The great Risk tournament is organized again. This year there are 1000 contestants. Each game in the tournament is played with 4 people. After each game, there will be one winner which goes to the next round. In the next round again groups of 4 players are formed. If one does not get an exact multiple of 4 in this way, some of the players in this round are selected randomly. These players can go to the following round automatically. In the end there will be one winner. How many games will be played?

Suppose: a = b

It then follows that
a * a = a * b
a^2 = a * b
a^2 - b^2 = a * b - b^2
(a - b) * (a + b) = (a - b) * b
a + b = b
a + a = a
2 * a = 1 * a
2 = 1

Is this derivation really correct? Or is there somewhere a mistake? If so, where?

Recently, somebody told: "My grandfather was born on the first sunday of the year. His seventh birthday was also on a sunday!" In which year was this grandfather born?