I totally disagree with the rebuttal to the trollface proof. It’s not a question of countability vs uncountability, it’s that the arc length of the limit is NOT the limit of the arc lengths.
The limit of the sequence of (piecewise linear) curves really *is* the circle, not just a countable approximation. But the limit of the length is not the length of the limit.

pgn674

Forever Alone Guy got the discrepancy wrong.
107 boys and 100 girls
For every 107 boys there are 100 girls, and for every 100 girls there are 107 boys.
107/(107+100)=0.516908213 -> 51.6908213% boys
100/(107+100)=0.483091787 -> 48.3091787% girls
Check: 51.6908213+48.3091787=100
For every 48.31 girls, there are 51.69 boys, so 51.6908213-48.3091787=3.3816426% of boys get no girls.
3.3816426% of 6,697,254,041 = 226,477,196

JoshuaI totally disagree with the rebuttal to the trollface proof. It’s not a question of countability vs uncountability, it’s that the arc length of the limit is NOT the limit of the arc lengths.

The limit of the sequence of (piecewise linear) curves really *is* the circle, not just a countable approximation. But the limit of the length is not the length of the limit.

pgn674Forever Alone Guy got the discrepancy wrong.

107 boys and 100 girls

For every 107 boys there are 100 girls, and for every 100 girls there are 107 boys.

107/(107+100)=0.516908213 -> 51.6908213% boys

100/(107+100)=0.483091787 -> 48.3091787% girls

Check: 51.6908213+48.3091787=100

For every 48.31 girls, there are 51.69 boys, so 51.6908213-48.3091787=3.3816426% of boys get no girls.

3.3816426% of 6,697,254,041 = 226,477,196