SolveMe Mobiles: "Master" difficulty

Strategy 12:

This strategy is all about finding the ratio of two shapes. In the puzzle below, if we mentally remove one hexagon and triangle from both sides, we see 2 hexagons equals 1 triangle. If we consider the puzzle in its initial state, but replacing the triangle on the left strand with two hexagons, we have 15 on the left strand, equally divided amongst 5 hexagons, and so the hexagon is 3. The triangle is twice the hexagon, and so the triangle is 6.

Strategy 13:

In the puzzle below, you'd probably start by breaking the 100 recursively, to figure out that circle = 5. There are no circles in the bottom layer of strands, and so it's difficult to proceed from here. However, we can see two strands have both triangles and droplets, and for both strands we know their totals. Thus, we can setup a system of equations, to relate these shapes to those totals. Let \(x\) be triangle, and \(y\) droplet. Then

$$\begin{align} & x + 3y = 25 \\ & 2x + y = 10 \end{align}$$

This system can be solved mentally. Double the first equation, subtract the second equation from it, and you'll get \(5y = 40.\) Solving this mentally gives \(y = 8.\) Plugging this into the second equation, it's obvious that \(x = 1.\) So triangle = 1 and droplet = 8.

Here's a more difficult puzzle where strategy 13 could be employed:

Use the left subtree to obtain the equations

$$\begin{align} & H + 3T = 22 \\ & 4H + T = 22 \end{align}$$

Solve the first equation for \(H\)

$$H = 22 - 3T$$

Now use this in the second equation

$$\begin{align} & 4(22 - 3T) + T = 22 \\ & 88 - 12T + T = 22 \\ & 66 - 11T = 0 \\ & 6 = T \end{align}$$

Plugging the value of \(T\) into the second equation gives \(H = 5.\)