I went along to my first meetup of the Edinburgh Cinema group and Elise the facilitator had generously baked a chocolate cake for everyone. This was my first knowing encounter with ganache as an icing alternative and I was pointed towards Beatty’s Chocolate Cake for the recipe of the deliciously moist cake.
Even if, like me, you’re not a Scot, you probably still feel a sense of a pre-ordained script for tonight’s Euro 2012 qualifier in Alicante. Scotland have that traditional glimmer of a chance at making it to the next stage. It is even in their hands – beat Spain and they are through to the play-offs.
But Scottish sport doesn’t usually work like that. The script generally goes like this:
The small glimmer of hope is consolidated by valiant and sometimes heroic efforts.
I always like a good real-world application of mathematics, in this case algebra.
If you’re organising a knockout tournament for something and the number of players or teams is not a neat power of two, how many players have to be drawn in round 1, so that round 2 is a neat power of two? I figured this out by solving this pair of equations:
a + b = n 0.5a + b = R
where a is the number of players to be drawn in round 1, b is the remainder of players to be drawn in round 2, n is the total no. of players and R is the largest power of two less than or equal to n.
Solving the equations
0.5a + (n-a) = R a + 2n - 2a = 2R 2n - a = 2R a = 2n - 2R a = 2(n - R)
91 players enter our tournament, how many must be drawn in round 1? n = 91, R = 64 a = 2 x (91 - 64) a = 54
So we draw 54 players in round 1, which produces 27 winners to meet the remaining 37 players in round 2.