## Relationship between Variables

Hello Everyone brave enough to tackle this:

if I have:

• N competitors
• (N-1 competitors for 1 competitor to compete against)
• X sudoku puzzles
• 1 competition consists of 10 sudoku puzzles
• X sudoku puzzles from which a competition of 10 sudoku puzzles are randomly chosen
• to choose the 1st sudoku puzzle (of 10) there are X choices
• to choose the 2nd sudoku puzzle (of 10) there are X-1 choices
• to choose the 3rd sudoku puzzle (of 10) there are X-2 choices
• to choose the 4th sudoku puzzle (of 10) there are X-3 choices
• to choose the 5th sudoku puzzle (of 10) there are X-4 choices
• etc...
• to choose the 10th sudoku puzzle (of 10) there are X-9 choices
• There are P 'one vs. one' competitions, in which the two competitors face the same 10 sudoku puzzles.
• There are Q 'one vs. group' competitions, a group consists of M competitors, in which all the competitors face the same 10 sudoku puzzles.
• There are R 'group vs. group' competitions, competing groups can be of different sizes, in which all the M competitors from the 2 different groups face the same 10 sudoku puzzles.
• there are, at any one moment, a varied number of 'one vs. one' competitions and a maximum of N/2 'one vs. one' competitions each involving the completion of 10 sudoku puzzles to be chosen at random from the bank of X sudoku puzzles
• there are, at any one moment, a varied number of 'one vs. group' competitions each involving the completion of 10 sudoku puzzles to be chosen at random from the bank of X sudoku puzzles
• there are, at any one moment, a varied number of 'group vs. group' competitions each involving the completion of 10 sudoku puzzles to be chosen at random from the bank of X sudoku puzzles
• at any one moment, there are a number of 'one vs. one', 'one vs. group' and 'group vs. group' competitions occuring
• each player never faces the same sudoku puzzle twice irrespective of competing as an individual or in a group, he or she always completes a new sudoku puzzle but that same sudoku puzzle can be played by any and all players as long as they too have not played this sudoku puzzle before.
• the same 10 sudoku puzzles can be offered to competitors in competitions P, Q and R occuring at the same time as long as all or any one of the competitors involved had not faced any one of these 10 sudoku puzzles previously.
Question 1: What equation/formula captures the relationship between variables N, X, M, P, Q and R. Assumptions can be made for the value for M, P, Q and R for a certain scenario?

Question 2: Which scenario results in the quickest depletion of the bank of X sudoku puzzles?