## optimultiplication [a riddle]

**T**he riddle of this week is about an optimisation of positioning the four digits of a multiplication of two numbers with two digits each and is open to a coding resolution:

Four digits are drawn without replacement from {0,1,…,9}, one at a time. What is the optimal strategy to position those four digits, two digits per row, as they are drawn, toward minimising the average product?

Although the problem can be solved algebraically by computing **E**[X⁴|x¹,..] and **E**[X⁴X³|x¹,..] I wrote three R codes to “optimise” the location of the first three digits: the first digit ends up as a unit if it is 5 or more and a multiple of ten otherwise, on the first row. For the second draw, it is slightly more variable: with this R code,

second<-function(i,j,N=1e5){draw drew=matrix(0,N,2) for (t in 1:N) drew[t,]=sample((0:9)[-c(i+1,j+1)],2) conmean=(45-i-j)/8 conprod=mean(drew[,1]*drew[,2]) if (i<5){ #10*i pos=c((110*i+11*j)*conmean, 100*i*j+10*(i+j)*conmean+conprod, (100*i+j)*conmean+10*i*j+10*conprod)}else{ pos=c((110*j+11*i)*conmean, 10*i*j+(100*j+i)*conmean+10*conprod, 10*(i+j)*conmean+i*j+100*conprod) return(order(pos)[1])}

the resulting digit again ends up as a unit if it is 5 (except when x¹=7,8,9, where it is 4) or more and a multiple of ten otherwise, but on the second row. Except when x¹=0, x²=1,2,3,4, when they end up on the first row together, 0 obviously in front.

For the third and last open draw, there is only one remaining random draw, which mean that the decision only depends on x¹,x²,x³ and **E**[X⁴|x¹,x²,x³]=(45-x¹-x²-x³)/7. Attaching x³ to x² or x¹ will then vary monotonically in x³, depending on whether x¹>x² or x¹<x²:

fourth=function(i,j,k){ comean=(45-i-j-k)/7 if ((i<1)&(j<5)){ pos=c(10*comean+k,comean+10*k)} if ((i<5)&(j>4)){ pos=c(100*i*comean+k*j,j*comean+100*i*k)} if ((i>0)&(i<5)&(j<5)){ pos=c(i*comean+k*j,j*comean+i*k)} if ((i<7)&(i>4)&(j<5)){ pos=c(i*comean+100*k*j,j*comean+100*i*k)} if ((i<7)&(i>4)&(j>4)){ pos=c(i*comean+k*j,j*comean+i*k)} if ((i>6)&(j<4)){ pos=c(i*comean+100*k*j,j*comean+100*i*k)} if ((i>6)&(j>3)){ pos=c(i*comean+k*j,j*comean+i*k)} return(order(pos)[1])}

Running this R code for all combinations of x¹,x² shows that, except for the cases x¹≥5 and x²=0, for which x³ invariably remains in front of x¹, there are always values of x³ associated with each position.

April 14, 2017 at 5:55 pm

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