In each family, the sum of the ages of the boys equals the sum of the ages of the girls, and the sum of the squares of the ages of the boys equals the sum of the the squares of ages of the girls. The sum of the ages of all children is 60.

What are the ages of the children in each family?

{boys:#[B1 B2 B3], girls:#[G1 G2 G3]}where the variables B1, B2 and B3 stand for the ages of the boys in descending order (i. e., B3 is the age of the youngest boy in the family), and the variables G1, G2 and G3 stand for the ages of the girls, also in descending order. This representation of a family avoids possible symmetries. The constraints that must hold for a family will be posted by the defined constraint makefamily. A solution is a pair consisting of Maria's and Clara's family.

fun age_list space = let val ages as #[a,b,c] = FD.intvarVec(space,3,#[(0,9)]) in FD.rel(space,a,FD.GR,b); FD.rel(space,b,FD.GR,c); ages end fun makefamily space = let val boysages = age_list space val girlsages = age_list space val tmp1 = FD.range(space,(0,243)) val tmp2 = FD.range(space,(0,243)) val ages as #[a,b,c,d,e,f] = Vector.concat([boysages,girlsages]) fun squares_sum(#[x,y,z],res)= post(space,FD(x)`* FD(x)`+ FD(y)`* FD(y)`+ FD(z)`* FD(z) `= FD(res),FD.DOM) in FD.distinct(space,ages,FD.DOM); post(space,FD(a) `+ FD(b) `+ FD(c) `= FD(d) `+ FD(e) `+ FD(f),FD.DOM); squares_sum(boysages,tmp1); squares_sum(girlsages,tmp2); FD.rel(space,tmp1,FD.EQ,tmp2); {boys= #[Vector.sub(ages,0),Vector.sub(ages,1), Vector.sub(ages,2)], girls = #[Vector.sub(ages,3),Vector.sub(ages,4), Vector.sub(ages,5)]} end fun family space = let val maria = makefamily space val clara = makefamily space val ageofmariasyoungestgirl = Vector.sub(#girls maria,2) val ageofclarasyoungestgirl = Vector.sub(#girls clara,2) val ages = Vector.concat([#boys (maria),#girls (maria), #boys(clara),#girls(clara)]) in Vector.app(fn x => FD.rel(space,x,FD.GR,ageofmariasyoungestgirl)) (#boys maria); FD.relI(space,ageofclarasyoungestgirl,FD.EQ,0); post(space,SUMV(ages) `= `60,FD.DOM); FD.branch(space,ages,FD.B_SIZE_MIN,FD.B_MIN); {maria,clara} end

The procedure age_list ensures that the elements in the list of boys or girls appear in descending order.

Andreas Rossberg 2006-08-28