In a popular dancing game there are 4 directions: up, down, left, right. In the game these arrows may appear once every quarter-note or eight-note. Also two arrows may appear at the same time in the same beat. Three or four arrows cannot.

In a song that is 1 minute long and plays at 1 bar(4 quarter notes) every 4 seconds, what are the total possible number of different dances that can exist.

Lemme take a shot at the solution

In a single beat, you can have either 2 eight-notes or 1 quarter note

For quarter notes there can be the following combinations
01 X rest (no arrow)
04 X One arrow
06 x Two arrows
or a total of 11 combinations

For eight notes there can be the following combinations for the 1st 1/2 of the beat
01 X rest (no arrow)
04 X One arrow
06 x Two arrows

I am assuming that if the 2nd 1/2 of the beat is a rest it will be the same as a combination using one quarter note (e.g. UP (1/8) REST (1/8)= UP (1/4) ) - This is the way these games work.

Hence, for a full single beat, there can be the following combinations of 2 eight notes
11 X 10 or 110 combinations

For one full beat there can be 110+11 = 121 Combinations

For 60 full beats there can be 121^60 or about 9.2709 x 10^124.

(On a side note, some of the dancing games I played had triplets... that was tough... )

Posted by delvin on 2003-02-16 16:00:17