nyuuzyou/wikis huggingface dataset:
Four Fours

__NOTOC__
== Puzzle ==
Using exactly $4$ instances of the number $4$, it is possible to write an expression for all positive integers from $0$ to $100$, using whatever arithmetical operations are necessary.
=== Further Results ===
== Solution ==
==== Four Fours: $0$ ====
==== Four Fours: $1$ ====
==== Four Fours: $2$ ====
==== Four Fours: $3$ ====
==== Four Fours: $4$ ====
==== Four Fours: $5$ ====
==== Four Fours: $6$ ====
==== Four Fours: $7$ ====
==== Four Fours: $8$ ====
==== Four Fours: $9$ ====
==== Four Fours: $10$ ====
==== Four Fours: $11$ ====
==== Four Fours: $12$ ====
==== Four Fours: $13$ ====
==== Four Fours: $14$ ====
==== Four Fours: $15$ ====
==== Four Fours: $16$ ====
==== Four Fours: $17$ ====
==== Four Fours: $18$ ====
==== Four Fours: $19$ ====
==== Four Fours: $20$ ====
==== Four Fours: $21$ ====
==== Four Fours: $22$ ====
==== Four Fours: $23$ ====
==== Four Fours: $24$ ====
==== Four Fours: $25$ ====
==== Four Fours: $26$ ====
==== Four Fours: $27$ ====
==== Four Fours: $28$ ====
==== Four Fours: $29$ ====
==== Four Fours: $30$ ====
==== Four Fours: $31$ ====
==== Four Fours: $32$ ====
==== Four Fours: $33$ ====
==== Four Fours: $34$ ====
==== Four Fours: $35$ ====
==== Four Fours: $36$ ====
==== Four Fours: $37$ ====
==== Four Fours: $38$ ====
==== Four Fours: $39$ ====
==== Four Fours: $40$ ====
==== Four Fours: $41$ ====
==== Four Fours: $42$ ====
==== Four Fours: $43$ ====
==== Four Fours: $44$ ====
==== Four Fours: $45$ ====
==== Four Fours: $46$ ====
==== Four Fours: $47$ ====
==== Four Fours: $48$ ====
==== Four Fours: $49$ ====
==== Four Fours: $50$ ====
==== Four Fours: $51$ ====
==== Four Fours: $52$ ====
==== Four Fours: $53$ ====
==== Four Fours: $54$ ====
==== Four Fours: $55$ ====
==== Four Fours: $56$ ====
==== Four Fours: $57$ ====
==== Four Fours: $58$ ====
==== Four Fours: $59$ ====
==== Four Fours: $60$ ====
==== Four Fours: $61$ ====
==== Four Fours: $62$ ====
==== Four Fours: $63$ ====
==== Four Fours: $64$ ====
==== Four Fours: $65$ ====
==== Four Fours: $66$ ====
==== Four Fours: $67$ ====
==== Four Fours: $68$ ====
==== Four Fours: $69$ ====
==== Four Fours: $70$ ====
==== Four Fours: $71$ ====
==== Four Fours: $72$ ====
==== Four Fours: $73$ ====
==== Four Fours: $74$ ====
==== Four Fours: $75$ ====
==== Four Fours: $76$ ====
==== Four Fours: $77$ ====
==== Four Fours: $78$ ====
==== Four Fours: $79$ ====
==== Four Fours: $80$ ====
==== Four Fours: $81$ ====
==== Four Fours: $82$ ====
==== Four Fours: $83$ ====
==== Four Fours: $84$ ====
==== Four Fours: $85$ ====
==== Four Fours: $86$ ====
==== Four Fours: $87$ ====
==== Four Fours: $88$ ====
==== Four Fours: $89$ ====
==== Four Fours: $90$ ====
==== Four Fours: $91$ ====
==== Four Fours: $92$ ====
==== Four Fours: $93$ ====
==== Four Fours: $94$ ====
==== Four Fours: $95$ ====
==== Four Fours: $96$ ====
==== Four Fours: $97$ ====
==== Four Fours: $98$ ====
==== Four Fours: $99$ ====
==== Four Fours: $100$ ====
=== Lemmata ===
== Glossary ==
== Historical Note ==
== Sources ==
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Classic Problems
Category:Arithmetic
Category:Four Fours
Four Fours/0

== Puzzle ==
Using exactly $4$ instances of the number $4$, the task is to write an expression for $0$, using whatever arithmetical operations you consider necessary.
== Solutions ==
== General Solutions ==
Let $\map {e_n} m$ be an expression which uses $n$ instances of $4$ to make $m$.
Then we have:
== Also see ==
:
Category:Four Fours
Category:Four Fours/0
Four Fours/1

== Puzzle ==
Using exactly $4$ instances of the number $4$, the task is to write an expression for $1$, using whatever arithmetical operations you consider necessary.
== Solutions ==
== General Solutions ==
Let $\map {e_n} m$ be an expression which uses $n$ instances of $4$ to make $m$.
Then we have:
== Also see ==
:
Category:Four Fours
Category:Four Fours/1
Four Fours/1/General Solutions/1

== Puzzle ==
== Solution ==
Let $\map {e_n} m$ be an expression which uses $n$ instances of $4$ to make $m$.
Then we have:
:$1 = \dfrac {\map {e_2} m} {\map {e_2} m}$
where $m \in \N$ is arbitrary.
Category:Four Fours/1
Four Fours/1/General Solutions/2

== Puzzle ==
== Solution ==
Let $\map {e_n} m$ be an expression which uses $n$ instances of $4$ to make $m$.
Then we have:
:$1 = \dfrac {\map {e_1} m} {\map {e_3} m}$
where $m \in \N$ is arbitrary.
Category:Four Fours/1
Four Fours/100/Solutions/2

== Puzzle: Four Fours: $100$ ==
== Solution ==
:$100 = \paren {\dfrac 4 {.4} }^{\tfrac 4 {\sqrt 4} }$
== Sources ==
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/100
Four Fours/13/Mistake

== Source Work ==
:
:Four Fours
::Solution
== Mistake ==
:The writer added the following postscript: "Another simple solution for $13$ is: $13 = \dfrac {4!} 4 + \dfrac 4 4$."
In fact $\dfrac {4!} 4 + \dfrac 4 4 = 7$.
It is probable that the above is a misprint for this solution:
== Sources ==
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Mistakes/Mathematical Bafflers
Four Fours/13/Solutions/4

== Puzzle: Four Fours: $13$ ==
== Solution ==
:$13 = \map \Gamma 4 \times \sqrt 4 + \dfrac 4 4$
where $\Gamma$ is the gamma function: $\map \Gamma 4 = 6$
== Sources ==
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/13
Four Fours/17/Solutions/1

== Puzzle: Four Fours: $17$ ==
== Solution ==
:$17 = 4 \times 4 + \dfrac 4 4$
== Sources ==
* : Arithmetical and Algebraical Problems: Digital Puzzles: $58$. -- The Two Fours
* : Solutions: $(46)$
* : Arithmetical and Algebraical Problems: Digital Puzzles: $109$. The Two Fours
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/17
Four Fours/19/Solutions/2

== Puzzle: Four Fours: $19$ ==
== Solution ==
:$19 = 4! - 4 - \dfrac 4 4$
== Sources ==
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/19
Four Fours/2/Solutions/1

== Puzzle: Four Fours: $2$ ==
== Solution ==
:$2 = \dfrac 4 4 + \dfrac 4 4$
== Sources ==
* : Solutions: $(46)$
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/2
Four Fours/31/Solutions/1

== Puzzle: Four Fours: $31$ ==
== Solution ==
:$31 = 4! + 4 + \sqrt {\dfrac 4 {. \dot 4} }$
== Sources ==
* : Solutions: $(46)$
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/31
Four Fours/33/Solutions/2

== Puzzle: Four Fours: $33$ ==
== Solution ==
:$33 = \dfrac 4 {. \dot 4} + \paren {\sqrt 4 + \sqrt 4}!$
== Sources ==
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/33
Four Fours/33/Solutions/4

== Puzzle: Four Fours: $33$ ==
== Solution ==
:$33 = \map \Gamma 4 + \dfrac {\map \Gamma 4} {\sqrt 4} + 4!$
== Sources ==
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/33
Four Fours/4/Solutions/3

== Puzzle: Four Fours: $4$ ==
== Solution ==
:$4 = \sqrt {\dfrac 4 {.\dot 4} } + \dfrac 4 4$
where $.\dot 4$ is interpreted as $0.44444 \ldots = \dfrac 4 9$
== Sources ==
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/4
Four Fours/5/Solutions/2

== Puzzle: Four Fours: $5$ ==
== Solution ==
:$5 = \sqrt {\dfrac 4 {.\dot 4} } + \dfrac 4 {\sqrt 4}$
== Sources ==
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/5
Four Fours/50/Solutions/3

== Puzzle: Four Fours: $50$ ==
== Solution ==
:$50 = 44 + 4 + \sqrt 4$
== Sources ==
* : Arithmetical and Algebraical Problems: Digital Puzzles: $58$. -- The Two Fours
* : Solutions: $(46)$
* : Arithmetical and Algebraical Problems: Digital Puzzles: $109$. The Two Fours
Category:Four Fours/50
Four Fours/63/Solutions/1

== Puzzle: Four Fours: $63$ ==
== Solution ==
:$63 = \dfrac {4!} {.4} + \sqrt {\dfrac 4 {. \dot 4} }$
== Sources ==
* : Solutions: $(46)$
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/63
Four Fours/66/Solutions/2

== Puzzle: Four Fours: $66$ ==
== Solution ==
:$66 = \dfrac {4!} {.4} + \dfrac 4 {\sqrt {. \dot 4} }$
== Sources ==
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/66
Four Fours/67/Solutions/2

== Puzzle: Four Fours: $67$ ==
== Solution ==
:$67 = \dfrac {4! + 4} {. \dot 4} + 4$
== Sources ==
* : Solutions: $(46)$
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/67
Four Fours/73/Solutions/2

== Puzzle: Four Fours: $73$ ==
== Solution ==
:$73 = \dfrac {4! \times \sqrt 4 + \sqrt {. \dot 4} } {\sqrt {. \dot 4} }$
== Sources ==
* : Solutions: $(46)$
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/73
Four Fours/75/Solutions/2

== Puzzle: Four Fours: $75$ ==
== Solution ==
:$75 = \dfrac {4! \times \sqrt 4 + \sqrt 4} {\sqrt {. \dot 4} }$
== Sources ==
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/75
Four Fours/78/Solutions/2

== Puzzle: Four Fours: $78$ ==
== Solution ==
:$78 = \dfrac {4! + 4! \times {. \dot 4} } {\sqrt {. \dot 4} }$
== Sources ==
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/78
Four Fours/79/Solutions/1

== Puzzle: Four Fours: $79$ ==
== Solution ==
:$79 = \paren {\dfrac 4 {. \dot 4} }^{\sqrt 4} - \sqrt 4$
== Sources ==
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/79
Four Fours/81/Solutions/2

== Puzzle: Four Fours: $81$ ==
== Solution ==
:$81 = \paren {\dfrac 4 {. \dot 4} }^{\tfrac 4 {\sqrt 4} }$
== Sources ==
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/81
Four Fours/83/Solutions/1

== Puzzle: Four Fours: $83$ ==
== Solution ==
:$83 = \paren {\dfrac 4 {. \dot 4} }^{\sqrt 4} + \sqrt 4$
== Sources ==
* : Solutions: $(46)$
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/83
Four Fours/85/Solutions/1

== Puzzle: Four Fours: $85$ ==
== Solution ==
:$85 = \paren {\dfrac 4 {. \dot 4} }^{\sqrt 4} + 4$
== Sources ==
* : Solutions: $(46)$
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/85
Four Fours/85/Solutions/3

== Puzzle: Four Fours: $85$ ==
== Solution ==
:$85 = \paren {\dfrac {\map \Gamma 4} {\sqrt 4} }^4 + 4$
== Sources ==
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/85
Four Fours/87/Solutions/1

== Puzzle: Four Fours: $87$ ==
== Solution ==
:$87 = 4! \times 4 - \dfrac 4 {. \dot 4}$
== Sources ==
* : Solutions: $(46)$
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/87
Four Fours/89/Solutions/1

== Puzzle: Four Fours: $89$ ==
== Solution ==
:$89 = \dfrac {4! + \sqrt 4} {.4} + 4!$
== Sources ==
* : Solutions: $(46)$
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/89
Four Fours/91/Solutions/1

== Puzzle: Four Fours: $91$ ==
== Solution ==
:$91 = 4! \times 4 - \dfrac {\sqrt 4} {.4}$
== Sources ==
* : Solutions: $(46)$
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/91
Four Fours/93/Solutions/1

== Puzzle: Four Fours: $93$ ==
== Solution ==
:$93 = 4! \times 4 - \sqrt {\dfrac 4 {. \dot 4} }$
== Sources ==
* : Solutions: $(46)$
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/93
Four Fours/99/Solutions/1

== Puzzle: Four Fours: $99$ ==
== Solution ==
:$99 = 4! \times 4 + \sqrt {\dfrac 4 {. \dot 4} }$
== Sources ==
* : Solutions: $(46)$
* : $1$. Say it with Letters: Algebraic Amusements: Four Fours
Category:Four Fours/99
Four Fours/Lemmata/One Four

== Puzzle ==
$4$ can be used on its own to make the following:
==== One Four: $\dfrac 4 {10}$ ====
==== One Four: $\dfrac 4 9$ ====
==== One Four: $\dfrac 2 3$ ====
==== One Four: $1$ ====
==== One Four: $2$ ====
==== One Four: $3$ ====
==== One Four: $4$ ====
==== One Four: $6$ ====
==== One Four: $9$ ====
==== One Four: $24$ ====
==== One Four: $64$ ====
== Glossary ==
Category:Four Fours
Category:Four Fours/Lemmata/One Four
Four Fours/Lemmata/One Four/3/Solutions/1

== Puzzle: One Four: $3$ ==
== Solution ==
:$3 = \map \pi {\map \Gamma 4}$
where:
:$\map \Gamma 4$ denotes the Gamma function of $4$: $\map \Gamma 4 = 6$
:$\pi$ denotes the prime-counting function:
::the primes smaller than $6$ are: $2$, $3$, $5$.
Category:Four Fours/Lemmata/One Four
Four Fours/Lemmata/One Four/64/Solutions/1

== Puzzle: One Four: $64$ ==
== Solution ==
:$64 = \floor {\surd \surd \surd \surd \surd \surd \surd \surd \surd \floor {\surd \surd \surd \surd \surd \surd \surd \surd \surd \floor {\surd \surd \surd \surd \surd \surd \surd \surd \surd \surd \surd \surd \surd \floor {\surd \surd \surd \surd \surd \surd \surd \surd \floor { \surd \surd \surd \surd \surd \surd \surd \surd \surd \surd \surd \floor {\surd \floor {\surd \floor {\surd \surd \surd \surd \surd \paren {4!} !} !} !} !} !} !} !} !}$
where:
:$x!$ denotes the factorial of $x$
:$\floor x$ denotes the floor function of $x$.
== Proof ==
Note that in the below, factorial has a higher binding priority than the square root function.
That is, $\surd x!$ means $\sqrt {\paren {x!} }$ and not $\paren {\sqrt x} !$.
We have that:
Then:
Then:
The pattern continues.
We see that the $k$th floor function evaluates to the integer $a_k$ such that:
:${a_k}^{2^t} < a_{k - 1} ! < \paren {a_k + 1}^{2^t}$
where $t$ is the number of square root signs between the floor function delimiters.
Laborious evaluation then allows us to construct the following table:
:$\begin {array} {crrlll}
k & a_k & t & {a_k}^{2^t} & a_{k - 1} ! & \paren {a_k + 1}^{2^t} \\
0 & 24 & & & & \\
1 & 5 & 5 & 0.23 \times 10^{23} & 0.62 \times 10^{24} & 0.79 \times 10^{25} \\
2 & 10 & 1 & 100 & 120 & 121 \\
3 & 1904 & 1 & 36 \, 25216 & 36 \, 28800 & 39 \, 29025 \\
4 & 442 & 11 & 0.67 \times 10^{5417} & 0.42 \times 10^{5419} & 0.68 \times 10^{5419} \\
5 & 6673 & 8 & 0.1062 \times 10^{979} & 0.1097 \times 10^{979} & 0.1104 \times 10^{979} \\
6 & 577 & 13 & 0.4 \times 10^{22619} & 0.9 \times 10^{22623} & 0.5 \times 10^{22625} \\
7 & 422 & 9 & 0.14 \times 10^{1344} & 0.25 \times 10^{1344} & 0.49 \times 10^{1344} \\
8 & 64 & 9 & 0.58 \times 10^{924} & 0.21 \times 10^{926} & 0.16 \times 10^{928} \\
\end {array}$
From Floor of Root of Floor equals Floor of Root, we have:
:$\ds \floor {\sqrt {\floor x} } = \floor {\sqrt x}$
so we need only to take the floor function of the entire result, and before the factorial is evaluated.
The result follows.
Note in passing that $\floor {\surd 10} = 3$, allowing us an expression for $3$ using one $4$:
:$3 = \floor {\surd \floor {\surd \floor {\surd \surd \surd \surd \surd \paren {4!} !} !} }$
from $(2)$ above.
Similarly, as $3! = 6$, we have a similar expression for $6$:
:$6 = \floor {\surd \floor {\surd \floor {\surd \surd \surd \surd \surd \paren {4!} !} !} }!$
== Sources ==
* 
Category:Four Fours/Lemmata/One Four
Four Fours/Lemmata/One Four/9/Solutions/1

== Puzzle: One Four: $9$ ==
== Solution ==
:$9 = \map \pi {4!}$
where:
:$4!$ denotes the factorial of $4$
:$\pi$ denotes the prime-counting function:
::the primes smaller than $24$ are: $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$.
Category:Four Fours/Lemmata/One Four
Four Fours/Lemmata/Three Fours

== Puzzle ==
Three instances of $4$ can be used to make the following:
==== Three Fours: $0$ ====
==== Three Fours: $1$ ====
==== Three Fours: $2$ ====
==== Three Fours: $3$ ====
==== Three Fours: $4$ ====
==== Three Fours: $5$ ====
==== Three Fours: $6$ ====
==== Three Fours: $7$ ====
==== Three Fours: $8$ ====
==== Three Fours: $9$ ====
==== Three Fours: $10$ ====
==== Three Fours: $11$ ====
==== Three Fours: $12$ ====
==== Three Fours: $13$ ====
==== Three Fours: $14$ ====
==== Three Fours: $15$ ====
==== Three Fours: $16$ ====
==== Three Fours: $17$ ====
==== Three Fours: $18$ ====
==== Three Fours: $19$ ====
==== Three Fours: $20$ ====
==== Three Fours: $21$ ====
==== Three Fours: $22$ ====
==== Three Fours: $23$ ====
==== Three Fours: $24$ ====
==== Three Fours: $25$ ====
==== Three Fours: $26$ ====
==== Three Fours: $27$ ====
==== Three Fours: $28$ ====
==== Three Fours: $29$ ====
==== Three Fours: $30$ ====
==== Three Fours: $32$ ====
==== Three Fours: $33$ ====
==== Three Fours: $34$ ====
==== Three Fours: $36$ ====
==== Three Fours: $40$ ====
==== Three Fours: $42$ ====
==== Three Fours: $44$ ====
==== Three Fours: $45$ ====
==== Three Fours: $46$ ====
==== Three Fours: $48$ ====
==== Three Fours: $50$ ====
==== Three Fours: $52$ ====
==== Three Fours: $53$ ====
==== Three Fours: $54$ ====
==== Three Fours: $55$ ====
==== Three Fours: $56$ ====
==== Three Fours: $58$ ====
==== Three Fours: $60$ ====
==== Three Fours: $61$ ====
==== Three Fours: $62$ ====
==== Three Fours: $63$ ====
==== Three Fours: $64$ ====
==== Three Fours: $65$ ====
==== Three Fours: $70$ ====
==== Three Fours: $72$ ====
==== Three Fours: $78$ ====
==== Three Fours: $81$ ====
==== Three Fours: $84$ ====
==== Three Fours: $90$ ====
==== Three Fours: $96$ ====
== Glossary ==
Category:Four Fours
Category:Four Fours/Lemmata/Three Fours
Four Fours/Lemmata/Two Fours

== Puzzle ==
Two instances of $4$ can be used to make the following:
==== Two Fours: $0$ ====
==== Two Fours: $1$ ====
==== Two Fours: $2$ ====
==== Two Fours: $3$ ====
==== Two Fours: $4$ ====
==== Two Fours: $5$ ====
==== Two Fours: $6$ ====
==== Two Fours: $7$ ====
==== Two Fours: $8$ ====
==== Two Fours: $9$ ====
==== Two Fours: $10$ ====
==== Two Fours: $12$ ====
==== Two Fours: $15$ ====
==== Two Fours: $16$ ====
==== Two Fours: $18$ ====
==== Two Fours: $20$ ====
==== Two Fours: $22$ ====
==== Two Fours: $23$ ====
==== Two Fours: $24$ ====
==== Two Fours: $25$ ====
==== Two Fours: $26$ ====
==== Two Fours: $28$ ====
==== Two Fours: $30$ ====
==== Two Fours: $36$ ====
==== Two Fours: $44$ ====
==== Two Fours: $48$ ====
==== Two Fours: $54$ ====
==== Two Fours: $60$ ====
==== Two Fours: $64$ ====
==== Two Fours: $96$ ====
==== Two Fours: $120$ ====
== Glossary ==
Category:Four Fours
Category:Four Fours/Lemmata/Two Fours
Four Fours/Lemmata/Two Fours/64/Solutions

== Puzzle: Two Fours: $64$ ==
== Solutions ==
=== Solution $1$ ===
=== Solution $2$ ===
=== Solution $3$ ===
=== Solution $4$ ===
=== Solution $5$ ===
=== Solution $6$ ===
Category:Four Fours/Lemmata/Two Fours
Four Fours/Lemmata/Two Fours/64/Solutions/1

== Puzzle: Two Fours: $64$ ==
== Solution ==
:$64 = \paren {\sqrt 4}^{\map \Gamma 4}$
where $\map \Gamma 4$ denotes the Gamma function of $4$.
== Also presented as ==
This is how it appears according to :
:$64 = \sqrt {4^{\map \Gamma 4} }$
== Sources ==
* 
Category:Four Fours/Lemmata/Two Fours
Four Fours/Lemmata/Two Fours/64/Solutions/2

== Puzzle: Two Fours: $64$ ==
== Solution ==
:$64 = \sqrt {\paren {\sqrt {\sqrt 4} }^{4!} }$
== Proof ==
:$\sqrt {\paren {\sqrt {\sqrt 4} }^{4!} } = \sqrt {\sqrt {2^{24} } } = \sqrt {2^{12} } = \sqrt {4096} = 64$
== Also presented as ==
's presentation of this (of which the above is a reproduction) is not the neatest.
This is how it appears according to :
:$64 = \surd \surd \surd \paren {4^{4!} }$
== Sources ==
* : Solutions: $58$. -- The Two Fours
* 
* : Answers: $109$. The Two Fours
Category:Four Fours/Lemmata/Two Fours
Four Fours/Lemmata/Two Fours/64/Solutions/3

== Puzzle: Two Fours: $64$ ==
== Solution ==
:$64 = 4^{\floor {\ln 4!} }$
where $\floor {\, \cdot \,}$ denotes the floor function.
== Proof ==
== Sources ==
* 
Category:Four Fours/Lemmata/Two Fours
Four Fours/Lemmata/Two Fours/64/Solutions/4

== Puzzle: Two Fours: $64$ ==
== Solution ==
:$64 = \map \phi {\map \phi {4^4} }$
where $\phi$ denotes the Euler $\phi$ function.
== Proof ==
== Sources ==
* 
Category:Four Fours/Lemmata/Two Fours
Four Fours/Lemmata/Two Fours/64/Solutions/5

== Puzzle: Two Fours: $64$ ==
== Solution ==
:$64 = \paren {\map \phi {4!} }^\sqrt 4$
where $\phi$ denotes the Euler $\phi$ function.
== Proof ==
== Sources ==
* 
Category:Four Fours/Lemmata/Two Fours
Four Fours/Lemmata/Two Fours/64/Solutions/6

== Puzzle: Two Fours: $64$ ==
== Solution ==
:$64 = 4^{\sqrt {\map {\pi} {4!} } }$
where $\map \pi \cdot$ represents the prime-counting function.
== Proof ==
The primes less than $4! = 24$ are:
:$2, 3, 5, 7, 11, 13, 17, 19, 23$
so $\map {\pi} {4!} = 9$.
Thus:
:$4^{\sqrt {\map {\pi} {4!} } } = 4^{\sqrt 9} = 64$
== Sources ==
* 
Category:Four Fours/Lemmata/Two Fours
Four Kepler-Poinsot Polyhedra

== Theorem ==
There exist exactly four Kepler-Poinsot polyhedra:
:$(1): \quad$ the small stellated dodecahedron
:$(2): \quad$ the great stellated dodecahedron
:$(3): \quad$ the great dodecahedron
:$(4): \quad$ the great icosahedron.
== Proof ==
== Historical Note ==
== Sources ==
* : $9$
* : $9$
* 
* 
Category:Kepler-Poinsot Polyhedra
Category:4
Four Straight Lines are Proportional iff Similar Parallelepipeds formed on them are Proportional

== Theorem ==
== Proof ==
:
Let $AB$, $CD$, $EF$ and $GH$ be four straight lines in proportion, so that:
:$AB : CD = EF : GH$
Let similar and similarly situated parallelepipeds $KA$, $LC$, $ME$ and $NG$ be described on $AB$, $CD$, $EF$ and $GH$ respectively.
It is to be demonstrated that:
:$KA : LC = ME : NG$
From :
:$KA : LC = AB^3 : CD^3$
For the same reason:
:$ME : NG = EF^3 : GH^3$
We have that:
:$AB : CD = EF : GH$
Therefore:
:$KA : LC = ME : NG$
Suppose that:
:$KA : LC = ME : NG$
Then from :
:$KA : LC = AB^3 : CD^3$
For the same reason:
:$ME : NG = EF^3 : GH^3$
We have that:
:$KA : LC = ME : NG$
Therefore:
:$AB : CD = EF : GH$
== Sources ==
* : Book $\text{XI}$. Propositions
Category:Parallelepipeds
Fourier's Theorem/Integral Form

== Theorem ==
Let $f: \R \to \R$ be a real function which satisfies the Dirichlet conditions on $\R$.
Then:
:$\dfrac {\map f {t^+} + \map f {t^-} } 2 = \ds \int_{-\infty}^\infty e^{2 \pi i t s} \paren {\int_{-\infty}^\infty e^{-2 \pi i t s} \map f t \rd t} \rd s$
where:
:$\map f {t^+}$ and $\map f {t^-}$ denote the limit from above and the limit from below of $f$ at $t$.
=== Continuous Point ===
== Proof ==
== Sources ==
* : Chapter $2$: Groundwork: The Fourier transform and Fourier's integral theorem
Category:Fourier Transforms
Fourier's Theorem/Integral Form/Continuous Point

== Theorem ==
Let $f: \R \to \R$ be a real function which satisfies the Dirichlet conditions on $\R$.
Let $f$ be continuous at $t \in \R$.
Then:
:$\ds \map f t = \int_{-\infty}^\infty e^{2 \pi i t s} \paren {\int_{-\infty}^\infty e^{-2 \pi i t s} \map f t \rd t} \rd s$
== Proof ==
At a point of continuity we have:
The result follows from Fourier's Theorem: Integral Form.
== Sources ==
* : Chapter $2$: Groundwork: The Fourier transform and Fourier's integral theorem
Category:Fourier Transforms
Fourier's Theorem/Lemma 1

== Lemma for Fourier's Theorem ==
Let $\psi$ be a real function defined on a closed interval $\closedint a b$.
Let $\psi$ be piecewise continuous with one-sided limits on $\closedint a b$.
Then:
:$\ds \lim_{N \mathop \to \infty} \int_a^b \map \psi u \sin N u \rd u = 0$
== Proof ==
We are given that $\psi$ is piecewise continuous with one-sided limits on $\closedint a b$.
Therefore, there exists a finite subdivision $\set {x_0, x_1, \ldots, x_m}$ of $\closedint a b$, where $x_0 = a$ and $x_m = b$, such that for all $i \in \set {1, 2, \ldots, m}$:
:$\psi$ is continuous on $\openint {x_{i - 1} } {x_i}$ 
:$\ds \lim_{x \mathop \to {x_{i - 1} }^+} \map \psi x$ and $\ds \lim_{x \mathop \to {x_i}^-} \map \psi x$ exist.
From the corollary to Sum of Integrals on Adjacent Intervals for Integrable Functions:
:$\ds \int_a^b \map \psi u \sin N u \rd u = \sum_{r \mathop = 0}^{m - 1} \int_{x_r}^{x_{r + 1} } \map \psi u \sin N u \rd u$
Then:
The last integral is bounded.
Thus $(1)$ is less than $\dfrac {M_r} N$ for $M_r \in \R$.
Let $M = \max \set {\size {M_0}, \size {M_1}, \dotsc, \size {M_{m - 1} } }$.
Then:
:$\ds \size {\int_a^b \map \psi u \sin N u \rd u} < \dfrac {M m} N$
As $M$ and $m$ are finite:
:$\ds \lim_{N \mathop \to \infty} \dfrac {M m} N = 0$
Hence the result.
== Sources ==
* : Chapter Two: $\S 2$. Some Important Limits: Lemma $(1)$
Category:Fourier's Theorem