Final answer:
When rolling two six-sided dice, there are 10 distinct ways to roll such that the product of the numbers is a perfect square, considering only the perfect squares that can be obtained by numbers 1 to 6.
Step-by-step explanation:
To determine how many ways two dice can be rolled such that the product is a perfect square, we first acknowledge that when two fair six-sided dice are rolled, there are a total of 36 possible outcomes, since each die has 6 faces, and 6 times 6 gives us 36 outcomes.
Perfect squares between 1 and 36 that can be obtained by multiplying two numbers between 1 and 6 are 1 (1x1), 4 (2x2), 9 (3x3), 16 (4x4), 25 (5x5), and 36 (6x6).
Now, let's list out the combinations of rolled numbers that yield these perfect squares:
- 1: (1,1)
- 4: (1,4), (2,2), (4,1)
- 9: (1,9), (3,3), (9,1)
- 16: (1,16), (2,8), (4,4), (8,2), (16,1)
- 25: (1,25), (5,5), (25,1)
- 36: (1,36), (2,18), (3,12), (4,9), (6,6), (9,4), (12,3), (18,2), (36,1)
However, since we are constrained by our dice only showing numbers 1 through 6, we omit combinations that contain numbers greater than 6.
Adjusting for this, our actual combinations that are plausible with two six-sided dice are:
- 1: (1,1)
- 4: (1,4), (2,2), (4,1)
- 9: (1,9), (3,3), (9,1) - Removing those containing a 9
- 16: (2,8), (4,4), (8,2) - Removing those containing a 16 and correcting for the 8 with 1 being impossible
- 25: (5,5) - Removing those containing a 25
- 36: (6,6) - Removing those containing a 36
Summing up these combinations, we have 1 + 3 + 1 (only (3,3) is possible) + 3 + 1 + 1 = 10 ways.
Therefore, there are 10 distinct ways to roll the two dice such that the product of the numbers is a perfect square.