## Le Monde puzzle [#1707]

**A** geometric Le Monde mathematical puzzle:

*Given a pizza of diameter 20cm, what is the way to cut it by two perpendicular lines through a point distant 5cm from the centre towards maximising the surface of two opposite slices?**Using the same point as the tip of the four slices, what is the way to make four slices with equal arcs in four cuts from the tip again towards maximising the surface of two opposite slices?*

**F**or both questions, I did not bother with the maths but went itself to a discretisation of the disk, counting the proportion of points within two opposite slices and letting the inclination of these slices move from zero to π/2. Unsurprisingly, for the first question, the answer is π/4, given that there is no difference between both surfaces at angles 0 and π/2. My R code is as follows, using (5,0) as the tip:

M=100 surfaz=function(alpha){ surfz=0 cosal=cos(alpha);sinal=sin(alpha) X=Y=seq(-10,10,le=M) Xcosal=(X-5)*cosal Xsinal=(X-5)*sinal for (i in 1:M){ norm=sqrt(X[i]^2+Y^2) scal1=Xsinal[i]+Y*cosal scal2=-Xcosal[i]+Y*sinal surfz=surfz+sum((norm<=10)*(scal1*scal2>0))} return(4*surfz/M/M/pi)}

The second puzzle can be solved by a similar code, except that the slice area between two lines has to be determined by a cross product:

surfoz=function(alpha,ploz=FALSE){ sinal=sin(alpha);cosal=cos(alpha) X=Y=seq(-10,10,le=M) frsterm=cosal*(10*cosal-5)+sinal*(10*sinal-5) trdterm=cosal*(10*cosal+5)+sinal*(10*sinal+5) surfz=0 for (i in 1:M){ norm=sqrt(X[i]^2+Y^2) scal1=(10*(Y[i]-5)*cosal-(10*sinal-5)*X)*frsterm scal2=-(-10*(Y[i]-5)*sinal-(10*cosal-5)*X)*frsterm scal3=(-10*(Y[i]-5)*cosal+(10*sinal+5)*X)*trdterm scal4=-(10*(Y[i]-5)*sinal+(10*cosal+5)*X)*trdterm surfz=surfz+sum((norm<=10)* ((scal1>0)*(scal2>0)+ (scal3>0)*(scal4>0)))} return(4*surfz/M/M/pi)}

a code that shows that *all* cuts lead to identical surfaces for bot sets of slices. A fairly surprising result!

July 28, 2017 at 10:43 pm

Thanks Christian for your pertinent comments on the weekly Le Monde Logic Game. As usual this one was less simple as thought from the wording of the problem: the pizzaiolo cutting.

I found the same as you using standard geometry. The answer to the second question was as you said somewhat surprising: see a note on that at https://www.researchgate.net/publication/318755694_Logic_Game_1017