Math Toy Puzzle #1 Solution
Last week, I posted about a math toy that my wife purchased. I also posted three puzzles related to this toy. The first 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
Randomly (and independently) spin each of the six dials of the toy. Look at the mathematical statement that appears on the top of the toy. What is the probability that the mathematical statement is correct?
Here is how I would solve this problem, using some bookkeeping and some sample space management. Consider the red, orange, and yellow dials. We can organize the sample space for these three dials by producing ten 10 x 10 tables. Each 10 x 10 table would represent one of the mathematical operators on the orange dial. There would be four tables using the plus operator, three tables using the subtraction operator, and three tables using the multiplication operator.
The 4 addition tables would look like this:
...where the bold numbers in the first column represent the digit that shows up on the red dial, and the bold numbers in the first row represent the digit that shows up on the yellow dial. The "1/100" in each cell is the probability that the blue and purple dials show the correct answer given the red, orange, and yellow dials. If the red dial is 5, the orange dial is "+", and the yellow dial is 7, the probability that the correct sum is displayed on the blue and purple dials is 1/100.
The 3 multiplication tables would look the same, but we'd put a little "x" sign in the upper left corner.
The 3 subtraction tables would look a bit different, since we can't have a negative difference.
For the cells containing zeroes, it is impossible to yield a correct mathematical statement when the number on the red dial is less than the number on the yellow dial (and the orange dial shows subtraction).
Since every cell of the 10 tables is equally likely to be picked, we can compute the probability of a correct mathematical equation as follows:
P(correct equation) = P(valid expression left of the equal sign) x P(correct answer on right side of equals sign)
P(correct equation) = (865/1000) x (1/100) = 865/100,000 = 0.00865 = 0.865%
Does this look correct? Let me know in the comments!
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