1. More Word Problems! Help!!

6. A 2-by-4-by-8 rectangular solid is painted red. It is cut into unit cubes and reassembled into a 4-by-4-by-4 cube. If the entire surface of this cube is red, how many painted unit-cube faces are hidden in the interior cube?

7. If a year has 364 days, then the same calendar could be used every year by only changing the year. A "regular" year has 365 days and a leap year has 366 days. The year 2000 has a leap year and leap years occur every 4 years between the years 2000 and 2100. Claudia has calendar for 2009. What will be the next year that she can use this calendar by merely changing the year?

8. A 6-question True-False test has True as the correct answer for at least 2/3 of the questions. How many different true/False answer patterns are possible on an answer key for this test?

2. Hello, Brent!

It's simple arithmetic . . .

6. A 2-by-4-by-8 rectangular solid is painted red.
It is cut into unit cubes and reassembled into a 4-by-4-by-4 cube.
If the entire surface of this cube is red,
how many painted unit-cube faces are hidden in the interior cube?
Code:
                 8
* - - - - - - *
/             /|
4 /             / | 2
/             /  |
* - - - - - - *   *
|             |  /
2 |             | / 4
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8
The total surface area is:

. . $\displaystyle \begin{array}{cccccc}\text{top/bottom:} & 2\times(8\times4) &=&64 \\ \text{left/right:} & 2\times(4\times2) &=& 16 \\ \text{front/back:} & 2\times(8\times2) &=&32 \\\end{array}\quad\Rightarrow\quad 112$

There are 112 red faces on the sixty-four unit cubes.

Code:
          * - - - - *
/         /|
4/         / |
/         /  | 4
* - - - - *   |
|         |   *
4 |         |  /
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4
The total surface area of the cube is: .$\displaystyle 6 \times (4\times4) \:=\:96$

There are 96 red faces on the outside of the cube.

Therefore, there are: .$\displaystyle 112 - 96 \:=\:\boxed{16}$ red faces hidden inside.