Here is a game you can set up on a spreadsheet. Enter an arbitrary matrix M somewhere.
Three by three is a good way to start.
Enter a 3 component column vector v and use the mmult command (or do it out yourself) to compute Mv and for each component of v compute the ratio of
Calculate the variance of these ratios (that is, the sum of their squares minus the square of their sum).
The players can take turns generating the original M and v; then they take turns modifying v by changing one of its components.
If the variance of the ratios decreases the player scores a point, otherwise loses one. The game ends when the variance becomes negligible, say less than 10-10.
The ratios then will be more or less the same and hence the eigenvalue associated with the eigenvector produced.
If you get too good at this, you can try with a 5 by 5 matrix, though it is boring to enter one at the start.