Question

How many ways are there to roll doubles or a sum divisible by three on two dice?

Answer 1

To find the ways to roll doubles or a sum divisible by three on two dice, we can consider the following cases:

1) Doubles: There are 6 ways to roll doubles since we have 6 possible outcomes when both dice show the same number (1-1, 2-2, 3-3, 4-4, 5-5, 6-6).

2) Sum divisible by 3: In this case, we need to determine the pairs of numbers that add up to a multiple of 3.

For the first die:
- When the first die shows 1, the second die can show 2 or 5.
- When the first die shows 2, the second die can show 1, 4, or 6.
- When the first die shows 3, the second die can show 1, 4, or 6.
- When the first die shows 4, the second die can show 2 or 5.
- When the first die shows 5, the second die can show 1, 4, or 6.
- When the first die shows 6, the second die can show 2 or 5.

Therefore, there are 2 pairs for case 1, 3 pairs for cases 2-5, and 2 pairs for case 6, giving us a total of 2 + 3(4) + 2 = 16 pairs.

To find the total number of ways to roll doubles or a sum divisible by three, we add the number of ways from both cases:

Total ways = Doubles + Sum divisible by 3
= 6 + 16
= 22

Hence, there are 22 ways to roll doubles or a sum divisible by three on two dice.