Johnny is looking for his ideal girfriend, which (according to him; and not meant to favor any preconceived biases) must be red-haired, green-eyed, slender, and tall.
He knows four women: Adele, Betty, Carol, and Doris.
Here is a list of caveats:
# Only one of the four women has all four characteristics that Johnny requires.
# Only three of the women are both green-eyed and slender.
# Only two of the women are both red-haired and tall.
# Only two of the women are both slender and tall.
# Only one of the women is both green-eyed and red haired.
# Adele and Betty have the same color eyes.
# Betty and Carol have the same color hair.
# Carol and Doris have different builds.
# Doris and Adele are the same height.
Which one of the four women satisfies all of Johnny's requirements?
groebner({r[a]^2+r[a],
g[a]^2+g[a],
s[a]^2+s[a],
t[a]^2+t[a],
r[b]^2+r[b],
g[b]^2+g[b],
s[b]^2+s[b],
t[b]^2+t[b],
r[c]^2+r[c],
g[c]^2+g[c],
s[c]^2+s[c],
t[c]^2+t[c],
r[d]^2+r[d],
g[d]^2+g[d],
s[d]^2+s[d],
t[d]^2+t[d],
r[a]*g[a]*s[a]*t[a]+(r[b]*g[b]*s[b]*t[b]+1)*(r[c]*g[c]*s[c]*t[c]+1)*(r[d]*g[d]*s[d]*t[d]+1),
r[b]*g[b]*s[b]*t[b]+(r[c]*g[c]*s[c]*t[c]+1)*(r[d]*g[d]*s[d]*t[d]+1)*(r[a]*g[a]*s[a]*t[a]+1),
r[c]*g[c]*s[c]*t[c]+(r[d]*g[d]*s[d]*t[d]+1)*(r[a]*g[a]*s[a]*t[a]+1)*(r[b]*g[b]*s[b]*t[b]+1),
r[d]*g[d]*s[d]*t[d]+(r[a]*g[a]*s[a]*t[a]+1)*(r[b]*g[b]*s[b]*t[b]+1)*(r[c]*g[c]*s[c]*t[c]+1),
(g[a]*s[a]+1)+g[b]*s[b]*g[c]*s[c]*g[d]*s[d],
(g[b]*s[b]+1)+g[c]*s[c]*g[d]*s[d]*g[a]*s[a],
(g[c]*s[c]+1)+g[d]*s[d]*g[a]*s[a]*g[b]*s[b],
(g[d]*s[d]+1)+g[a]*s[a]*g[b]*s[b]*g[c]*s[c],
r[a]*t[a]*r[b]*t[b]+(r[c]*t[c]+1)*(r[d]*t[d]+1),
r[a]*t[a]*r[c]*t[c]+(r[b]*t[b]+1)*(r[d]*t[d]+1),
r[a]*t[a]*r[d]*t[d]+(r[b]*t[b]+1)*(r[c]*t[c]+1),
r[b]*t[b]*r[c]*t[c]+(r[a]*t[a]+1)*(r[d]*t[d]+1),
r[b]*t[b]*r[d]*t[d]+(r[a]*t[a]+1)*(r[c]*t[c]+1),
r[c]*t[c]*r[d]*t[d]+(r[a]*t[a]+1)*(r[b]*t[b]+1),
s[a]*t[a]*s[b]*t[b]+(s[c]*t[c]+1)*(s[d]*t[d]+1),
s[a]*t[a]*s[c]*t[c]+(s[b]*t[b]+1)*(s[d]*t[d]+1),
s[a]*t[a]*s[d]*t[d]+(s[b]*t[b]+1)*(s[c]*t[c]+1),
s[b]*t[b]*s[c]*t[c]+(s[a]*t[a]+1)*(s[d]*t[d]+1),
s[b]*t[b]*s[d]*t[d]+(s[a]*t[a]+1)*(s[c]*t[c]+1),
s[c]*t[c]*s[d]*t[d]+(s[a]*t[a]+1)*(s[b]*t[b]+1),
r[a]*g[a]+(r[b]*g[b]+1)*(r[c]*g[c]+1)*(r[d]*g[d]+1),
r[b]*g[b]+(r[c]*g[c]+1)*(r[d]*g[d]+1)*(r[a]*g[a]+1),
r[c]*g[c]+(r[d]*g[d]+1)*(r[a]*g[a]+1)*(r[b]*g[b]+1),
r[d]*g[d]+(r[a]*g[a]+1)*(r[b]*g[b]+1)*(r[c]*g[c]+1),
g[a]+g[b],
r[b]+r[c],
s[c]+s[d]+1,
t[d]+t[a]},
{r[a],g[a],s[a],t[a],
r[b],g[b],s[b],t[b],
r[c],g[c],s[c],t[c],
r[d],g[d],s[d],t[d]},lex,2)
=
References
http://groups.google.com/group/sci.math.symbolic/browse_frm/thread/6315e82bf75a5c6/ Google Groups: View Thread "Helping Johnny to find his ideal woman (a problem in Set Theory, and Mathematica)"