Le Monde puzzle [#954]

By xi’an

A square Le Monde mathematical puzzle:

Given a triplet (a,b,c) of integers, with a? Can you find the triplet (a,b,c) that produces the sum a+b+c closest to 1000?

This is a rather interesting challenge and a brute force resolution does not produce interesting results. For instance, using the function is.whole from the package Rpmfr, the R functions

ess 
and
quest1
do not return any solution when a=1,2,3,4,5
Looking at the property that a+b,a+c,b+c, and a+b+c are perfect squares α²,β²,γ², and δ². This implies that
a=(δ+β)(δ-β), b=(δ+γ)(δ-γ), and c=(δ+α)(δ-α)
with 1
a=4γ, b=2γ+1, and c=(γ+1+α)(γ+1-α)
which leads to only two terms to examine. Hence writing another R function
abc=function(al,ga){
 a=4*ga
 b=2*ga+1
 k=(ga+al+1)*(ga-al+1)
 return(c(a,b,k))}

and running a check for the smallest values of α and γ leads to the few solutions available:
> for (ga in 3:1e4) 
for(al in 1:(ga-2)) 
if (ess(abc(al,ga))) print(abc(al,ga))
[1] 80 41 320
[1] 112 57 672
[1] 192 97 2112
[1] 240 121 3360
[1] 352 177 7392
[1] 416 209 10400
[1] 560 281 19040
[1] 640 321 24960
[1] 816 409 40800
[1] 912 457 51072

Filed under: Kids, R Tagged: Alice and Bob, is.whole, Le Monde, mathematical puzzle, R, recursive function, Rmpfr 

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