---
excalidraw-plugin: parsed
tags:
  - excalidraw
excalidraw-open-md: true
---


> [!question] Question
> Find numbers from 0 to 100, but only using 4 times the digit "4" (and any operator)
> 
> Trouver les nombres de 100, mais seulement en utilisant 4 fois le chiffre 4 (et des opérateurs)


# Ideas / Idées
You can always use up a 4 using : "$\sqrt{ \sqrt{ x^{4} } } = x$" (does nothing). This allows formulas with only 3 or 2 times the 4.
On peut toujours utiliser un 4 avec : "$\sqrt{ \sqrt{ x^{4} } } = x$" (ne fait rien). Cela permet des formules avec seulement 2 ou 3 fois le 4.

# Solutions

## $n = 0$
- $4 - 4 + 4 - 4$
- $44 - 44$
## $n=1$

## $n = 41$

- $41 = 44 - \sqrt{ 4 } - (\lim\limits_{ n \to \infty } \sqrt[n]{4})$

## $n = 71$

- $71 = (\Gamma(4) + \Gamma(\sqrt{ 4 })) \cdot (\Gamma(\sqrt{ 4 }) \cdot \Gamma(\sqrt{ 4 }))$
- $71 = (4! + (\sqrt{ 4 })!) \cdot ((\sqrt{ 4 })! \cdot (\sqrt{ 4 })!)$

# General solution / Solution générale

## 1

$\displaystyle\int _{-4}^{4}  \dfrac{\underbrace{(x \times x)+(x \times x)+\cdots + (x \times x)}_{6n \text{ times }  x}}{4^{4}} \, dx = n$

## 2

$\displaystyle\sqrt{ 4 } \cdot\frac{\ln \left( \frac{\overbrace{\sqrt{ \sqrt{ \sqrt{ \sqrt{ \cdots 4 } } } }}^{n \text{ times }} }{\ln(4)} \right)}{\ln(4)}$

> [!definition] Logarithm / Logarithme
> $\log_{n}(n^{x}) = x$

> [!definition] Natural logarithm / Logarithme naturel
> $\ln(x) = \log_{e}(x)$
> So : / Alors :
> $\ln(e^{x}) = x$

## 3

$0 = 4 - 4$
$1 = (\sqrt{ 4 })!$
$2 = \sqrt{ 4 }$
$3 = \dfrac{4!}{\sqrt{ 4 }}$
$4 = $

> [!definition] Factorial / Factorielle
> $n! = 1 \times 2 \times 3 \times \cdots \times n$

> [!definition] Double factorial / Double factorielle
> $n!! = 1 \cancel{\times 2} \times 3 \cancel{\times 4} \times \cdots \times n$ if $n$ is odd / si $n$ est impair
> $n!! = \cancel{1 \times} 2 \times \cancel{3 \times} 4\times \cdots \times n$ if $n$ is even / si $n$ est pair

> [!definition] Concatenation
> New operator / Nouvel opérateur :
> $4 \cdot 4 = 44$
> $71 \cdot 92 = 7192$

## 2

$\displaystyle\log_{\log_{4}(\sqrt{ 4 })} \left( \log_{4} \left( \underbrace{\sqrt{ \sqrt{ \sqrt{ \sqrt{ \dots \sqrt{ 4 } } } } }}_{n \text{ times}} \right) \right) = n$

#### Why ? / Pourquoi ?

$\log_{4}(\sqrt{ 4 }) = \frac{1}{2}$

$\log_{4}(4) = 1$
$\log_{4}(4^{2}) = 2$
$\vdots$
$\log_{4}(4^{p}) = p$

$\log_{4}(\sqrt{ 4 }) = \log_{4}\left( 4^{\frac{1}{2}} \right) = \frac{1}{2}$
$\log_{4}(\sqrt{ \sqrt{ 4 } }) = \log_{4}\left( 4^{\frac{1}{4}} \right) = \frac{1}{4}$
$\log_{4}(\sqrt{ \sqrt{ \sqrt{ 4 } } }) = \log_{4}\left( 4^{\frac{1}{8}} \right) = \frac{1}{8}$
$\log_{4}(\sqrt{ \sqrt{ \sqrt{ \sqrt{ 4 } } } }) = \log_{4}\left( 4^{\frac{1}{16}} \right) = \frac{1}{16}$
$\vdots$
$\displaystyle\log_{4}\left( \underbrace{\sqrt{ \sqrt{ \sqrt{ \dots \sqrt{ 4 } } } }}_{n \text{ times}} \right) = \log_{4} \left( 4^{\frac{1}{2^{n}}} \right) = \frac{1}{2^{n}}$

$\displaystyle\log_{\frac{1}{2}} \left( \frac{1}{2 \times n} \right) = n$

![[kangourou 2024-07-15 problèmes du jour.svg]]

%%
# Excalidraw Data

## Text Elements
Velocity
Vitesse ^H0Z9Cp37

Acceleration ^y4qpuUdL

Position ^UY2ok27f

derivative
dérivée ^dW5ZQscu

derivative
dérivée ^S24o7Dxr

integral
intégrale ^dpD4ekhS

integral
intégrale ^7kTHCP9V

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```
%%