Math Toy Puzzle #2

Two weeks ago, I posted about a math toy that my wife purchased. I also posted three puzzles related to this toy. The second puzzle is reprinted below:

Recently, my wife bought the following wooden toy from a clearance bin at Target. It's a set of six dials, rainbow-colored, that are connected by a wooden axle.

The dials are shaped like decagons, and each side of the dial is painted as follows:

• The red dial: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• The orange dial: The following mathematical operators, in this order: +, -, x, +, -, x, +, -, x, +. Note that there are consecutive plus signs on the dial.
• The yellow dial: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• The green dial: The = sign appears on all sides
• The blue dial: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• The purple dial: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Spin the dials so that you form ten mathematical statements, one on each of the ten sides of the toy simultaneously. What is the maximum number of correct mathematical statements that you can display at the same time?

I don't have a firm solution to the problem, but I do have some insights that may help you to come up with the solution.

What are the maximum number of addition statements that can be true simultaneously? There are four plus signs on the toy, but note that the valid sums can run from 00 (for 0 + 0) to 18 (for 9 + 9). Therefore, we can only have, at most, two valid addition statements--one that sums to a single digit and requires zero for the blue dial, and one that sums to a number between 10 and 18 and uses one for the blue dial.

This is only possible if the two plus signs are next to each other on the toy. And, it turns out you can do that. For example, you can spin the dials of the toy such that, say, 4 + 5 = 0_ on one side, and 5 + 6 = 1_ on the next side. However, we run into a problem. Let's set the dials of the toy such that a + b = 0c, where a and b are any digits that sum to a single digit number. Because the values on the dials are in a specific order, the next side of the toy will display (a + 1) + (b + 1) = 10 + (c + 1). The 10 represents the blue dial's 1. Checking various values of a and b and c show that it is impossible for two addition expression to be correct on the toy at the same time. Therefore, the maximum number of addition statements that can be true simultaneously is one, and the blue dial must be set to 0 or 1.

What are the maximum number of subtraction statements that can be true simultaneously? This is much easier to answer, since valid subtraction statements must set the blue dial to 0. Any other value leads to an impossible subtraction result. Therefore, the maximum is one, and the blue dial must be set to 0.

Given the previous two results, is it possible for an addition statement and a subtraction statement to be true at the same time on the toy? No. For this to happen, a subtraction statement must use the blue dial's 0, and the addition statement must use the blue dial's 1. However, it is impossible, given the order of the orange dial, to have a subtraction statement immediately before an addition statement. Therefore, either an addition statement or a subtraction statement can be true on the toy, but never both.

That leaves the three multiplication sides, so it may be possible for a maximum of four correct statements to appear on the toy. However, this is where I get a bit stuck. A multiplication statement with a blue dial's 9 is automatically incorrect, since the largest possible product is 9 x 9 = 81. (In fact, any side of the toy using the blue dial's 9 is automatically incorrect.) There is only one way to use a blue 8 in a valid way (the aforementioned 9 x 9 = 81), but a quick check of the toy shows that this would be the only correct side. There are only two ways to use a blue 7 in a valid multiplication (9 x 8 = 72 and 8 x 9 = 72), but another check of this permutation reveals a single correct side.

Maybe it's time for me to use a computer and brute force? Let me know in the comments if you have any other ideas.