# Bongard Problem #121 (solution)

#### Left: Under each VV
there is a circle; under each ΛΛ there is
a triangle; under each VΛ there is a square; and under each ΛV, as
well as under "left-overs", there is nothing.

#### Right: Under each ΛΛ there is a circle; under each VV
there is a triangle; under each ΛV there is a square; and under each VΛ, as well as
under "left-overs", there is nothing.

**Note 1:** Some people do not like the idea of
"left-overs". I agree, the problem would be a bit neater if those were missing.

**Note 2:** Abe Dillon suggested an alternative and simpler solution: “Left:
a triangle appears on the bottom if and only if there are more
Λ-shapes than
V-shapes. Right: not so.”
Although Dillon’s solution is probably not the one the designer intended
(judging from the exact positioning of the shapes directly above the others that
they correspond to), it does appear a bit shorter, even when its right-side
answer is analyzed into its logical description: “Right:
either a triangle does not appear on the bottom, or (exclusively) the number of
V-shapes is greater than or equal to the number of
Λ-shapes.”

*Back to problem #121*