Walking wrap up, 2025

December 2024

My original walking goal this year was 2,809 miles. This was to honor the perfect square 2025 = 45². 2,809 is the only square number (= 53²) between 2,800 and 2,900--my arbitrary goal range for the past few years--and it seemed like a manageable number after last year's excessive 2,865.

Too manageable. In autumn 2024, my doctor recommended that I work with a personal trainer because my knees had started making disconcerting crunching sounds when I climbed stairs. Crunch, crunch, crunch. Last December, I started working with a trainer at Empower Fitness, which is a 5.5-mile walk from home. In theory I was supposed to be strengthening muscles other than the ones I use excessively for walking, but of course I walked there and back twice a week almost every week that I was in town, and those miles added up. My knees still crunch when I climb stairs, but I've gone from barely being able to lift a 50-lb box of clay without hurting my back to deadlifting 120 pounds, so that's an improvement.

By September 2025, I thought I was more likely to land on 2,850, the triangular number I had anticipated in last year's wrap-up as a follow-up to 2024's sphenic number. 

T_n = n*(n+1)/2 
T₇₅ = 75*(75+1)/2 = 2850
Image courtesy of Melchoir in Wikipedia

Still, that seemed like blowing the only opportunity in my lifetime to walk a square number of miles in a square year. The next time a year will be square is 2116, and I don't expect to live to be 149 years old. I can walk 2,850 miles any ol' year.

I upped my goal to 2,916 miles, and enjoyed the not-quite-palindromic, reversible-numbers ring of walking 54² miles in the year 45².

I had to jack up my miles in mid-October to reach my new goal (and to bank miles for the Durham Pottery Tour), and I over-jacked. Then we spent five days on Ocracoke over Thanksgiving, where I noticed that if I jacked up my miles even more, I could make November a rare 300+ mile month. This left me a mere 155 miles short of 2,916 by the first day of December.

I confidently walked the numeral on Dec. 9--then reached it in miles on Dec. 23

Once in the habit, it's hard to cut back. I imagined my problem-solving friend N's voice in my head saying, "your goal is 2,916 recorded miles. Stop recording when you reach it." When I told her in person, she agreed her imaginary voice gave good advice. Did I heed it? Of course not.

Instead, I upped my goal to 2,965 recorded miles, a number that's nomenclaturally relevant because it's a centered square. Centered squares are figurate numbers (numbers that can be represented by dots laid out in geometric patterns) that have a single dot in the center, surrounded by dots laid out in successive outlines of squares: 

25 dots, one center dot surrounded by three successive square outlines. Image by Stefan Friedrich Birkner

13 dots demonstrating a nifty property of centered square numbers. Image by Stefan Friedrich Birkner

As the second dots-image above shows, centered squares can be expressed as sums of two consecutive perfect squares: 13 = 3² + 2². In the image above that, 25 = 4² + 3²; in my updated aspirational miles for this year, 2,965 = 39² + 38². In contrast to perfect squares, which are relevant to the mathematics of assorted disciplines, centered square numbers are of interest mainly for recreational mathematics.

While I was exploring finding a mileage higher than 2,916 that still relates to squares, I noticed that 2,965 is 49 miles more than 2,916; that is, 2,965 = 54² + 7². This made me wonder whether all positive integers can be expressed as sums of some small quantity of perfect squares (vs. e.g. tritely summing 1² until you reach the number you seek). I asked the interwebs, and lo: every positive integer can indeed be expressed as the sum of at most four perfect squares, a result known as Lagrange's Four-Square Theorem.

For the special case of a positive integer being the sum of exactly two perfect squares, as is the case for 2,965, see Fermat's theorem on the sum of two squares and this fun Sum of Two Squares calculator. Turns out 2,965 is the sum of only two combinations of two perfect squares: 54² + 7² and 39² + 38².

After learning all that, I overshot 2,965 too, because we decided to go back to Ocracoke with friends during the last week of December, and the very best thing for a walkaholic to do on Ocracoke is to scour the entire length of the beach multiple times looking for scotch bonnets. I blew past 2,965 on Dec. 28, with three entire days of walking left to go--but thanks to the four-square theorem, I figured any integer's-worth of miles could be a celebration of squares.

Mission accomplished

Whereas last year I came precipitously close to violating my 2023 goal to not be quite so obsessive about walking, this year I just went ahead and violated it. 

Final totals for 2025: 10 scotch bonnets (3 over Thanksgiving, 7 [!] in December) and 55² recorded miles, a new record for me. And if it so happened that the siren song of the scotch bonnet and the annual Ocracoke bird count called to me on this sunny last day of 2025--well then, I am fortunate to have wise friends like N.*

Next year's goal: 2026 is a "semiprime," a.k.a. "biprime." Semiprime numbers have only two factors, excluding 1 and the number itself, e.g. 2,026 = 2 * 1,013. (Incidentally, 2,809 = 53 * 53 is also semiprime--the prime factors don't need to be distinct.) Walking a semiprime number of miles has been on my to-do list as a follow-up to walking a sphenic number, and there are several options between 2,800 and 2,900.

And the year after that? 2027 is prime.

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2025 walking highlights:

Longest single walk: 31.22 miles. This was my annual American Tobacco Trail long walk on January 18. I mentioned to a friend that two years ago I had extended the 24.5-mile route to 26.2 so I could say I had walked a marathon, and that last year I had brought that up to 28. She noted that three more miles would be 50k. So I walked my first and probably only 50k ever. Turns out that once you hit ~26 miles using a 24.5-mile path and then some, there's not much entertainment value in going five miles farther. I don't know how ultramarathoners do it.

Not doing this again

Number of weeks topping 100 miles: 1 (early June).

Most miles in a single Monday-Sunday week: 101, thanks to a combination of hiking the Malerweg and intentionally walking some extra miles in Berlin to reach 100+.

Number of months topping 300 miles: 3 (June, July, and November). Haven't managed that since 2022. Hooray for long summer hikes and November beaches.

Most miles in a single month: 313.8 (November).

Favorite multi-day walk: Rorschach to Gündlischwand in 15 days.

Favorite single-day walk: This is a new category this year, so I don't have to choose between the Malerweg and Switzerland for the multi-day walk. On June 2, S and I hiked up Kaiserkrone in Saxony and gazed across Nebelmeer 217 years into the past.

Afterward, S caught a train back to Dresden, and I hiked the section of the Malerweg between Rathen and Liebethaler Grund. Bestill my beating heart...

Caspar David Friedrich was here.

Favorite photo: see "favorite single-day walk" above. Runners-up:

Obersee at dawn on day 6 of our hike from Rorschach to Gündlischwand

Stormy day on the Wörthsee after sailing the model boat.

Nothing else new to report. All of the walking philosophy from the preceding walking-intense years still stands, as does the warning from my PCP that knee cartilage has a limited lifespan, so don't overdo it. Ha.

What does 2026 hold in store? Less walking than in 2025. Another American Tobacco Trail marathon. Another car-free February. And a 17-day hut-to-hut hike from the Dolomites to the Adriatic. 

Here's to great walking in 2026!

*Below, ChatGPT's attempt at representing the figurate number of recorded miles with which I actually ended 2025. This illustrates that (1) ChatGPT isn't great at representing figurate numbers, because going by the number of dots in the border, the star is short by ~984 dots (and some of the "dots" are less dots than dot-like), and (2) someone has zero self-restraint when it comes to walking on undeveloped beaches.

Walking wrap up, 2024

For reasons explained elsewhere, my goal in 2024 was to walk a sphenic number of recorded miles in the vicinity of my 2023 mileage. 

I closed last year with 2,814 miles. This year, I aimed for 2,821 ( = 7 * 13 * 31). I figured if I overshot by a mile, I'd still end with a sphenic number (2,822 = 2 * 17 * 83). And if I overshot that by a mile, I'd land on a semiprime (2,823 = 3 * 941)--not ideal, but whatevs.¹

I was behind schedule in November, but a hiking-focused Thanksgiving trip to Tucson helped put 2,821 back in the realm of obtainable, so I stepped up my miles in December. 

I hit 2,821 on the Atlantic coast of Shackleford Banks, NC, on December 27.

Likewise 2,822...

...and 2,823.

So much for that.

I considered giving myself a new sphenic goal, or a semiprime, or a prime, and then Wikipedia told me that 2,850 is a triangular number. It isn't sphenic (although some triangular numbers are), but I figure a triangular number is in the spirit of the word sphenic--a sphenic number's etymological equilateral cousin, so to speak--so 2,850 became my new goal.²

T_n = n*(n+1)/2
T₇₅ = 2850 = 75*(75+1)/2
(Image courtesy of Melchoir on Wikipedia)

By the time I decided T₇₅ should be my goal, I had three days and 11 miles left, so of course I overshot that too.

Final recorded mileage for 2024 = 2865 miles. 2865 = 3 * 5 * 191, a sphenic number.

My last walk of 2024 was 0.2 miles longer than necessary

This is way more miles than I intended, and getting there brought me precipitously close to violating my 2023 goal to not be quite so obsessive about walking. But walking keeps me sane when SAD kicks in, so it's all good.

Next year's goal: in honor of 2025 = 45², I'll initially aim for 2,809 = 53². Overshooting options include the triangular³ number 2,850 or a semiprime.

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Some 2024 walking highlights:

Longest single walk: 28.09 miles. This was my annual American Tobacco Trail long walk on February 4, with a little extra distance to ensure it was longer than last year's ATT marathon. (I did the ATT again in mid-April, but didn't walk quite as far.) 

Number of weeks topping 100 miles: just 1 (in early June). I had 4 in 2023. I felt like I took more rest days this year than last, but apparently I did better keeping up with my weekly mileage goal (50 miles) and more forcefully made up for the lapses (18 weeks under 50 miles this year, vs. 20 last year).

Most miles in a single week: 100.3, which I only reached because I felt obligated to go for a 1.75 mile walk at 10pm on June 9. 

Number of months topping 300 miles: 2 (June and July), same as last year.

Most miles in a single month: 337.3 (June)

Favorite multi-day walk: Sterzing to Feltre in 13 days. S and I started in Sterzing, where we left off in 2023, and declared the Alps crossed when we arrived in Feltre. Feltre is described on various websites as being in the foothills of the Alps (although those hills still looked pretty mountainous to us), and is where the Alta Via 2 delle Dolomiti terminates.

Favorite photo: Mein geliebter Wanderer ueber dem Nebel(isch)meer, taken on Day 9 on the descent from Rifugio delle Selle:

There were some fine runners up:

Day 5, on Seceda

Day 4, en route to Rifugio Resciesa

Day 12, atop Mt. Pavione. The photo does justice neither to the expanse of globeflowers nor to the double cirque.

Day 8, heroic S literally moving mountains to help me cross a river of snow-melt

Nothing else new to report. All of the walking philosophy from the preceding walking-intense years still stands.

What does 2025 hold in store? Another American Tobacco Trail marathon. Another car-free February. And hopefully the Malerweg--a loop hike in the footsteps of Caspar David Friedrich (and other Romantic-era artists), to see the geology that inspired paintings like Der Wanderer ueber dem Nebelmeer--and a 2-week point-to-point trek in German-speaking Switzerland.

Here's to great walking in 2025!

My serifs weren't as good this year as last...

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¹I walked a 4-almost-prime number last year, so walking a sphenic number this year was supposed to make it obvious to aim for a semiprime in 2025 and a prime in 2026. I'm all about long-term goals.

²In case you missed it, the word "sphenic" comes from the Greek word σφήνα, meaning "wedge." 

³Of course, there are also square numbers (like 2025 and 2809), and pentagonal numbers, and hexagonal and etc. polygonal numbers, all of which are fun to think about as mileage goals, as well as polyhedral numbers that my brain has a harder time wrapping itself around.

Walking wrap up, 2023

My goal in 2023 was to walk less than in 2022, since my 2022 mileage (3,024 miles) required an obsessiveness I didn't want to repeat. 

Goal achieved! By 4pm on New Year's Eve, I had reached 2,813 recorded miles, which is indeed less than 3,024, but still enough to say "I walked a lot."

Then I thought, "but wouldn't it be cool to have a mile total that's divisible by 3? And by 2?" So at 10pm, after dinner with friends yet before sitting around a fire pit with them waiting for midnight, S and I excused ourselves to walk a tidy figure 8 around our block and an adjacent block, and I ended the year with 2,814 miles.

2,814 is indeed divisible by 3 and by 2--and by 7 and 67. Wouldja look at that--it's the product of four distinct prime numbers! What fun! I wonder how many of those are there in the vicinity of 2,800?¹ 

Had I been paying more attention, I might have happily stopped walking at 2,813 miles, since 2,813 is a semiprime--the product of just two prime numbers (2813 = 29 * 97). By definition, those prime numbers don't have to be distinct--so the squares of prime numbers are also semiprimes.

What about products of three prime numbers? A positive integer having three distinct prime factors is called a sphenic number (by definition, no squares are allowed). Once you go from three to four prime factors, you enter the territory of k-almost primes, where k is the number of prime factors. Squares are allowed, so 2,814, with its four distinct prime factors, is 4-almost prime, but so is 16 (16 = 2 * 2 * 2 * 2). I have mixed feelings about this.

I think for 2024 I should aim for my mileage total to be a sphenic number.³

I asked ChatGPT for a list of sphenic numbers between 2,800 and 3,200, and learned that ChatGPT is as incompetent at listing sphenic numbers as it is at reading roadmaps. This both gives me hope and stokes fear: AI will never replace people, but reliance on AI will probably end civilization as we know it.

To the sphenic list rescue: The On-Line Encyclopedia of Integer Sequences! Had I stopped at 2,810 miles last year, I would have nailed it.

For those of you dear readers who are still hanging in there, here's one last, exciting tidbit tangentially related to the number 2,814. I find it delightful how easy it is to see that 7 is a factor of 2,814, compared to e.g. the less immediately obviously 7-factorable number 2,226 (also a 4-almost-prime with four distinct factors). So I asked Google if there's an easy way to quickly determine whether a number is divisible by 7, AND THERE IS!² Oh my goodness, that extra figure-8 mile at 10pm on New Year's Eve was worthwhile!

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Here are the 2023 walking stats I'm most proud of:

Longest single walk: 26.81 miles. I finally walked a marathon! Twice, actually, on January 15 and January 27. The first was my annual American Tobacco Trail long walk. The second was a repeat for a friend who had wanted to go the first time.

Number of weeks topping 100 miles: 4 (2 in June, 1 in July, 1 in November)

Most miles in a single week: 108 (July)

Number of months topping 300 miles: 2 (June and July)

Most miles in a single month: 361 (June)

Favorite multi-day walk: In June, S and I went out the front door of the familial cottage in southern Bavaria and spent 11 days hiking to Italy.

Favorite photo: Hiking up from Hallerangerhaus to the Lafatscher saddle on Day 6 of that multi-day hike: 

Nothing else new to report. All of the walking philosophy from the preceding walking-intense years still stands.

What does 2024 hold in store? Another American Tobacco Trail marathon. Another car-free February. Hopefully completing our multi-day trek southward across the Alps, from our 2023 end point to where the mountains yield to flat terrain. 

Here's to great walking in 2024!

¹Between 2,800 (my minimum mileage goal last year) and 2,850, there are three almost-primes that have four distinct prime factors: 2,805 (5 * 3 * 11 * 17); 2,814 (2 * 3 * 7 * 67); and 2,838 (2 * 3 * 11 * 43).

²To quickly determine whether a number is divisible by 7: 

(1) remove the last digit of the number; 

(2) double that digit; then 

(3) subtract that from the truncated original number. 

If the result is divisible by 7, then the original number is divisible by 7. 

For example, 2,226:

(1) remove the 6;

(2) double it (6 * 2 = 12); then

(3) subtract 12 from the truncated number (222 - 12 = 210). 

210 is divisible by 7, so 2,226 is divisible by 7. 

My mathematician dad liked general proofs over anecdotal examples, so I'm including this link for him, wherein a general proof is offered.

(And about the above example, what fun: 210 = 2 * 3 * 5 * 7 -- the product of the four smallest prime numbers. 2,226 is likewise the product of four distinct primes -- 2 * 3 * 7 * 59. Clearly 2023 has given me food for thought for my 2024 walks.)

³I was interested in the etymology of the word sphenic, so I looked it up in the etymology bible, a.k.a. the Oxford English Dictionary--and there's no entry for the word at all. Tsk. Wikipedia to the rescue: the word is from Ancient Greek σφήν (sphḗn, “wedge”), + -ic.

Walking to Saxapahaw

Since returning from Germany last summer, I have been mourning the lack of easy long-distance day-hikes enabled by convenient mass transit. I finally decided to aim for some point-to-point hikes from Durham, figuring I could select destinations on bus or train lines.

In January, I started mapping out some potential routes. Maybe I could walk to Virginia! How many days would it take to reach the border, or Richmond, or Charlottesville? Turns out about 3 days, 10 days, and 10 days respectively. There's even a designated greenway--the 3,000-mile long East Coast Greenway--that passes through both Durham and Richmond, albeit mostly on roads and intended for bicyclists. 

I got as far as determining that finding places to stay en route would be a challenge. Then I discovered the ODX bus--the Orange-Durham Express--which runs from Durham to Hillsborough to Efland and back on weekdays during the early morning and early evening.

In February, the ODX enabled a 44-mile, 2-day walk starting from home:

    * Day 1: Durham to Hillsborough (19.5 miles), taking the ODX home (+1.5 miles). 

    * Day 2: ODX to Hillsborough (+1.4 miles), then walking to Saxapahaw (21.6 miles). 

There exists no mass transit home from Saxapahaw. S, who bicycles there on a semi-regular basis, kindly offered to pick me up by car. 

Day 1, Tuesday Feb. 21:

The walk from home to Hillsborough was hands down the most gloriously beautiful local walk I've taken since becoming a walkaholic. It knocked happening upon the ECWA Beaver Marsh Preserve on a snowy January day in 2021 down to second place.  

I needed to start our taxes, so my route began with a walk to Costco to buy software. From there, I continued past the strip malls...

For questioning potters...

...to Guess Road, and more commercial strips...

Because Du meine Seele, du mein Herz...

...eventually wending my way into the Croasdaille Farms subdivision.

I may or may not have walked on this path:

From Croasdaille, it was another few miles to the Laurel Bluffs Trail along the Eno River...

Stray daffodil

Trout lily

...then north along Cole Mill Road, past my second favorite fire hydrant in all of Durham (first favorite is described here).

Eno flowing beneath Cole Mill Road

Then the trash started, and didn't let up until the intersection with Pleasant Green. Here's a photo of someone's personal toxic waste dump at an especially egregious spot along Cole Mill Road:

This award-winning sculpture adorns the entrance to a private residence:

Open Hand, by Joe Coates

Eventually I landed in Eno River State Park. There's a short trail that begins just before the parking lot for Few's Ford, and it spits you out on the east side of the suspension bridge. That was an "ooh ahh!" moment for me, since I completely bypassed the bigger parking lot and the stepped trail down to the river. 

Ooh ahh! I'm east of the suspension bridge!

Cox Mountain Trail

Komoot, my route-planning app, told me to head west off of the Cox Mountain trail. Had it not told me to turn, I would not have noticed the un-signed trail. Clearly little used, and possibly non-maintained, it proved well worth the turn.

Old Eno homestead is gone, but daffodils remain.

The biggest, reddest open vein of clay I've ever seen runs up a steep hill

It's hard to make out, but the trail was an old forest road

The trail continued to the state park border and, with ominous signage, onto private land; so [TECHNICAL DIFFICULTIES] following [TECHNICAL DIFFICULTIES] and [TECHNICAL DIFFICULTIES], past a pottery, to where I finally arrived at the Mountains to Sea Trail--a.k.a. Hwy 70:

Vroom vroom. Yes, this stretch of highway is part of the MST.

After walking a mile or so north on 70, I missed the turn-off to stay on the MST along the Eno. Because I had just enough time to catch the 5:20pm bus home if I hurried, I chose not to backtrack to find the turnoff, and instead continued on the highway until St. Mary's Road, then turned left, past Ayr Mount and River Park Elementary and into scenic downtown Hillsborough. Shortly thereafter, I was one of only two passengers on the impressively speedy ODX bus back to Durham.

Day 2, Tuesday Feb. 28:

In theory, I was supposed to do this section the day after Day 1. But life and weather and blisters changed that. A week later, and on a day's notice, my friend TS joined me for the walk to Saxapahaw. I'd never been to Saxapahaw, and figured that arriving on foot would be a rewarding way to experience it.

T and I met shortly after dawn and skedaddled downtown to the bus station. T is one of the three fastest walkers I know (hat tips also to RM and CM), so we arrived with ten minutes to spare. 

The speedy ODX dropped us off across the street from the Weaver Street Co-op, where we were instructed to stretch--so we did.

We linked up with the MST behind the co-op. 

Mosaic in a parking garage behind the co-op: "Love Hillsborough"

The MST follows the Hillsborough River Walk for a bit:

The River Walk includes bridges, boardwalks, and paved trails...

...and tons of turkey vultures. 

Eventually we left the River Walk and headed west along a road to Efland. This road did not show Hillsborough at its finest. (I took a photo of some vulgar flags in front of a trailer home, but the image is too uncouth to post here.)

Eventually we approached I-40...

Cotton

The scenic Mountains-to-Sea Trail...

...and crossed to the south side of the interstate. It took another mile or two before we finally left the traffic and noise of the Efland I-40 interchange behind.

Our route followed an approximation of a future planned section of the MST. For now, it's an "alternate" route entirely on paved roads.

At the intersection of Chestnut Ridge Church Road and Buckhorn Road, we stopped at the INFO CEN ER, but learned very little:

We encountered only a handful of bicyclists and joggers over the course of the day.

Look! There's one!

For lunch, we paused near some houses on a scenic stretch of Bradshaw Quarry Road. With a healthy fear of trespassing and gun ownership in rural NC, I was game for sitting on a guard rail with our backs to the traffic, but T insisted on tromping off the road a bit. We found a fallen tree to sit on, and ate lunch surrounded by trout lilies. 

Then it was back to the country roads...

We passed a sign for Yellow Rake Road, although that was clearly a misnomer:  

Shortly beyond Pink Broom Road, it was baby burro season...

We crossed into Alamance County, which looked much like Orange County.

Over our entire route, we saw sign after sign that said "Private Property: No Trespassing." Sometimes these signs made sense, e.g. next to fishing ponds in which one might drown, and on Orange County Water and Sewer Authority land on which one might taint the drinking water. Still, we wondered what secret treasures must be hidden in these here hills to warrant so many signs, and we applauded multiple residential property owners for wisely making their land and wood piles look distinctly non-covetable.

This property looked covetable. What illicit activities would you engage in here?

By way of contrast, here's a welcoming, self-serve, honor-system, wee 24/7 open-door apiary store a few miles outside of Saxapahaw. I Venmo'd the owners $25 and added a 3-pound glass jar of local honey to my backpack, to gift to my honey-loving honey in appreciation of the upcoming ride home.

Signage told us when we arrived in Saxapahaw, although our official destination--the General Store--was still a few miles down the road.

After tanking up on seltzer and a chocolate chip cookie at the General Store, we meandered through Saxapahaw down to the Haw River. 

There's an island in the middle of the river, aptly named Saxapahaw Island, with a park aptly named Saxapahaw Island Park. We hiked a short loop trail around the park before heading back to the General Store to meet S for dinner.

And that was that. I was pleased that the ride home took a good 40 minutes, since it's always disheartening to have a seven-hour hike undone by a mere 20-minute car or train ride. (And for those of you paying attention, the drive to Hillsborough would have been about 30 minutes, which is also more than 20. Whew!)