Why is there n-1 different objects in a n by n matrix game like Bejeweled? For games that consists of a grid, and is similar to the concept like bejeweled: has an n by n matrix and n-1 different objects. 
What is the reason for this? Why not have more than n-1 different objects, or have less than n-1 objects. What is the logic behind this? 
Some Examples: 
DOTS have a 6 by 6 matrix, with 5 different colors of circles
Bejeweled has a 8 by 8 matrix with 7 different jewels
ANIPANG has a 7 by 7 matrix with 6 different animals


 A: Possible Guidance:
Developers of these types of games usually trial a few types of game before settling for the one that is not too easy and not too hard. There may be some neat mathematics behind this so lets do some visual speculation first.
Let's play with some $5\times5$ boards. I have written a program to randomly generate elements from $[0,n)= \Bbb Z_n$ where we will investigate the different $n$. Let us look at the case $n=4$.
$$
\left[
\begin{matrix}
0&0&2&2&3\\
1&1&3&0&1\\
2&0&3&3&3\\
3&0&2&1&0\\
1&0&1&1&1\\
\end{matrix}
\right]
$$
As you can see a game such as you describe, as far as I understand, would preferably not want all of these side by side elements. So we know that choosing a $n> 4$ is probably a good idea. What about the case $n=5$? Now there should be $5$ different elements and this is a random generation.
$$
\left[
\begin{matrix}
4&2&2&3&3\\
2&3&1&2&3\\
3&0&4&0&4\\
1&4&1&1&2\\
1&3&4&1&4\\
\end{matrix}
\right]
$$
Note that this should be the so called "optimum" for a size $5$ matrix. Now let's get our hands dirty with some statistical analysis of these objects and look for correlation (of some sort).
Firstly I randomly generate a matrix and count how many pairs there are. Pairs are defined to be immediately adjacent to each other. This can be up, down, left or right. For example the above matrix has A pair of twos on the top row. I do this $10\;000$ times for each size of matrix. We then take an arithmetic mean of the number of pairs.

On the horizontal axis we have the number of elements and on the vertical axis we have the mean number of pairs. Note that the lines are joined. Starting from the bottom we have size of $n=2$ all the way to $11$. The number of elements span from $2$ to $20$. Looking at the curves, I suspect that it is of the form $\frac{\log a}{\log x}$. I am now assuming they are looking at the derivative of these curves to find their optimum layout.
Unfortunately I do not have the statistical skills to show you a way to optimize this however I hope that this answer has given you a clue about what these people may be looking at. You should note however that these curves will all limit to zero number of pairs as the diversity of the elements increases. 
