A 12 by 12 by 12 cube has all its faces painted blue. It is then broken down to 1728 individual 1 by 1 by 1 cubes. The probability that a randomly chosen 1 by 1 by 1 cube has exactly 2 faces painted blue is \frac {a} {b} , where a, b are coprime. What is the value of a+ b?
There will be 12*(12 - 2) = 120 cubes with exactly 2 faces painted blue.
ie 120/1728 = 5/72
ie a + b = 5 + 72 =77
To find the probability that a randomly chosen 1 by 1 by 1 cube has exactly 2 faces painted blue, we need to determine the number of cubes with exactly 2 blue faces and divide it by the total number of cubes.
Let's break down the problem step by step:
Step 1: Determine the number of cubes with exactly 2 blue faces.
To find this, consider the cubes located at the edges and corners of the larger 12 by 12 by 12 cube. These cubes have exactly 2 blue faces since they are adjacent to two other cubes. There are 8 cubes at the corners of the larger cube and 12 cubes on each edge (excluding corners). So, the number of cubes with exactly 2 blue faces is 8 + (12 x 12) = 8 + 144 = 152.
Step 2: Determine the total number of cubes.
We know that the larger 12 by 12 by 12 cube is broken down into 1728 individual 1 by 1 by 1 cubes. Therefore, the total number of cubes is 1728.
Step 3: Calculate the probability.
The probability of randomly choosing a cube with exactly 2 faces painted blue is given by:
Probability = (Number of favorable outcomes) / (Total number of outcomes)
Number of favorable outcomes = 152 (from Step 1)
Total number of outcomes = 1728 (from Step 2)
Therefore, Probability = 152 / 1728 = 19 / 216.
The value of a + b is therefore 19 + 216 = 235.