Balance Puzzles

A balance puzzle is a type of logic puzzle where the goal is to make all components equal, or balanced. For example, a scale is balanced when both sides have equal weight on them. It is important to determine the relationships between elements of a balance puzzle. For example, when trying to balance a scale, it is important to know the weight of the objects being added to each side. Knowing how to set up algebraic expressions, isolate variables, and solve algebraic equations are key skills for solving these problems. It can be useful to approach balancing puzzles using algebra and systems of linear equations.

In order to balance the scale, how many green squares would need to be added to one end of the scale if the other end had $3$ purple circles on it?

Since it takes $2$ green squares to balance a scale that has $1$ purple circle on the other end, we know that we need $2$ green squares for every purple circle. Since there are $3$ purple circles, we need $3 \times 2 = 6$ green squares in order to balance the scale.

To model this problem with equations, we get

$\begin{aligned} p &= 2g\\ 3p &= Xg\\ 3(2g) &= 6g =Xg\\ X &= 6.\ _\square \end{aligned}$

Balancing Equations

If a scale is balanced, that means that the combined type and quantity of elements on one side of the scale have equal weight to the combined type and quantity of elements on the other side of the scale. For example, if a red ball weighs twice as much as a blue ball, then it is true that the blue ball weighs half as much as the red ball. In other words, if $1$ red ball is placed on one side of a scale, in order to balance the scale, $2$ blue balls must be placed on the other end.

Balancing puzzles can be modeled as linear systems of equations since it is possible to derive linear relationships about elements on the scale (as mentioned above). Each scale will yield one algebraic expression, so if a problem shows multiple scales, multiple relationships can be derived. The equations determined from the scales can be combined to determine the relationships between objects that are not even being weighed against each other. Consider a problem with two scales: one that shows $2$ blue balls for every $1$ red ball, and another that shows $3$ green squares for every $1$ blue ball. From these two equations, a relationship between green squares and red balls can be derived. Since there are $2$ blue balls for every $1$ red ball, and $3$ green squares for every $1$ blue ball, there are $2 \times 3$ green balls for every $1$ red ball.

Describe the relationships between the shapes in the illustrations below using algebraic expressions.

We can get one equation per scale. Since there are two scales, we will get two equations.

From the scale on the left: 3 Green Squares = 1 Purple Circle.

From the scale on the right: 1 Green Square = 2 Red Triangles. $_\square$

The example above yields two linear equations. Since there are three variables--Red Triangles, Green Squares, Purple Circles--it is possible to derive the relationships between each pair of variables. The equations already state the relationship between Red Triangles and Green Squares (there are two Red Triangles for every Green Square) and the relationship between Green Squares and Purple Circles (there are three Green Squares for every Purple Circle). These two equations can be combined to derive the relationship between Purple Circles and Red Triangles.

Using the two equations from the example above, write an equation that shows the relationship between Red Triangles and Purple Circles.

Here we will abbreviate Green Squares to $G$ , Purple Circles to $P$ , and Red Triangles to $R$ .

Here are the first two equations:

$2R = 1G\quad \text{ and }\quad 3G = 1P.$

Both of these equations contain a $G$ term, so we can isolate $G$ in one equation, and plug that result in for the $G$ in the second equation:

$\begin{aligned} 1G &= 2R\\ 3(2R) &= 1P\\ 6R &= 1P. \end{aligned}$

The relationship between Purple Circles and Red Triangles is (6 Red Triangles) = (1 Purple Circle), or there are six Red Triangles for every Purple Circle. $_\square$

Mobile Balancing

Mobile balancing puzzles contain shapes that hang from a mobile. If one shape is heavier than another, the mobile will shift so that the heavier shape hangs lower than the lighter shape. In mobile balancing problems, the relative weights of the shapes (which can be determined from how the shapes hang) can be used to show relationships between shapes.

Determine the relationship between the green triangle and the red square from this mobile. (Assume that the fulcrum is at the center of each rod.)

Because the mobile is balanced (flat), we can tell that the green triangle and red square have equal weight. Therefore, to balance a mobile, $1$ red square is needed for every $1$ green triangle (and vice versa). $_\square$

What can you tell about the relationship between blue circles and green triangles using the mobile below? (Assume that the fulcrum is at the center of each rod.)

We can tell that the mobile is unbalanced because the mobile is tilted. Because the green triangle is lower than the blue circle, we know that the green triangle is heavier than the blue circle. $_\square$

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