the question is: “two dot cubes are tossed. The total number of dots on the two top faces is 6. What is the total of the dots on the bottom faces of the two dot cubes?”
and the answer makes even less sense: ” The total number of dots on the opposite sides of a cube is 7. So the two dot cubes contain a total of 7 x 2, or 14 dots, on the top and bottom faces. Since 6 dots appear on the top faces, there must be 14 – 6, or 8 dots, on the bottom faces.”
can anyone explain what this means? where are they getting these numbers?
A standard die (or dot cube) is arranged as follows:
1 and 6 are on opposite sides.
2 and 5 are on opposite sides.
3 and 4 are on opposite sides.
As you can see, opposite sides always add to be 7. Thus if you roll two of them, the sum of the top and bottom sides will be 14.
If you have 6 showing on the top, the value on the bottom will have to be 8 to make a total of 14.
If you were to roll a 1 and a 5, the opposite sides would be 6 and 2 = 8
If you were to roll a 2 and a 4, the opposite sides would be 5 and 3 = 8.
If you were to roll a pair of 3s, the opposite sides would be a pair of 4s = 8.
These all take advantage of the layout of the numbers on a die always being a sum of 7 on opposite sides.
If you mean standard dice, then if you look at a die, the opposite sides of a die always equals 7:
6 and 1 are opposites
5 and 2 are opposites
4 and 3 are opposites
So if you are throwing two dice, the sum of both tops and both bottoms will be 14 (7 * 2)
If the tops total 6, then the bottoms total:
14 – 6 = 8
The sum of the bottoms of the two dice will be 8.