*[heard in the métro this morning]*

“…les équations à deux inconnues ça va encore, mais à trois inconnues, c’est trop dur!”

*[“…systems of equations with two unknowns are still ok, but with three variables it is too hard!”]*

an attempt at bloggin, nothing more…

**I** fought with my L^{α}T_{ε}X compiler this morning as it did not want to deal with my code:

\begin{algorithmic}[1] \STATE N=1000 \STATE $\hat\pi=0$ \FOR {I=1,N} \STATE X=RDN(1), Y=RDN(1) \IF {$\text{X}^2+\text{Y}^2<1$} $\hat\pi$ = $\hat\pi +1$ \ENDIF \ENDFOR \RETURN 4*$\hat\pi/$N \end{algorithmic}

looking on forums for incompatibilities between beamer and algorithmic, and adding all kinds of packages, to no avail. Until I realised one \STATE was missing:

\begin{algorithmic}[1] \STATE N=1000 \STATE $\hat\pi=0$ \FOR {I=1,N} \STATE X=RDN(1), Y=RDN(1) \IF {$\text{X}^2+\text{Y}^2<1$} \STATE $\hat\pi$ = $\hat\pi +1$ \ENDIF \ENDFOR \RETURN 4*$\hat\pi/$N \end{algorithmic}

(**T**his is connected with my AMSI public lecture on simulation, obviously!)

**T**his year, my daughter started writing algorithms in her math class (she is in *seconde*, which could correspond to the 10th grade). The one she had to write down last weekend was Buffon’s neddle and the approximation of π by Monte Carlo (throwing cannon balls was not mentioned!). Here is the short R code I later wrote to show her the outcome (as the class has not yet learned a computer language):

n=10^6 counter=0 #uniforms over the unit square ray=runif(n)^2+runif(n)^2 #proportion within the quarter circle conv=cumsum((ray<1))/(1:n) plot(conv,type="l",col="steelblue",ylim=c(pi/4-2/sqrt(n), pi/4+2/sqrt(n)),xlab="n",ylab="proportion") abline(h=pi/4,col="gold3")

and here is an outcome of the convergence of the approximation to π/4: