# multiplying the bars

February 24, 2020
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The latest Riddler makes the remark that the expression

|-1|-2|-3|

has no unique meaning (and hence value) since it could be

| -1x|-2|-3 | = 5   or   |-1| – 2x|-3| = -5

depending on the position of the multiplication sign and asks for all the possible values of

|-1|-2|…|-9|

which can be explored by a recursive R function for computing |-i|-(i+1)|…|-(i+2j)|

```vol<-function(i,j){x=i
if(j){x=c(i-(i+1)*vol(i+2,j-1),abs(i*vol(i+1,j-1)+i+2*j))
if(j>1){for(k in 1:(j-2))
x=c(x,vol(i,k)-(i+2*k+1)*vol(i+2*k+2,j-k-1))}}
return(x)}
```

producing 40 different values for the ill-defined expression. However, this is incorrect as the product(s) hidden in the expression only involve a single term in vol(i,j)… I had another try with the decomposition of the expression vol(i,j) into a first part and a second part

```prod<-function(a,b) a*b[,1]+b[,2]

val<-function(i,j){
x=matrix(c(i,0),ncol=2)
if(j){x=rbind(cbind(i,prod(-(i+1),val(i+2,j-1))),
cbind(abs(prod(-i,val(i+1,j-1))-i-2*j),0))
if(j-1){for(k in 2:(j-1)){
pon=val(i,k-1)
for(m in 1:dim(pon)[1])
x=rbind(x,cbind(pon[m,1],pon[m,2]+prod(-(i+2*k-1),val(i+2*k,j-k))))}}}
return(x)}
```

but it still fails to produce the right version.

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