**A**nother X’mas Le Monde mathematical puzzle:

*A regular dice takes the values 4, 8 and 2 on three adjacent faces. Summit values are defined by the product of the three connected faces, e.g., 64 for the above. What values do the three other faces take if the sum of the eight summit values is 1768? *

**H**ere is the simple R code I used to find a solution:

summi=function(x){
#(x[1],x[2],x[3]) opposed to (4,8,2)
sum(outer(c(2,x[1]),outer(c(8,x[2]),c(4,x[3]),"*"),"*"))}
faces=matrix(sample(1:20,3*10^4,rep=T),ncol=3)
resum=apply(faces,1,summi)
sol=faces[resum==1768,]

with the result:

> sol
[,1] [,2] [,3]
[1,] 2 18 13
[2,] 2 18 13
[3,] 2 18 13
[4,] 6 5 13

which means the missing faces are (6,5,13) since the puzzle also imposed all faces were different. The following histogram of the sample of sums shows a reasonable gamma G(1.9,1763) fit.

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