### More Mathematics Puzzles

Since it looks like quite a few of my readers are a **lot** sharper than the average Joe, I feel liberated to step up the difficulty slightly this time round. Here are three questions that are easy enough to state so that anyone can understand what they're about, but tough to take a bit of effort to solve.

- The product N of three positive integers is 6 times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of N.
- Let N be the greatest integer multiple of 8, no two of whose digits are the same. What is the remainder when N is divided by 1000?
- Define a
*good word*as a sequence of letters that consists only of the letters A, B and C - not all of these letters need appear in a given sequence - and in which A is never immediately followed by B, B is never immediately followed by C, and C is never immediately followed by A. How many seven-letter good words are there?

ADDENDUM: To quell the appetites of those who feel the problems above were not in the least challenging, here are two more for your consumption. If these ones strike you as being as easy as the previous ones, I'll be extremely impressed!

- Find, as a function of
*n*, the sum of the digits of9 x 99 x 9999 x ... x (10 ^{2n}-1),

where each factor has twice as many digits as the previous one. - A computer screen shows a 98 x 98 chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minimum number of mouse clicks needed to make the chessboard all one color.

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