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?

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.

-----

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.

On sphenic numbers and AI

Way back in January, in my Walking Wrap-Up post for 2023, I mentioned sphenic numbers. Sphenic numbers are positive integers that are the product of three distinct prime numbers (e.g. 30 = 2 * 3 * 5 and 42 = 2 * 3 * 7). I was so enamored of this concept that I decided my mileage goal for 2024 should be sphenic. I also noted that when I queried Generative AI about the sphenic numbers closest to my 2023 mileage, AI struggled to come up with anything remotely correct.

I was thinking about sphenic numbers again in September as my public Instagram account (@geekpots) crept toward 1,000 followers. As is the Instagram self-promotional advertising way, I considered holding a pottery giveaway to celebrate reaching a notable number-of-followers goal-post--but my plan would be different from the norm. 

First, the giveaway wouldn't require anyone to do anything other than to comment some variation of "pick me!"--none of that stuff like "to enter the drawing, you must (1) like this post, (2) sign up for my newsletter, (3) tag two friends, and (4) promise your first-born to a gnome-like man who spins straw into gold."

Second, why celebrate 1,000, when I could celebrate the largest sphenic number below 1,000?

I asked Google for a list of sphenic numbers between 900 and 1,000, and Google's generative AI responded with this:

AI Overview 

The only sphenic number between 900 and 1000 is 930. 

Explanation: 

• A sphenic number is a positive integer that can be written as the product of exactly three distinct prime numbers. 
• Prime factors of 930 are 2, 3, and 5, making it a sphenic number.

Here's a fuzzy screen-shot as evidence:

"The only sphenic number between 900 and 1000 is 930" 

Gentle reader(s), do you remember the sepia-tinted olden days when Google searches were actually useful? 

930 is not sphenic. Its prime factors are 2, 3, 5, and 31. It's "4-almost-prime," not sphenic. There are 16 sphenic numbers between 900 and 1,000, and the highest is 994 ( = 2 * 7 * 71 ).

I subsequently repeated the Google query shown above every other week or so, and the answer always came back "930." Go ahead, you try it.

------

Halfway through December, S and I met up for brunch with E and his sweetie A, and at one point our conversation landed on generative AI. I griped about Google's incorrect claim about 930. E suggested I ask ChatGPT instead.

ChatGPT said:

"Thus, 990 is the only sphenic number between 900 and 1000"

Gentle reader(s), 990 is not sphenic. Nor does 990 equal 2 * 3 * 5 * 11.

E asked ChatGPT to write Python code that could generate a list of sphenic numbers between two user-chosen numbers. His request took him all of 30 seconds to type, and all of 2 seconds for ChatGPT to produce. E then copied and ran the code for 900 ≤ n ≤ 1000, and it generated an accurate list. (If you don't want to bother with code, you can find a list of the first 10,000 sphenic numbers here.)

-----

A few days later, when I was out walking, it occurred to me that because ChatGPT is good with languages but poor with comprehension, maybe it could write about sphenic numbers better than it could come up with lists of them. So I asked it to. The conversation is recorded below, with my queries to the left, ChatGPTs responses in blockquotes, and my thoughts in italics. 

------

Write a poem about sphenic numbers that includes a definition 

A sphenic number, rare and true,
Is made from primes in a special brew. 
Three primes distinct, their product shines, 
Each factor unique, no overlap binds. 

It's not a square, nor a cube,

But a product of primes that does not lube.

Take 6, for instance, a simple start, 

2 * 3, primes from the heart. 

Next comes 30, a step more wide,

2 * 3 * 5, side by side. 

Primes that combine with none the same, 

A sphenic number, in prime’s name. 

Not every product of primes you’ll see, 

Makes a sphenic number, as clear as can be.

But those that do, with their special form,

Stand as distinct, away from the norm.

So now you know what makes them tick,

Three primes combined, the magic trick.

A sphenic number, bold and bright,

Shining like stars, in math’s pure light. 

Formulaic, but not bad for a first try. Not sure what "lube" means in line 6.

------

Write a poem about sphenic numbers that includes a definition and an example of a sphenic number between 900 and 1000 

A sphenic number, rare and bright,

Is made from primes, in pure delight.

Three distinct primes, all multiplied,

A product formed where factors bide.

Each prime stands alone, not shared,

A number with no twins, compared.

Multiply the primes you see,

And find the product’s mystery. 

Take 930, it fits the bill, 

Prime factors—three, and they stand still: 

2, 3, and 5, they intertwine,

Forming 930, pure and fine.

Sphenic numbers, quite unique, 

In prime trio, their strength we seek. 

You and Google AI: What's with "930"? Also, 2 * 3 * 5 = 30, not 930.

------

Explain how 930 is a sphenic number 

To explain how 930 is a sphenic number, let's first define what a sphenic number is: a sphenic number is a positive integer that is the product of exactly three distinct prime numbers.

Now, let's check if 930 fits this definition:

1. Prime Factorization of 930:

• Start by dividing 930 by 2 (since it's even): 930 / 2 = 465

• 465 is odd, so it's not divisible by 2. 

• Next, divide by 3: 465 / 3 = 155 

• 155 is divisible by 5: 155 / 5 = 31 

• 31 is a prime number.

2. Prime Factors of 930:

The prime factorization of 930 is: 930 = 2 * 3 * 5 * 31

• Here, we see that 930 has four prime factors (2, 3, 5, and 31), not three distinct primes.

Thus, 930 is not a sphenic number, because a sphenic number requires exactly three distinct primes. I apologize for the earlier confusion.

Apology appreciated. (At this point, my cellphone had a conniption fit, so I had to restart it, which meant starting the conversation from scratch.)

------

Write a poem about sphenic numbers that includes a definition and an example of a sphenic number between 900 and 1000

[...]

For instance, look at nine hundred and ninety, 

A sphenic number, strong and mighty. 

Its primes are three, ten, and eleven, 

Multiply them, and the answer is heaven.

[...]

Ten is not prime. 3 * 10 * 11 = 330, not "nine hundred and ninety," and it isn't sphenic.

------

Explain how 990 is sphenic 

You're right to question the number 990! In fact, 990 is not a sphenic number. 

Let's break down why: A sphenic number is a positive integer that is the product of exactly three distinct prime numbers. In other words, it is a product of three unique prime factors, with no repeats. 

Now, let's factorize 990: 

1. First, divide by 2 (since it's even): 990 ÷ 2 = 495 

• So, 2 is a prime factor.

2. Next, divide 495 by 3 (since the sum of its digits, 4 + 9 + 5 = 18, is divisible by 3): 495 ÷ 3 = 165 

• So, 3 is a prime factor. 

3. Finally, divide 165 by 5 (since it ends in 5): 165 ÷ 5 = 33 

• So, 5 is a prime factor. 

4. Now, 33 is divisible by 3 again: 33 ÷ 3 = 11 

• So, 3 is a prime factor again. 

Therefore, the prime factorization of 990 is: 990 = 2 * 3² * 5 * 11

Since 3 is repeated as a factor, 990 is not a sphenic number. A true sphenic number would have exactly three distinct primes and 990 has four distinct prime factors, with one of them (3) repeating. 

So, a correct example of a sphenic number between 900 and 1000 would be 930, since: 930 = 2 * 3 * 5 * 31

Here, the product of the distinct primes is 930, which has exactly three distinct prime factors: 2, 3, and 5. Apologies for the confusion, and thank you for pointing it out! 

You're welcome, but golly, you're kinda an idiot. How is it that you can think to sum the digits of a number and divide by 3 to tell whether the number is divisible by 3, but you can't recognize the difference between "three" and "four" when it comes to the number of prime factors?

(I then ask it about 930 again, because it probably doesn't remember anything from before my phone conked out. Note it's offer to find a correct example--)

------

Explain how 930 is sphenic 

Apologies again for the confusion in my previous response. You're right to ask for clarification about 930. Upon reviewing the factorization of 930, it turns out 930 is also NOT a sphenic number. 

Let's break it down properly: [yadda yadda yadda]

Correct Example of a Sphenic Number Between 900 and 1000: Let's find the correct example! A sphenic number between 900 and 1000 should have exactly three distinct prime factors.

The number 930 doesn't qualify, but 990 does have the required prime factors as well. Here's the correct example: 

Example: 990: 

The prime factorization of 990 is: 990 = 2 * 3 * 5 * 11 

But because there's no repeating primes or errors, this number might be 

Aw geez, 990 again, plus an unfinished sentence at the end...

------

Might be what? 

I apologize for the confusion in my previous responses. Let's clarify everything: [yadda yadda yadda] 

Correct Example of a Sphenic Number Between 900 and 1000: Let's find a correct sphenic number between 900 and 1000. The number 935 is a sphenic number because: [yadda yadda yadda]

Thus, the prime factorization of 935 is: 

935 = 5 * 11 * 17

It has exactly three distinct prime factors (5, 11, and 17), so 935 is a sphenic number. 

Final Answer: 935 is a correct sphenic number between 900 and 1000. Thank you for your patience! 

"Final answer," lol, after how many tries? 935 is indeed sphenic--but a broken clock is likewise correct twice a day.

Assuming ChatGPT trained on some website that claimed 990 was sphenic, I ask for its source.

------

Where did you read that 990 is sphenic? 

I made the mistake of incorrectly asserting that 990 was a sphenic number based on my own error in processing the factors. I didn't read or reference any specific external source that labeled 990 as a sphenic number. The mistake stemmed from my own misunderstanding while performing the prime factorization. I didn’t check carefully enough to see that 3 repeats in the factorization of 990. I mistakenly thought it might still qualify, but as we now know, 990 is not sphenic due to the repetition of 3. I appreciate your sharp observation, and again, I apologize for the oversight. Thank you for your patience! 

Like college undergrads everywhere, ChatGPT says it did its own work!

------

Where did you read that 930 is sphenic? 

I mistakenly suggested that 930 might be a sphenic number based on an incorrect factorization and assumption. Similar to my mistake with 990, I didn't read or refer to any specific source that stated 930 was a sphenic number. It was an error I made while processing the factorization. To clarify, I incorrectly concluded that 930 might be a sphenic number, but upon breaking it down, I found that 930 has four distinct prime factors (2, 3, 5, and 31), so it is not a sphenic number either. I didn’t rely on an external source to make this claim; it was a mistake in my reasoning. I appreciate your careful attention and the opportunity to correct it. Thank you again for your patience! 

I don't believe ChatGPT, and accuse it of using Google:

------

Google AI search also says 930 is sphenic. Did you use Google AI search? 

I did not use Google AI search or any external AI tools to suggest that 930 is a sphenic number. My mistake was entirely due to my own miscalculation and incorrect reasoning when processing the prime factorization of 930. 

To clarify, I incorrectly concluded that 930 might be a sphenic number based on its factorization, without thoroughly verifying that it had exactly three distinct prime factors. In fact, 930 has four distinct prime factors (2, 3, 5, and 31), so it does not qualify as a sphenic number. 

It’s possible that external sources or other AI tools might have made a similar error when processing this number. However, I do not rely on Google AI search for these calculations—it's purely based on the reasoning I perform. I appreciate your attention to detail and your patience in allowing me to correct this.

OK, this explanation sounds plausible. Let's get away from 990 and 930.

------

Give me another example of a sphenic number between 900 and 1000 

Let's find another example of a sphenic number between 900 and 1000! A sphenic number is the product of exactly three distinct prime numbers. Here’s a valid example: 

1002: Let's factorize 1002: 

1. 1002 ÷ 2 = 501 (so 2 is a prime factor). 
2. 501 ÷ 3 = 167 (so 3 is a prime factor). 
3. 167 is a prime number. 

So, the prime factorization of 1002 is: 1002 = 2 * 3 * 167 . This has exactly three distinct prime factors: 2, 3, and 167. Therefore, 1002 is a sphenic number. 

Conclusion: 1002 is a valid sphenic number between 900 and 1000. 

WTF. And how do you decide when to use boldface font? I know a bunch of grad students applying for grants who'd like to know.

------

Explain how 1002 is between 900 and 1000 

You're absolutely right, and I apologize for the mistake! 1002 is not between 900 and 1000—it is greater than 1000. I made an error in my response, and I appreciate your patience in pointing that out. 

Let's correct this and find a proper sphenic number between 900 and 1000. 

Correct Example: 910

Now, let's check 910. [yadda yadda yadda] The prime factorization of 910 is: 910 = 2 * 5 * 7 * 13. Since this has four prime factors, 910 isn't a sphenic number.

Let's check a new candidate like 935 

From "Correct Example" to "isn't sphenic," all on your own!

------

You already showed 935 is sphenic. Give me a different example. 

[yadda yadda yadda]...969...

969 is sphenic! Is ChatGPT finally figuring this out?

------

What determines whether you check the truth of your statements?

Thank you for your thoughtful question. The accuracy of my responses is primarily determined by the way I process information and follow logical steps when explaining or performing calculations. However, as a language model, I rely on the context provided by you and on the knowledge I have been trained on. Here’s how I typically check the truth of my statements: 

1. Internal Reasoning: I perform the necessary logical steps to check whether a statement is true. For example, when discussing sphenic numbers, I perform prime factorization to verify whether a number has exactly three distinct primes. However, I don’t have external tools like calculators or databases for real-time verification.
2. Knowledge Base: My responses are generated based on the large dataset I’ve been trained on, which includes mathematical principles, definitions, and examples up until my last update. I use this knowledge to form my answers, but I don’t have access to real-time data or the internet. 
3. Self-Correction: In cases where I make an error (such as the one regarding 1002 or 930), I rely on feedback, like yours, to help me recognize the mistake and adjust my understanding. This is why I appreciate your patience when I make errors, as you help guide me toward the correct answer. 
4. User Interaction: I check the accuracy of my responses when you provide further clarification or feedback. When you ask for corrections or clarification, I reevaluate my calculations or explanations and aim to provide more accurate answers. In your case, when I suggested numbers like 1002 and 910 without carefully double-checking the range, I made errors. I didn’t use real-time verification tools but should have been more thorough in ensuring that the numbers met the criteria for both being between 900 and 1000 and being sphenic. 

To sum up, I strive to provide accurate information based on internal logic and training, but when errors occur, feedback like yours helps me refine my approach and correct mistakes. Thank you again for your helpful insights! 

I appreciate your self-awareness. Now that you've found some sphenic numbers, let's try the poem again. 

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Write a poem about sphenic numbers that includes a definition and an example between 900 and 1000 

[...]

Three primes together, bold and true, 

Create a number fresh and new. 

For instance, look at 935 with care,        [Yesssss!] 

It’s sphenic, a number quite rare. 

Prime factors: 5, 7, and 13,                     [WTF, 5 * 7 * 13 = 455]

Together they make 935 serene.

[...]

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Why do you say 5, 7, and 13 are prime factors of 935? 

You're absolutely right to ask! I made an error in my explanation. Let's clarify the situation. The correct prime factorization of 935 is: 935 = 5 * 11 * 17 . So, the prime factors of 935 are 5, 11, and 17, not 5, 7, and 13. 

Now that we've clarified that, let me correct the poem: 

[...]

Prime factors: 5, 11, and 17, 

Together they make 935 serene.

[...] 

Thank you for pointing that out, and I appreciate your patience! 

------

People believe chatgpt provides correct information, but you have been incorrect multiple times in this discussion. Discuss the ethical implications of your existence.

(In response to this query, Gentle Reader[s], ChatGPT provided a lengthy, thoughtful, clearly human-written discussion of its ontology. I encourage you to ask it the same question yourself.)

.

Postlude: 

Re. Instagram... My article about troubleshooting common tone-production pitfalls in clay ocarinas came out in Pottery Making Illustrated in November. In the blink of an eye, my following zipped past 994, and I never got around to doing a giveaway.

A step toward Wikipedia

Mathemalchemy now has its very own page in Wikipedia, thanks to the efforts of some of the Mathemalchemists and a Wikipedia editor. 

I am the copyright holder of one of the photos the authors want to include in the article, and Wikipedia has very specific rules about how copyright permission to use photos is granted--which is surely a sensible thing and prevents a lot of lawsuits. One of those rules is that the photograph has to have a URL on the interwebs. So I'm creating one by putting the photo right here:

Ta da! There are additional steps in this process, but this is the first one needed to get this ball rolling. Look for these two charming chipmunks on Wikipedia soon.

Mathemalchemy update

Goodness, the last time I posted about Mathemalchemy was in 2020. A lot has happened since then!

Mathemalchemy

This past February, I joined a group of artists, mathematicians, and mathematician-artists collaborating on a math-art installation with layer upon layer upon layer of mathematical and artistic thought, narrative, punning, and play. For me, during this summer of covid, it's been an opportunity both to get to know a bunch of really interesting, dedicated, talented, creative people and to explore new objets d'clay. I've prototyped herons and hagfish-inspired Fibonacci sea serpents for a Knotical scene, tortoise shells decorated with pentagonal and heptagonal tilings of the hyperbolic plane, and clay heads for an industrious feline baker of tessellating cookies, with more creatures coming soon. These seeds are stimulating adventures for me outside of the project too. The first heron prototypes, for example, flew off to the Sculpture in the Garden show at the North Carolina Botanical Garden, and successive generations rounded out a recent Claymakers Zoom class on wheel-thrown and altered birds and beasts.

The math-art project now has a website--mathemalchemy.org--and released a teaser trailer this past week. Check it out! 

Penguins meet Fibonacci serpent

Tessellating shell for Tess the tortoise

First batch of heads that looked more feline than
kangarooine or chipmunkine. Still working on it...

Juvenile wheel-thrown heron; the adults are taller and have metal legs.

Quarantine Lunchbox (2020)

Backyard State U had a covid trainwreck last month. Who could have imagined that behavioral scientists and epidemiologists actually know what they're talking about?

E heeded the poem, but some of his bright-eyed bushy-tailed first-year suite-mates did not. (And who can blame them, really? We know that kids do all sorts of risky stuff when they're away from home for the first time.) A suite-mate tested positive for covid. 

Thus E spent two full weeks in quarantine, as required by BSU's overwhelmed Student Housing, waiting to find out if he had covid too. He tested negative ten days in, so was either barely symptomatic early on or never had it. As soon as E was sprung from confinement, S helped him move out of his quarantine dorm and then out of his regular dorm. E might not be on campus anymore, but hoo boy, he's certainly getting an education this year.

While hunkered down, E received two meals each day, delivered at the same time by Dining Services, providing him with a grand total of 14 memorable hummus-veggie wrap lunches and 14 functional dinners. E, a budding environmental engineer, set to work redeploying the cardstock lunch containers. He prototyped a lot of paper airplanes, but he also made some art. If you want to know what 14 days in quarantine looks like, check out the exhibit below.

E

Quarantine Lunchbox

2020

Cardstock boxes, glue, metal tin

9.5" x 3.75" x 2.25"

Private collection

Gallery show at Claymakers

I'm excited to join Roberta Wood for a two-person exhibit at Claymakers (705 Foster Street, Durham, NC). The show opened today and runs through March 14. It includes many of my "mathier" pots--Klein bottles, intersecting sliced tori, nested spheroids, and zigzag transformation vessels among them--along with Roberta's beautifully geometric sculptures. I think our work goes well together. Please stop by if you're in the area!

Fame and glory where it counts

How do you make a geek glow? You not only make her Klein bottle the pin-up model for October but also put it on the cover of the American Mathematical Society 2020 Calendar of Mathematical Imagery and then send her a bunch of copies. My math professor dad would have enjoyed this!

JMM 2019

I'm delighted that this Klein bottle representation* found an appreciative audience at the 2019 Joint Mathematical Meetings (the American Mathematical Society and the Mathematical Association of America) in Baltimore, where it won an award for "best textile, sculpture, or other medium." Hooray for clay! The gallery of exhibited art is here.

(*Since a Klein bottle has only one side, it can't actually have holes; those come from the artistic license of living in 3D.)

Necco wafers

When I was in high school, my calculus teacher liked to talk about Necco Wafers. He described them as having a thickness of dh and showed that if you integrate the area of a Necco Wafer over a roll of height h, you get the volume of the roll. We were supposed to imagine the Necco Wafers as infinitessimally thin, and the simile helped us understand that little things add up.

I use the Necco Wafer simile when I teach pottery classes. When you make a cylinder on the wheel, you do several "pulls," thinning the wall with each pull and increasing the height of the cylinder. If you pull the clay up too quickly--before the wheel has an opportunity to spin around a sufficient number of times for each wall slice dh--the cylinder wall doesn't thin evenly, and you create thin sections that coil up the pot and make the wall and rim uneven. If students don't know what I mean by dh, I tell them about my high school calculus teacher and ask them to imagine their cylinder as a stack of Necco Wafers. Usually everyone gets the idea, although sometimes I have to explain to the young'uns what Necco Wafers are.

Last week, one of my students brought me a roll of Necco Wafers that she found in the retro-candy section of a store. Turns out the only thing Necco Wafers are really good for are similes. Tip: avoid eating them, but if you feel you must, DO NOT eat the green or purple ones.

Third generation is the charm: reflections on cherry pie and slide rules

When I was a kid, my siblings and I would ask my mother to tell us bedtime stories about her childhood. "Tell us a story about the Olden Days," we would say, night after night, and she would oblige with sepia-tinted tales about life on the Indiana farm or in Hyde Park, Chicago.

The stories that stuck with me most were wonderful adventures foiled by tragedy, like the time my mom and a friend decided to make a cherry pie, so they picked the cherries from the tree on the farm all by themselves, and they made a pie crust all by themselves, and they filled up that pie crust and baked that cherry pie all by themselves, and then no one in my mom's family would even taste it.

(Years later, when I brought up the story and wondered how her family could have been so callous, she said "oh, didn't I mention that the cherries were full of pits and worms, and I'm sure we didn't add any sugar, and we made the crust from flour and water and it was like grey paste? I didn't even eat any of it." I had somehow deleted all of that in my childhood indignation.)

Then there was the story about when my mother, as a little girl, got her very first helium balloon. This was after World War II, because during the war, all the latex was going to the war effort instead of to wonderful frivolous things like balloons. As she was crossing the street, with the balloon tied safely around her tiny wrist so it wouldn't float away (or so I imagined), that wonderful very first balloon popped. Popped! O cruel fate!

(Years later, after I had a child of my own, my mother observed that unlike my own son, my siblings and I had always cried and cried when balloons popped, and she had never understood what was up with that. I asked how we could not have cried, in light of her poignant story of post-war balloon loss. "It was just a balloon," she said.)

One memorable story was less tragic and less sepia-toned, at least to me if not to my mother. As I remember it, the story went something like this: "once upon a time, long ago in the olden days, when my sweet-sixteen birthday rolled around, I hoped for the usual things a 16-year-old girl wants for a birthday present. A pearl necklace, or a new bicycle, or maybe a pony. Something pretty or fun. But I didn't get a pearl necklace, or a new bicycle, or a pony. Instead, my parents gave me a slide rule."

Perhaps I am mis-remembering, and I didn't hear that story until my own sixteenth birthday was approaching. In any case, because my mother has a sense of humor, I too received a slide rule when I turned sixteen: a six-inch long ca. 1936 Keuffel & Esser Ever-There No. 4097D in a slim leather case. Fifteen months later, when my sister had her sixteenth birthday, she too was gifted with a slide rule ("only because it was tradition," she confirmed last night via email).

My sweet-sixteen Keuffel & Esser Ever-There No. 4097D

I am proud that the coming-of-age slide-rule tradition began with women, passed from my mother, as the first recipient, to her daughters. Family lore does not record whether my mother's brother received a slide rule for his sixteenth birthday, and as far as I know, my older brother's slide rules were not affiliated with his sixteenth. (Yes, "slide rules," plural. He reports, "I think I was given more than one because I think I lost more than one.")

My mother broadened the tradition to include menfolk and rites of passage other than birthdays when I married S. She felt it was important to properly welcome my husband into the family, so she gave him a big, fancy, vintage engineer's slide rule with a sturdy leather holster to call his own: a 14-inch long duplex Engineer's No. 1510 in fashionable plastic-coated bamboo, made in Japan, presumably by Relay/Ricoh, for the US market, no date.

S's hefty Engineer's No. 1510 is more than twice as long and twice as thick as my diminutive Keuffel & Esser Ever-There. Surely there's a Master's thesis waiting to be written about slide rules and gender.

My father was a math professor. He owned a slide rule and knew how to wield it. When I was young, I enjoyed playing with its smoothly gliding bars, but by the time I was old enough that a slide rule could have helped with school math, the world had begun its shift to TI-30s and wasn't looking back.

Neither my sister, my husband, nor I ever learned how to use our slide rules. My sister's probably migrated to my parents' attic decades ago. For the past 20 years, S and I have kept ours in their leather cases, tucked away in a desk drawer (his) and in a box in a closet (mine), unused.

Until last week. My son, just thirteen months shy of his sixteenth birthday, is learning about logarithms at school.

"Logarithms?" I said brightly, when he told me. "Let me show you something."

I opened the top right-hand drawer of our roll-top desk and pulled out his father's marital slide rule. We removed it from its holster, and E and I went online to learn how to use it.

"This is SO COOL," E said, as he figured one square root after another to within two significant digits; "when the power goes out after the Zombie Apocalypse, I can calculate numbers without a calculator! Let's do another!"

He asked why we had such a fascinating object. I told him about his grandmother's sixteenth birthday, how she had wanted a pearl necklace or a new bicycle or a pony, and how she had gotten a slide rule instead, and then I fetched my own sweet-sixteen slide rule.

"That is sooooo cute," he said, picking it up.

"Yeah. It's not as fancy as papa's though," I said.

"But it's really cute. It's so small!," he squeaked in admiration.

I put an arm around my delighted child, pointed to the two slide rules on the table, and quoted from the Woozy Works: "Someday, son," I said, "all this will be yours."

And thus it came to pass that a questionable gift for a bright-eyed girl on the cusp of womanhood almost sixty years ago is finally bearing fruit, two generations later. This metaphorical homemade cherry pie will not just be tasted: to paraphrase Sir Francis Bacon, it will be chewed and digested. And thirteen months from now, my son will receive a slide rule to call his own.

"One day, son, ALL THIS WILL BE YOURS." My grandfather Mitch shows my uncle Bob the splendor of the junk in the barn; from Woozy Works: The Nearly Complete and Mostly Unexpurgated Writings of the Blue Spanch (1996).

Bridges 2015 and Baltimore photo dump

I've been back from the Bridges conference for about a week. I enjoyed both the conference and Baltimore. The conference was held in the University of Baltimore's law building, in the Mt. Vernon neighborhood. While the location could hardly have been better, and while the law building was a delightfully math-artsy space for a math-art conference, the building lacked the large open halls of Towson (Bridges 2012). This might account for why there were no large-scale construction projects. I missed the teamwork and audacity of 2012's giant orange space-chips effort. Oh well.

I spent five very full days in Baltimore. I gave an ocarina-making workshop on Wednesday, the first day of the conference. The purpose of the workshop was twofold: (1) to demonstrate that one can produce a full-octave diatonic scale "thinking like a pianist" (seven-hole ocarina, opening successive holes to produce successive pitches in the scale) or "thinking like a mathematician" (four-hole ocarina, taking advantage of hole permutations); and (2) to proselytize clay. Interestingly, most of the workshop participants ended up thinking more like pianists than mathematicians; score one for the musicians. The workshop also gave me an excuse (not that I needed one) to visit Baltimore Clayworks (a topic for a later blog post) to have the ocarinas bisque-fired. I picked them up on Saturday and distributed them to their makers.

A bunch of mathematician-made functional ocarinas

A happy music-making mathematician

The conference had an art show split between three floors of the law building. I didn't take many photos because the photo quality here is better than anything I could manage. Photo dump is below. Talk highlights included Ingrid Daubechies on image processing for art restoration and art re-creation, and Craig Kaplan on half-toning (which culminated in a how-to for folding a modular origami half-toned Mona Lisa, and the suggestion that googly eyes might make a great half-toning medium).

While I was in Baltimore, still embracing the jetlag from our Germany trip, I got up around 5:45-6:00 a.m. every morning to go running. After the third day, it occurred to me that a city the size of Baltimore was likely to have 5Ks on a pretty regular basis, and lo, I found a 5K for Saturday morning. I finished the Casey Cares race in a respectable 20th place in the field of 50-59-year-olds, but only because the race volunteers weren't able to correctly subtract my birth year from 2015. I have no idea where I ranked among the official 40-49-year-olds, but they were probably all 30-39 years old anyway.

On Sunday morning, I took my camera with me on my run. Afterward, I headed to the Peabody Library, where conference attendees were introduced to some of the library's math-art-science gems.

PHOTO DUMP

I. BRIDGES

This lamp reminded me of my holes/shadows fetish

This torn-paper sphere by Liz Shreve was one of my favorite pieces in the show.

Wood is a lovely math-art medium

Fractal chess on a lenticular board

This room was so crammed full of workshop attendees, I took photos instead of trying to find a seat

Unit origami sonobes

Sonobes can be assembled into a variety of polyhedra

The law building exterior

Law building spiral staircase

Law building triangular staircase

II. OUT & ABOUT IN BALTIMORE (in no particular order)

The water in my VRBO apartment was generally clear, but one morning it turned rust-brown. Here it is on its way back to being more or less clear.

Park your bikes here

Park your dogs here

In front of Penn Station

Peabody Institute

Baltimore School of Beauty Culture

View from Federal Hill

Baltimore: where "science" is a verb

Girl on urchin

Washington Monument

Card catalog in Peabody Library

Peabody Library

Radiator grill in Peabody Library

Muybridge's woman walking downstairs influenced Duchamp

First edition Galileo Galilei, Istoria e dimostrazioni intorno alle macchie solari (1613)...

...proving sunspots were part of the Sun. Notice my clean fingertips. No gloves required at the library, just clean hands.