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.