Because this blogpost covers both
mountain biking and proving theorems, it is being simultaneously published in
Devlin’s more wide ranging blog profkeithdevlin.org.
Mountain biking is big in the San
Francisco Bay Area, where I live. (In its present day form, using
specially built bicycles with suspension, the sport/pastime was invented a few
miles north in Marin County in the late 1970s.) Though there are hundreds of
trails in the open space preserves that spread over the hills to the west of
Stanford, there are just a handful of access trails that allow you to start and
finish your ride in Palo Alto. Of those, by far the most popular is Alpine
Road.
My mountain biking buddies and I
ascend Alpine Road roughly once a week in the mountain biking season (which in
California is usually around nine or ten months long). In this post, I'll
describe my own long struggle, stretching over many months, to master one
particularly difficult stretch of the climb, where many riders get off and walk
their bikes.
[SPOILER: If your interest in
mathematics is not matched by an obsession with bike riding, bear with me. My
entire account is actually about how to set about solving a difficult math
problem, particularly proving a theorem. I'll draw the two threads together in
a subsequent post, since it will take me into consideration of how the brain
works when it does mathematics. For now, I'll leave the drawing of those
conclusions as an exercise for the reader! So when you read mountain biking,
think math.]
Alpine Road used to take cars all
the way from Palo Alto to Skyline Boulevard at the summit of the Coastal Range,
but the upper part fell into disrepair in the late 1960s, and the
two-and-a-half-mile stretch from just west of Portola Valley to where it meets
the paved Page Mill Road just short of Skyline is now a dirt trail,
much frequented by hikers and mountain bikers.
 |
| Alpine Road. The trail is washed out just round the bend |
A few years ago, a storm washed out
a short section of the trail about half a mile up, and the local authority
constructed a bypass trail. About a quarter of a mile long, it is steep,
narrow, twisted, and a constant staircase of tree roots protruding from the
dirt floor. A brutal climb going up and a thrilling (beginners might say
terrifying) descent on the way back. Mountain bike heaven.
There is one particularly tricky
section right at the start. This is where you can develop the key abilities you
need to be able to prove mathematical theorems.
So you have a choice. Read
Polya's classic book, or
get a mountain bike and find your own version of the Alpine Road ByPass Trail.
(Better still: do both!)
 |
| My mountain bike at the start of the bypass trail |
When I first encountered Alpine
Road Dirt a few years ago, it took me many rides before I managed to get up the
first short, steep section of the ByPass Trail.
 |
| What lies around that sharp left-hand turn? |
It starts innocently enough—because you cannot see what awaits just around that sharp left-hand turn.
After you have made the turn, you
are greeted with a short narrow downhill. You will need it to gain as much
momentum as you can for what follows.
 |
| The short, narrow descent |
I've seen bikers with extremely
strong leg muscles who can plod their way up the wall that comes next, but I
can't do it that way. I learned how to get up it by using my
problem-solving/theorem-proving skills.
The first thing was to break the
main problem—get up the insanely steep, root strewn, loose-dirt climb—into
smaller, simpler problems, and solve those one at a time. Classic Polya.
But it's Polya with a twist—and
by "twist" I am not referring to the sharp triple-S bend in the
climb. The twist in this case is that the penalty for failure is physical, not
emotional as in mathematics. I fell off my bike a lot. The climb is insanely
steep. So steep that unless you bend really low, with your chin almost touching
your handlebar, your front wheel will lift off the ground. That gives rise to
an unpleasant feeling of panic that is perhaps not unlike the one that many students
encounter when faced with having to prove a theorem for the first time.
 |
| If you are not careful, your front wheel will lift off the ground. |
The photo above shows the first
difficult stretch. Though this first sub-problem is steep, there is a fairly
clear line to follow to the right that misses those roots, though at the very
least its steepness will slow you down, and on many occasions will result in an
ungainly, rapid dismount. And losing momentum is the last thing you want, since
the really hard part is further up ahead, near the top in the picture.
Also, do you see that rain- and
tire-worn groove that curves round to the right just over half way up—just
beyond that big root coming in from the left? It is actually deeper and
narrower than it looks in the photo, so unless you stay right in the middle of
the groove you will be thrown off line, and your ascent will be over. (Click on
the photo to enlarge it and you should be able to make out what I mean about
the groove. Staying in the groove can be tricky at times.)
Still, despite difficulties in the
execution, eventually, with repeated practice, I got to the point of
being able to negotiate this initial stretch and still have some forward
momentum. I could get up on muscle memory. What was once a series of
challenging problems, each dependent on the previous ones, was now a single
mastered skill.
[Remember, I don't have
super-strong leg muscles. I am primarily a road bike rider. I can ride for six
hours at a 16-18 mph pace, covering up to 100 miles or more. But to climb a
steep hill I have to get off the saddle and stand on the pedals, using my body
weight, not leg power. Unfortunately, if you take your weight off the saddle on
a mountain bike on a steep dirt climb, your rear wheel will start to spin and
you come to a stop - which on a steep hill means jump off quick or fall. So I
have to use a problem solving approach.]
Once I'd mastered the first
sub-problem, I could address the next. This one was much harder. See that area
at the top of the photo above where the trail curves right and then left? Here
is what it looks like up close.
 |
| The crux of the climb/problem. Now it is really steep. |
(Again, click on the photo to get a
good look. This is the mountain bike equivalent of being asked to solve a
complex math problem with many variables.)
Though the tire tracks might
suggest following a line to the left, I suspect they are left by riders coming
down. Coming out of that narrow, right-curving groove I pointed out earlier, it
would take an extremely strong rider to follow the left-hand line. No one I
know does it that way. An average rider (which I am) has to follow a zig-zag
line that cuts down the slope a bit.
Like most riders I have seen—and
for a while I did watch my more experienced buddies negotiate this slope to get
some clues—I start this part of the climb by aiming my bike between the two
roots, over at the right-hand side of the trail. (Bottom right of picture.)
The next question is, do you go
left of that little tree root nipple, sticking up all on its own, or do you
skirt it to the right? (If you enlarge the photo you will see that you most
definitely do not want either wheel to hit it.)
The wear-marks in the dirt show
that many riders make a sharp left after passing between those two roots at the
start, and steer left of the root protrusion. That's very tempting, as
the slope is notably less (initially). I tried that at first, but with
infrequent success. Most often, my left-bearing momentum carried me into
that obstacle course of tree roots over to the left, and though I sometimes
managed to recover and swing out to skirt to the left of that really big
root, more often than not I was not able to swing back right and avoid running
into that tree!
The underlying problem with that
line was that thin looking root at the base of the tree. Even with the above
photo blown up to full size, you can't really tell how tricky an obstacle it
presents at that stage in the climb. Here is a closer view.
 |
| The obstacle course of tree roots that awaits the rider who bears left |
If you enlarge this photo, you can
probably appreciate how that final, thin root can be a problem if you are out of
strength and momentum. Though the slope eases considerably at that point, I—like many riders I have seen—was on many occasions simply unable to make it
either over the root or circumventing it on one side—though all three
options would clearly be possible with fresh legs. And on the few occasions I
did make it, I felt I just got lucky—I had not mastered it. I had got the
right answer, but I had not really solved the problem. So close, so often. But,
as in mathematics, close is not good enough.
After realizing I did not have the
leg strength to master the left-of-the-nipple path, I switched to taking the
right-hand line. Though the slope was considerable steeper (that is very clear
from the blown-up photo), the tire-worn dirt showed that many riders chose that
option.
Several failed attempts and one or
two lucky successes convinced me that the trick was to steer to the right of
the nipple and then bear left around it, but keep as close to it as possible
without the rear wheel hitting it, and then head for the gap between the tree
roots over at the right.
After that, a fairly clear
left-bearing line on very gently sloping terrain takes you round to the right
to what appears to be a crest. (It turns out to be an inflection point rather
than a maximum, but let's bask for a while in the success we have had so far.)
Here is our brief basking point.
 |
| The inflection point. One more detail to resolve. |
As we oh-so-briefly catch our
breath and "coast" round the final, right-hand bend and see the
summit ahead, we come—very suddenly—to one final obstacle.
 |
| The summit of the climb |
At the root of the
problem (sorry!) is the fact that the right-hand turn is actually sharper than
the previous photo indicates, almost a switchback. Moreover, the slope
kicks up as you enter the turn. So you might not be able to gain sufficient
momentum to carry you over one or both of those tree roots on the left that you
find your bike heading towards. And in my case, I found I often did not have
any muscle strength left to carry me over them by brute force.
What worked for me is making an
even tighter turn that takes me to the right of the roots, with my right
shoulder narrowly missing that protruding tree trunk. A fine-tuned approach
that replaces one problem (power up and get over those roots) with another one
initially more difficult (slow down and make the tight turn even tighter).
And there we are. That final little
root poking up near the summit is easily skirted. The problem is solved.
To be sure, the rest of the ByPass
Trail still presents several other difficult challenges, a number of which took
me several attempts before I achieved mastery. Taken as a whole, the entire
ByPass is a hard climb, and many riders walk the entire quarter mile. But
nothing is as difficult as that initial stretch. I was able to ride the rest
long before I solved the problem of the first 100 feet. Which made it all the
sweeter when I finally did really crack that wall.
Now I (usually) breeze up it,
wondering why I found it so difficult for so long.
Usually? In my next post, I'll use
this story to talk about strategies for solving difficult mathematical
problems. In particular, I'll look at the role of the subconscious in being
able to put together a series of mastered steps in order to solve a big
problem. For a very curious thing happened after I took the photos to
illustrate this post. I walked back down to collect my bike from the ByPass
sign where I'd left it, and rode up to continue my ride.
It took me four attempts to
complete that initial climb!
And therein lies one of the biggest
secrets of being able to solve a difficult math problem.