# SolveMe Mobiles: "Explorer" difficulty

This page can help you learn how to solve puzzles 1-60, using mental math alone.

I was able to solve the first 120 puzzles mentally, writing down only the values of each shape as I went. You should be able to use these strategies to do the same.

Terminology:

Call shapes for which their value is known "known shapes," and shapes for which their value is unknown "unknown shapes." Doing so will reduce bloat in the text below.

Strategy 1:

If you see the same shape on both sides, and its value is known, mentally remove both occurrences of the shape, also removing the sum of the two amounts from the number at the top of the mobile, if necessary. Here's a puzzle where this strategy could be used:

Strategy 2:

Mentally remove equal amounts from both sides, also removing the same amount from the number at the top of the mobile, if necessary. For example, in the puzzle below, I notice the left side is 8, and the right side is at least 5. Thus I can remove 5 from both sides mentally. Then I have fewer numbers to remember, and the numbers I do have to remember are as small as possible.

Strategy 3:

Temporarily remove the same unknown shape from both sides, mentally, to relate a known shape, to some other unknown shape. Here's a puzzle where this strategy could be used:

In this case, we mentally remove the square to relate the heart, whose value is known, to the crescent, whose value is unknown.

Strategy 4:

When the total of a strand is known, and the value of a shape along that strand is also known, mentally remove the shape and its value from the total of the strand. For example, in the puzzle below, the total of the right strand is 28 / 2 = 14. The value of each square is 1. Thus, mentally remove the squares, and remove 2 from the strand total. This allows us to find the value of each circle, by focusing on the right strand, and thus, solve the puzzle.

Strategy 5:

If the total \(t\) of a strand is known, and only one shape appears on the strand, \(n\) times, then the value of the shape is \(t / n.\) For example:

Strategy 6:

Recursively distribute the strand total evenly amongst its subtrees. For example, in the puzzle below, the 12 is split into 6 on both sides. On the right strand, we split 6 into 3 and 3.

Strategy 7:

If you know the total of one strand, and you know all but one shape on the other strand, then you have a missing addend problem. For example, in the puzzle below, the total of the left strand is 8, and the total of the right strand is the crescent, plus 3. Thus, we have 8 = crescent + 3, and thus, crescent = 5.

Strategy 8:

Recursively bubble strand totals upwards. For example, in the puzzle below, the value of the trapezoid is 6. Thus, the subtree total is 6 * 2 = 12. Bubbling once more, we find the total of the tree is 12 * 2 = 24. This allows us to find the value of the droplet before even considering the strand containing the heart. Using this strategy isn't necessary for this example, but it's something to keep in mind.

Going further:

Here's a problem that's not offered by the SolveMe Mobiles website, but is one I find interesting: Assuming shapes have nonzero whole number values, what solution would minimize the total for the mobile pictured below?

Consider the right subtree in isolation. Its left strand must be a nonzero multiple of 3, and its right strand must be a nonzero multiple of 2. Thus, the minimum for each strand must be lcm(3, 2) = 6. The subtree must therefore be 6 * 2 = 12. Now consider the original mobile. The left strand must be a nonzero multiple of 2. The right subtree must be a nonzero multiple of 12. lcm(2, 12) = 12, so the left strand can also be 12. Now that we know the total of each strand, we can easily solve the mobile, to find square = 6, diamond = 2, crescent = 3.