Kash Patel math

At a press conference on Tuesday, April 21, 2026, FBI director Kash Patel said, “This FBI director has been on the job twice as many days as every director before me. What that means is I've taken half as many days off as those before me. What that means is I've taken a third less vacation than those before me.” (Source: https://www.instagram.com/reels/DXal0fgDxd9/ )

Question:

How many days has Kash Patel worked compared to every FBI director before him?

Assumptions:

* Kash Patel is comparing his days on the job to the average of the number of days on the job of each individual director before him over the same number of days. That is, Kash Patel is NOT comparing his days on the job to the sum of days on the job of all preceding FBI directors. (For a list of previous FBI directors and duration of service, see https://www.britannica.com/topic/directors-of-the-Federal-Bureau-of-Investigation-2227605 )

* Because Kash Patel distinguishes between days off and vacation days, days off and vacation days are mutually exclusive.

* The sum of days worked, days off, and vacation days = Kash Patel's tenure so far at the FBI. Kash Patel was sworn in on Feb. 21, 2025. At the time of the press conference on April 21, 2026, Kash Patel had been FBI director for 424 days.

* The number of vacation days Kash Patel has taken since his swearing in = 17. (Source: https://www.forbes.com/sites/conormurray/2026/04/19/kash-patel-promises-hell-sue-the-atlantic-on-monday-after-report-of-excessive-drinking/ )

Definitions of terms:

Dk = Kash Patel’s days on the job
Ok = Kash Patel’s days off
Vk = Kash Patel’s vacation days
Dp = average of Kash Patel’s predecessors’ days on the job
Op = average of Kash Patel’s predecessors’ days off
Vp = average of Kash Patel’s predecessors’ vacation days

Equations:

(a) Dk + Ok + Vk = 424
(b) Dp + Op + Vp = 424
(c) Dk = 2Dp
(d) Ok = Op/2
(e) Vk = Vp – Vp/3 = 2Vp/3
(f) Vk = 17

Solve for Dk, Ok, Vk, Dp, Op, and Vp:

From (e) and (f),
Vp = 3Vk/2
= 17 * 3 / 2
Vp = 25.5

From (a) and (b),
Dk + Ok + Vk = Dp + Op + Vp
Dk = Dp + Op + Vp - Ok - Vk
= Dp + Op + 25.5 - Ok – 17
= Dp + Op - Ok + 8.5

From (d),
Dk = Dp + Op - Op/2 + 8.5
= Dp + Op/2 + 8.5

From (c), Dk = 2Dp
2Dp = Dp + Op/2 + 8.5
Dp = Op/2 + 8.5

From (b),
Dp + Op + Vp = (Op/2 + 8.5) + Op + 25.5
= 3Op/2 + 34 = 424
3Op/2 = 424 - 34 = 390
Op = 2 * 390 / 3 = 260
Ok = Op/2 = 130

Dp = 424 - Op - Vp
= 424 - 260 - 25.5
Dp = 138.5
Dk = 2 * 138.5 = 277

Conclusion:

For every 424 days in office, Kash Patel has worked 277 days; taken off 130 days; and taken 17 vacation days. For every 424 days in office, Kash Patel’s predecessors on average worked 138.5 days; took off 260 days; and took 25.5 vacation days.

Normalized per year:

Days worked per year: Kash Patel 238.5, Predecessors 119.3
Days off per year: Kash Patel 111.9, Predecessors 223.8
Vacation days per year: Kash Patel 14.6, Predecessors 22

Discussion questions:

* Do you think Kash Patel's assessment of his predecessors' work is accurate? Why or why not? How might you check?
* How would these numbers change if Kash Patel meant "business days" instead of "days"?
* How would these numbers change if weekends are included in "days off"?
* How would these numbers change if (a) "days off" are counted as "vacation days"? (b) If "vacation days" are counted as "days off"?
* Kash Patel was confirmed in the senate by a vote of 51-49. How many votes too many was that?
* J. Edgar Hoover was FBI Director for 48 years and 358 days. How many days total does Kash Patel think Hoover (a) worked, (b) took off, and (c) used for vacation?

Die Durhaemmer

When I first started walking with greater intention (absichtswandern?) back in 2021, one of the motivations was that those trans-continental trails I wanted to traverse weren't going to hike themselves. Turns out I was thinking about trans-continental trails on the wrong continent, but whatevs. 

As S approaches retirement and we contemplate spending more months every year in Germany, I realized that my German language skills weren't going to improve themselves. My German is adequate, but not great, and if I'm going to have any kind of social or work life in Germany, it behooves me to be better than adequate.

Thus was born Die Durhaemmer, a German-language review club. The name sorta means The Durhamites--or The Minor-Key Hammers. We're using a free online textbook and reviewing one unit or so per week. Participation varies, but we're hobbling along with a core group of attendees after six months, and I continue to meet new and interesting people connected by a shared language interest.

Alongside Die Durhaemmer, I've started using an app called Seedlang, which was created by the same folks who appear in many of the videos used by the online textbook. They have enough of a sense of humor and offer enough grammar explication that I paid for a year's subscription, and I've been refreshing and newly learning swaths of useful vocabulary. The app incorporates enthusiastic verbal reinforcement from its makers: every time you finish an exercise, someone pops onto the screen and says something like "way to go!"--which is how I learned Du bist der Hammer! means "You're da bomb!" So now we have a third meaning for Die Durhaemmer.

I'm looking forward to inserting some of my new vocab into conversations this summer, such as:

Die Fortbewegungsmittel (pl. -) - means of transportation. Same number of syllables as in English. Vroom vroom!

Die Bedienungsanleitung (pl. -en) - service manual. Why use two words when you can just use one?

Der Tacho (pl. -s) - speedometer. How have I gotten by in Germany for the past 36 years without knowing the word speedometer? Plus it sounds like a mix between Taco and Tycho, like Tycho Brahe the astronomer, which makes it fun to say.

My favorite new word so far is Die Gleichung. Gleich means "same." Equations express equivalences, things that are the quantitatively the same on both sides of the equals sign. The word "equation" in English offers a similar conceptual self-awareness as die Gleichung. In fact, there's a shared conceptual origin in both languages, even if the words gleich and "equal" themselves don't share any etymological roots. I find these sorts of connections delightful.   

My current favorite new sentence, courtesy of the app: Was ist die schnellste Geschwindigkeit, die du je gafahren bist? ("What's the fastest speed you've ever driven?") So many useful grammatical tidbits and reminders to cull from that, plus a casual way to segue iinto using the word Tacho.

Fun facts are rapidly accruing, like why is it das Wort (the word) but die Antwort (the answer), when gender is usually determined by the gender of the final root word (wort)? The interwebs say it's because the formerly neuter noun Antwort hung around so frequently with the feminine noun die Frage (the question), that users gradually made Antwort feminine.

Speaking of linguistic evolution, here's a fun fact about many weak masculine nouns (which I love--masculine nouns are the only ones that can be weak, and whatever their weakness is causes them to add an -n to the end when they're in any case but nominative, because who knows why). Many weak nouns, such as der Friede (peace), are becoming permanently -n-ed (-n-ified?): speakers often now use der Frieden instead of der Friede when peace is the subject. Cool.

I learned at last week's Durhaemmer meeting that one linguistic evolution is the gradual abandonment of the genitive case, which apparently increasingly circumvented using von (of) with the dative case. Genitive is generally used to indicate a noun that is of something, e.g. "of good courage" in "Sei gutes Muts" ("Be of good courage"--thanks, Der Tod und das Maedchen!), or "the pleasure of the miller" in "Das Wandern ist des Mullers Lust" ("Hiking is the pleasure of the miller"--thanks, Die Schoene Muellerin!). As enthusiastic as I am about linguistic shifts, this one makes me a little sad, as one of my motivations for starting Die Durhaemmer was to correctly deploy the genitive case. We won't even get to it until the free German 102 online textbook, but here it is, already dying. Dang. And yet, to paraphrase the dying John Adams' last words ("Jefferson still lives"), at least we still have the dative. (Dative is the to/for case--e.g. Ich gebe dem Hund den Ball--"I give to the dative dog the accusative ball"). We won't get to dative until the next textbook either, but Die Durhaemmer are a precocious lot, and we're already dativing.

Gentle reader, as you can tell from this post (if you've stuck with it this far), German is giving me as intellectually vast a rabbit hole as I've found physically with walking. The immediate impact is that I speak German a tad more precisely but much more slowly (click click click goes my brain). Looking forward to speeding things up and putting refreshed skills good use this summer.

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Update: I don't think E has read this post, but he happened to email a long word yesterday that he came across in his research: das Fernwasserleitungsnetz (long distance water distribution network).

Already behind...

Strava says I'm currently 156 miles behind my 2026 mileage goal (the biprime 2,866); and I haven't met my monthly goal (200 miles/month) even once this year.

I explain this lapse as a combination of weather and bugs. We had two successive weekends of ice and snow in Durham at the end of January. Then S got a cold, I got a cold, S got another cold, I got Covid, S got Covid, I was felled by spring allergies, and S was felled by spring allergies. We've been coughing for a good two months at this point and we're both ready for our Eustachian tubes to drain.

This isn't an excuse, just a placeholder to myself when I have to step up the miles at the end of the year.

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.

36 hours in NYC

Cross post!

Expanding the Bavarian curse repertoire

"Herrschaftzeiten," mumbles S. He is squinting at his cellphone. I recognize from the tone of his voice that this is a Bavarian curse, but despite living with S for 33 years, I've never heard this one.

--"Her satt zeichen?" I ask. "What's that?"

"The weather," he says.

--"No, I mean what's 'her satt zeichen'? You've been cursing a lot this semester, but I haven't heard that one yet."

"Herrschaftzeiten."

--"Herrschaft Zeiten? Lordship's times? That's a curse?"

"Herrschaftzeiten. Times of the superiors. It's what you say when your superiors are on your nerves."

--"Uh oh, something up at work?"

"No, it's the weather."

--"Hunh?"

"It's supposed to rain."

--"What does the weather have to do with Lordships? I mean, why that particular curse?"

"I dunno. It's just what you say."

I pull out my laptop and head to Google translate. It tells me Herrschaftzeiten means "reign times."* But S and the curse are Bavarian. He finds a Bayerisch translation.

"It comes from Herr, schau auf die Seite." 

--"Lord, look askance." That makes more sense. 

We decide to make a list of Bavarian curses for future reference. It turns out that despite all of his cursing, S thinks his repertoire is limited to only five. (As I was typing this, first it was just four, then just five, then just six, and now eight. Time will tell.)

S's comprehensive (?) list of Bavarian curses

Hoast mi! - Hast du mich verstanden - Do you understand me?! 

        * (I always thought this one meant "Hoerst du mich?"--Hear me!--but S says otherwise. And to be fair, S doesn't say this--I do. It's one of the first German curse phrases S taught me because it was cute to hear the oblivious USAmerican say it, and it makes Bavarians laugh.)

Himmel, Arsch, und Zwirrn - Heaven, ass, and twine.

So a Kas' - What a cheese. 

So was bloedes - Idiotically stupid

        * (I always thought this one mean "bloody" in the British curse sense, but I was wrong.)

Sau dumm - Pig dumb.

Hejgott sakra - Herr Gott Sakra - Lord God, Sacrament.

Herrschaftzeiten - Lord look askance.

Zefix - Crucifix.

Bonus list: Curses S's mom H "used all the time"

Kruzetiakn - Kruzi Turken - S thought this meant "crucify the Turks," but Kruze apparently refers to Kuruzen, a Hungarian Protestant sect that united with the Turks against Catholics in the late 1600s. 

Depp - Deppeda - So a Dep - What a dope.

Dridschla - Slowpoke, e.g. that jerk who's driving aggravatingly slowly right in front of you.

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For additional Bavarian curses, see Wikipedia's collection of Bavarian Curses at https://bar.wikipedia.org/wiki/Spruch:Boarische_Fluachsammlung. (The title of the page is, naturally, Bavarian for "Bavarian Curse Collection." In high German, one would write "Bayerische Fluchsammlung.")

-------------

*Reign and rain are homophones, although that has nothing to do with any of this.

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.