Calvin is playing a game of Dungeons and Dragons. In order to make it across the river, he needs to throw six 4-sided dice, and have their sum be a multiple of 5. How many different dice throws result in Calvin making it across the river?
Details and assumptions
The dice are considered distinct from each other.
A 4-sided dice is a (tetrahedron) dice which yields values of 1, 2, 3, 4.
x=largest sum possible with 2 six-sided dice 598+x= ?
To find out how many different dice throws result in Calvin making it across the river, we need to find all the possible combinations of throwing six 4-sided dice.
First, let's analyze the possible sums of the faces of a single die. Since each die has four sides with values 1, 2, 3, and 4, the possible sums range from 1 (1+1+1+1) to 16 (4+4+4+4).
To determine if a sum is a multiple of 5, we need to see if the remainder when dividing the sum by 5 is zero.
Since we're throwing six dice, each die can contribute a value between 1 and 4 to the sum. Thus, the range of the sum we're interested in is from 6 (1+1+1+1+1+1) to 24 (4+4+4+4+4+4).
Now, let's go through each possible sum from 6 to 24 and check if it's a multiple of 5. If it is, it means that specific combination of dice will allow Calvin to make it across the river. If not, Calvin will have to try a different combination.
6: Not a multiple of 5
7: Not a multiple of 5
8: Not a multiple of 5
9: Not a multiple of 5
10: A possible sum (1+1+1+1+1+5)
11: Not a multiple of 5
12: A possible sum (1+1+1+1+1+7)
13: Not a multiple of 5
14: A possible sum (1+1+1+1+1+9 or 1+1+1+1+2+4)
15: A possible sum (1+1+1+1+1+11 or 1+1+1+1+2+6)
16: A possible sum (1+1+1+1+1+13 or 1+1+1+1+2+8 or 2+2+2+2+2+2)
17: Not a multiple of 5
18: A possible sum (1+1+1+1+1+19 or 1+1+1+1+2+14 or 1+1+1+1+3+9 or 1+1+1+2+2+11 or 1+1+2+2+2+10 or 1+2+2+2+2+9)
19: Not a multiple of 5
20: A possible sum (1+1+1+1+1+23 or 1+1+1+2+2+13 or 1+1+2+2+2+12 or 1+2+2+2+2+11)
21: A possible sum (1+1+1+1+1+27 or 1+2+2+2+3+11 or 1+2+2+3+3+7 or 1+2+3+3+3+6 or 2+2+2+2+2+11 or 2+2+2+2+3+10 or 2+2+2+3+3+8 or 2+2+3+3+3+7 or 2+3+3+3+3+4)
22: Not a multiple of 5
23: Not a multiple of 5
24: A possible sum (2+2+2+3+3+12 or 2+2+3+3+4+10 or 2+3+3+3+4+7 or 3+3+3+3+3+9)
So, there are 9 different dice throws that result in Calvin making it across the river.