All four scales shown are balanced.
One possible replacement for the ? is
(A) (B) [[IMAGE1]] (C) [[IMAGE2]] (D) [[IMAGE3]] (E) [[IMAGE4]]
, 2012
Solution
In turn, we may use each of the three known scales to find a way to balance a circle, a diamond and a triangle.
Since many answers are possible, we must then check our solution to see if it exists among the five answers given.
First consider the scale at the top right.
A diamond and a circle are balanced by a triangle.
If we were to add a triangle to both sides of this scale, then it would remain balanced and the right side of this scale would contain what we are trying to balance, a circle, a diamond and a triangle.
That is, a circle, a diamond and a triangle are balanced by two triangles.
However, two triangles is not one of the five answers given.
Next, consider the scale at the top left.
A triangle and a circle are balanced by a square.
If we were to add a diamond to both sides of this scale, then it would remain balanced and the left side of this scale would contain what we are trying to balance, a circle, a diamond and a triangle.
That is, a circle, a diamond and a triangle are balanced by a square and a diamond.
This answer is given as one of the five answers.
In the context of a multiple choice contest, we do not expect that students will verify that the other four answers do not balance a circle, a diamond and a triangle. However, it is worth noting that it can be shown that they do not.
Want a route through all this instead of an archive? The track
puts 2,444 problems in a working order, from Junior Challenge level to the IMO shortlist.