# {*no-status title-slide} // commentaire -  {no} - Navigate using the arrow keys (right/left) or the buttons on the sides -  {no} -  {no} - - - # The Galette cutting problem - We have a Galette (King's Cake) - It contains a bean - We want to cut it - in $N$ equal slices - without hitting the bean - What is the probability to achieve this?

# The King's cake (Galette) {libyli} - @anim: g.brown + - $R_G$ {params} // radius of the galette - @anim: .slices + - $N$ {params} // number of pieces to cut - @anim: .feve1 + - $r$ {params} // radius of the bean (a circle) - @anim: .border + .R + - $R = R_G - r$ // possible radius for the bean center - @anim: .feve2 | .blue | .feve3 | .green | -.feve - @anim: %+class:showthem:.thecontrols - @anim: %+class:foggy:.content - @anim: .rhombus .theta + - $\theta = \frac{360}{N}$ // angle of a slice - @anim: %-class:showthem:.thecontrols - @anim: %viewbox:svg .zoomRhombus - @anim: .rhombus .shape | .rhombus .a | .rhombus .height + - $\sin(\theta) = \frac{r}{a}$
$a = \frac{r}{\sin(\theta)}$ // side of the rhombus - @anim: %+class:nothome:.rhombus .theta | .rhombus .dashed | .rhombus .x + - $x = a + a \cos(\theta)$ // horizontal position of the rhombus corner - @anim: %viewbox:svg .zoomSlice | -.rhombus - @anim: .area .split | .rectwr | .rectwr .r, .rectwr .x | .rectwr .w + - $w$ // width of the rectangle - @anim: .area .beta .diag + - $R^2 = w^2 + r^2$ // diagonal of the rectangle (Pythagore) - $w = \sqrt{R^2 - r^2}$ // width of the rectangle - @anim: .tri + - $A_t = \frac{r w}{2} {}$ *et $A_h = \frac{r x}{2} {}$* // area of the upper rectangles - @anim: .upperleft - @anim: .area .beta .angle + - $\sin(\beta) = r / R$
donc : $\beta = \arcsin(r/R)$ // angle of the blue slice - @anim: .area .beta .zone + - $A_b = \pi R^2 \cdot \beta / 360$ // area of the slice with angle $\beta$ - $A_p = \pi R^2 \cdot \theta / 360$ // area of the complete slice (with angle $\theta$) - $A = A\_b + A\_t - A\_h$ // white area - $A = \pi R^2 \cdot \frac{\beta}{360} + \frac{r w}{2} - \frac{r x}{2} {}$ // ... developped - $p = (A\_p - 2 A) / A_p {}$ // probability of not cutting the bean - $p = 1 - \frac{2 A}{A_p} {}$ // seen differently - @anim: %+class:showthem:.thecontrols - @anim: %dur:1000 + %viewbox:.zoomAll - Thanks!

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