I GREATLY DISAGREE WITH THE CONCEPT that you control your own destiny. Sure, there are plenty
of things you control, but there are many events that impact your life that you have absolutely no control
over. For a personal example, a few months before I entered the world, my dad started interviewing for full-time jobs, and had two good prospects. One big opportunity was in southern California; the other was in the
“Show Me State.” If my dad took the job in Orange County, it’s clear my life would be very, very different than
it is right now.
If I grew up in California, adjacent to the beach, I would have
learned to surf at an early age. My whole childhood would have
been devoted to catching waves, and before turning 20 I would
have joined the World Surf League—the prestigious, professional
tour of surfing. Instead of writing actuarial puzzles, I’d be out there
on the tour performing better barrels than Gabriel Medina, having
cooler hair than John John Florence, and offending more people
than Mick Fanning. Ah, what might have been!
It’s virtually impossible to be a pro surfer without daily access
to good waves, so my dreams died before they began when my
parents ended up here in the great Midwest. We traded police
chases, earthquakes, and fierce traffic for Blues hockey, toasted
ravioli, and jokes about what high school you attended. Maybe
we did get the better end of the bargain after all.
Even though I’m not a pro, I still enjoy watching the World
Surf League on television. One of the enjoyments of watching is
that surfing has one of the most mathematical scoring rubrics of
any sport out there. For the uninitiated, most surf competitions
have heats, where two surfers go head-to-head to see who is better.
Heats usually last for 35 minutes, and although a competitor can
ride as many waves as (s)he wants during those 35 minutes, only
the sum of your top two scores count to see who wins. If your
top two waves are a 7 and a 6, then you have a heat score of 13,
no matter what took place on all the other waves. And thanks to
the priority system, there is no “sharing” of waves; each wave gets
ridden by at most one surfer.
A few days ago my friend Park and I started watching the latest
stop on the World Surf League tour. Park made the comment that
these surfers are so good, anyone out there can ride any wave,
and it’s all pure luck who wins or loses depending on who rides
which wave. Unfortunately, there’s a lot of truth to this comment,
as the best in world can ride almost any wave that the ocean spits
out. If a wave has a potential score of a 9, you’d better believe that
any pro out there will score exactly a 9 on the wave. Not a point
lower, not a point higher.
Of course, if I were a pro surfer, I’d have a secret plan. See,
whereas most surfers are notoriously picky when it comes to which
waves to ride, as a math person, I know it’s just a numbers game,
and would make sure to ride at least one more wave during the
heat than my opponent. The question then becomes: How would
this impact my chances of winning a heat?
1. Let’s assume that I ride six waves in a heat, and my opponent rides
five. Let’s also assume that it’s totally random and uncorrelated
as to what the scoring potential of each wave is. Each wave has
a whole-number scoring grade between 1 and 10, with 1 being
the lowest score and 10 being the highest. It’s totally random and
unknowable what the score of the wave will be beforehand, as
there’s exactly a 10 percent chance it will be any number between
1 and 10. Given all of these assumptions, what’s the probability
of me winning, losing, and tying the heat?
2. To further reward gnarly high scores, the Big Wave Tour slightly
modifies the scoring system so your top wave counts double.
Hence, if my top two scores are a 7 and a 4, my heat score is now
18 (two times 7 plus 4) instead of 11. If all the other assumptions
from above still hold, what is the new probability of me winning,
losing, or tying the heat?
Solutions to Polyominoes
Problem 1: How many distinct ways can tetrominoes tile a 4x4
square? Two tilings are considered distinct if one cannot be made
to look like the other by a sequence of rotations or flips about its
horizontal, vertical, or diagonal axes.
There are 117 tilings of the 4× 4 square using tetrominoes. They
fall into the following 22 distinct classes:
Class 1 2 4 8
4 of a kind BBBB LLLL, IIII,
2 pair IBBI IIBB, ILLI,
Pair of L’s ILBL, ILLB IBLL, ILLB,
Pair of T’s ILTT, IT TL,