an even number of times, and will end up closed.
Category:Locker Problem
Loculus of Archimedes/Examples/Elephant

== Example of Use of Loculus of Archimedes ==
Using the shapes in the loculus of Archimedes exactly once each, make the shape of an elephant:
:Loculus-of-Archimedes-Elephant.png
== Solution ==
== Sources ==
* : Loculus of Archimedes: $17$
Category:Loculus of Archimedes