There are three groups of stones: in the first – 10, in the second – 15, in the third – 20. During one turn, you are allowed to split any pile into two smaller ones; the one who cannot make a move loses.
Two players take turns to put rooks on a chessboard so that the rooks cannot capture each other. The player who cannot make a move loses.
On a board there are written 10 units and 10 deuces. During a game, one is allowed to erase any two numbers and, if they are the same, write a deuce, and if they are different then they can write a one. If the last digit left on the board is a unit, then the first player won, if it is a deuce then the second player wins.
Four football teams play in a tournament. There’s the Ulams (\(U\)), the Vandermondes (\(V\)), the Wittgensteins (\(W\)) and the Xenos (\(X\)). Each team plays every other team
exactly once, and matches can end in a draw.
If a game ends in a draw, then both teams get \(1\) point. Otherwise, the winning team gets
\(3\) points and the losing team gets
\(0\) points. At the end of the
tournament, the teams have the following points totals: \(U\) has \(7\), \(V\)
has \(4\), \(W\) has \(3\) and \(X\) has \(2\).
Work out the results of each match, including showing that there’s no other way the results could have played out.