iiinova — where your digital self keeps evolving

Bill Shillito @solidangles.bsky.social · 31/08/2026

Which of these is named "Kiki", and which of these is named "Bouba"?

A summation sign, the Greek capital letter sigma.

An integration sign, an elongated letter S.

Bill Shillito @solidangles.bsky.social · 05/06/2026

Besides, arguments (1) and (2) work just as well to show that ...9999 (with "infinitely many nines" to the left) equals -1. How weird is that?

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Bill Shillito @solidangles.bsky.social · 21/01/2026

Teaching logic tomorrow ... with Clue!

Suppose you've made the following deductions:

* The murderer must be Miss Scarlett or Colonel Mustard.
S ∨ M

* If Miss Scarlett did it, it was in the kitchen with the lead pipe.
S → (K ∧ L)

If your friend shows you the kitchen card, what can you conclude?
¬K

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Bill Shillito @solidangles.bsky.social · 02/01/2026

As my contribution to sharing subtraction strategies, here's how I do subtraction nowadays, using negative numbers! Made this up on the fly way back when I first started tutoring because I couldn't remember how to do multiple borrowing anymore and I never liked the lie of "you can't take 3 from 2".

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Bill Shillito @solidangles.bsky.social · 01/01/2026

Deviled eggs for New Year's Eve!

Left: Everything bagel deviled eggs, with smoked salmon and chives.
Right: Dua Belibis deviled eggs, with green onion, cilantro, and chopped peanuts.

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Bill Shillito @solidangles.bsky.social · 22/12/2025

Most underrated video game villain.

A blue tornado with wide white eyes named Mr. Glitch, who was the villain of the Mathman segment from the children's educational show Square One TV.

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Bill Shillito @solidangles.bsky.social · 07/12/2025

Students who take AP Statistics are 3x more likely to be able to tell the difference between correlation and causation.

Screenshot of a College Board social media post promoting AP Precalculus. Large text reads “Students who take precalculus are 3x more likely to earn a college degree.” Below the text is an icon of a diploma with a ribbon. The bottom text says, “See why more schools are making AP Precalculus part of their math pathway.”

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Bill Shillito @solidangles.bsky.social · 18/11/2025

Which of these is an "A"? Why? #Grading Image source: www.mpusd.net/apps/pages/i...

A graphic titled “Student Goal: Ride the Bike Independently” shows four mastery levels with simple stick-figure illustrations of children on bicycles. From bottom to top, the levels are as follows:

* Level 1, “Emerging Mastery,” shows a child riding a bike with training wheels. The text reads, "The student is riding a bike, but only with the use of training wheels."

* Level 2, “Partial Mastery,” shows an adult steadying a child on a bike. The text reads "The student is pedaling well and staying upright, as long as someone is assisting."

* Level 3, “Sufficient Mastery,” shows a child riding a bike alone. The text reads, "The student is successful at riding the bike."

* Level 4, “Extensive Mastery,” shows a child doing a wheelie on a bike. The text reads, "The student can not only ride the bike independently, but also performs stunts!"

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Bill Shillito @solidangles.bsky.social · 21/10/2025

This was the exact moment I learned it's called "maths" across the pond.

A cropped screenshot of the ending credits to Puggsy on the Sega Genesis. The text reads:

AND REMEMBER THAT

11125 SQRD MINUS 181
13307 SQRD MINUS 712
21385X21386 SUB 1875

IS A SILLY MATHS
QUESTION

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Bill Shillito @solidangles.bsky.social · 11/10/2025

Ah yes, the four basic operations: addition, subtraction, checkmark, and multiplication.

A close-up of a parking meter with a digital display reading “SELECT SPACE LEFT / RIGHT.” Below the screen are colorful buttons: a blue plus, a blue minus, a green check mark, and a red X.

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Bill Shillito @solidangles.bsky.social · 09/10/2025

UGH, more mistakes that of course I only catch after posting ... here's the corrected graphic. One of these days I'll manage to post something completely right the first time. 😅

A diagram on a dark green background showing how the quadratic mean relates to variance and standard deviation.

Top left: {−6, 1, 5}, with a rightward arrow labeled “QM” leading to ≈ 4.546 (standard deviation*).

A downward arrow labeled “( )²” points to {36, 1, 25}.

From {36, 1, 25}, a rightward arrow labeled “AM” points to 20⅔ (variance*).

An upward arrow labeled “√( )” connects 20⅔ back to 4.546.
At the bottom, the label reads “Quadratic Mean.”

A footnote adds: “*Technically this is for a population variance and standard deviation. If this were for a sample, we’d actually divide by n - 1 instead of n to account for underestimating the variability, so the sample standard deviation would be √31 ≈ 5.568.”

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Bill Shillito @solidangles.bsky.social · 09/10/2025

It turns out we can instead use the quadratic mean, aka the root mean square (RMS). Squaring makes all the deviations positive, and square rooting at the end gets us back to our original units. (BTW, here's a video on why squaring really is the best choice: www.youtube.com/watch?v=q7se...) [15]

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Bill Shillito @solidangles.bsky.social · 09/10/2025

Let's look at one more application. In statistics, we often want to describe not just the center but the spread of a data set. The first thing most people try is averaging the deviations — but the overs will always cancel out the unders and give zero. So what can we do? [14]

A slide on a dark red background showing a data set and its mean and deviations.

Data: 10, 17, 21.

Arithmetic mean: AM{10, 17, 21} = (10 + 17 + 21) / 3 = 16.

Deviations listed:
10 − 16 = −6
17 − 16 = 1
21 − 16 = 5

On the right, under “Average deviation,” it shows AM{−6, 1, 5} = 0…?!, showing that the deviations cancel out.

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Bill Shillito @solidangles.bsky.social · 09/10/2025

It turns out these are both examples of a more general idea called an "f-mean." Here's the basic idea: 1. Apply some function f to your data. 2. Find the arithmetic mean. 3. Undo f with the inverse function f⁻¹. The geometric mean uses f(x) = log x, and the harmonic mean uses f(x) = 1/x. [13]

$A diagram on a brown background illustrating the general process for finding an f-mean. Top left: {a, b}, with a rightward arrow labeled “M_f” pointing to M_f{a, b} on the right. A downward arrow labeled “f( )” leads from {a, b} to {f(a), f(b)}. From {f(a), f(b)}, a rightward arrow labeled “AM” points to AM{f(a), f(b)}. An upward arrow labeled “f⁻¹( )” connects AM{f(a), f(b)} back to M_f{a, b}. At the bottom, the label reads “f-Mean.”$

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Bill Shillito @solidangles.bsky.social · 09/10/2025

Again, why did this work? Well, in physics, you learn that frequency and wavelength are inversely proportional. So can we do as follows: 1. Take the reciprocals of the data. 2. Find the arithmetic mean. 3. Undo the reciprocal with another reciprocal. That's the harmonic mean. [10]

A diagram on a dark purple background showing how the harmonic mean relates to reciprocals and the arithmetic mean.

Top left: {1/5, 1/7}, connected by a rightward arrow labeled “HM” to 1/6 on the right.

A downward arrow labeled “1/( )” leads from {1/5, 1/7} to {5, 7}.

From {5, 7}, a rightward arrow labeled “AM” points to 6.

An upward arrow labeled “1/( )” connects 6 back to 1/6.

At the bottom, the label reads “Harmonic Mean.”

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Bill Shillito @solidangles.bsky.social · 09/10/2025

It turns out we need to use the harmonic mean: flip the string fractions upside-down, add, and divide 2 (the number of values) by the result. So I have to place my finger to make the vibrating string 2/3 as long. (BTW, if you lightly touch here, you get what's called a "string harmonic!") [9]

A formula for the harmonic mean on a dark purple background. It shows “HM{1, ½} = 2 / (1/1 + 1/(½)) = 2/3,” with the label “Harmonic Mean” in bold.

Below is a violin with four labeled points along one string:

“A440 (open)” near the scroll for the full-length string,

A dotted line with an X about 1/4 of the way long the string, showing the incorrect location for E660.

“E660!” about 1/3 of the way along the string, that is, making the string 2/3 of its usual length.

“A880” halfway between the scroll and the bridge, indicating the octave point where the string length is halved.

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Bill Shillito @solidangles.bsky.social · 09/10/2025

When I used to play viola in high school (alto clef represent!) I tuned to A440. An octave up the A string (making it half as long) is A880, and a perfect fifth is E660. (Almost — more on that later.) Where should I put my finger to play that E? It's NOT halfway to the octave mark! [8, lol]

A viola on a dark purple background with three labeled points along the A string:

Near the scroll, “A440 (open)” marks the full string length.

Along the neck, “E660?” indicates the conjectured location for the note E.

“A880” marks the halfway point between the scroll and the bridge, one octave above the open A.

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Bill Shillito @solidangles.bsky.social · 09/10/2025

The trick is to use logarithms to extract the exponents! We do a three-step procedure: 1. Take logarithms of the value. 2. Find the arithmetic mean. 3. Undo the logarithm with an exponential function. What we get is the geometric mean. (Fun exercise: check this algebraically!) [5]

A diagram on a dark blue background showing how the geometric mean relates to logarithms and the arithmetic mean.

Top left: {10⁵, 10⁷}, connected by a rightward arrow labeled “GM” to 10⁶ on the right.

A downward arrow labeled “log( )” leads from {10⁵, 10⁷} to {5, 7}.

From {5, 7}, a rightward arrow labeled “AM” points to 6.

An upward arrow labeled “10( )” connects 6 back to 10⁶.

At the bottom, the label reads “Geometric Mean.”

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Bill Shillito @solidangles.bsky.social · 09/10/2025

We can get a more reasonable estimate with what's called the geometric mean: multiply the values and take a root (here a square root, since there were 2 numbers). This gives a much better estimate — as you can check on Wikipedia. But why did this work? How would anyone think to try this? [3]

A formula for the geometric mean on a dark blue background. It shows “GM{250, 4000} = √(250 × 4000) = 1000,” with the label “Geometric Mean” in bold below.

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Bill Shillito @solidangles.bsky.social · 09/10/2025

Let's start with ol' reliable: the arithmetic mean. The average. What it does is find the number that's in the "center" of a given data set by "redistributing" the sum evenly across all the values. But as anyone who's studied triangles knows, there may be multiple kinds of "center!" [1]

A formula for the arithmetic mean on a dark red background. It shows “AM{5, 7} = (5 + 7)/2 = 6,” with the label “Arithmetic Mean” in bold below.

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Bill Shillito @solidangles.bsky.social · 09/10/2025

I've been thinking a lot about means recently! Probably some combination of (1) teaching our data analysis unit in our STEM 101 course at Oglethorpe and (2) all of @howiehua.bsky.social's great posts surrounding Mean Girls Day. I'd like to show how some of these various means are related. 🧵 [0]

A four-panel graphic showing formulas for four types of means. Each panel includes the formula and the name of the mean in bold text below.

Top left (red background): Arithmetic Mean, labeled “AM = (x₁ + ⋯ + xₙ) / n.”

Top right (dark blue): Geometric Mean, labeled “GM = ⁿ√(x₁ ⋅ … ⋅ xₙ).”

Bottom left (purple): Harmonic Mean, labeled “HM = n / (1/x₁ + ⋯ + 1/xₙ).”

Bottom right (dark green): Quadratic Mean, labeled “QM = √((x₁² + ⋯ + xₙ²) / n).”

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Bill Shillito @solidangles.bsky.social · 25/09/2025

At least Italian hasn't caved though! They still use the long scale. The extra thousand becomes "-ardo" (borrowed from French) at the end of the word but it still is easy enough to understand. And the term "milione" makes sense: the "-one" suffix makes it "big thousand". Magnifico! (5/5)

$A chart showing the Italian long scale names for large numbers, written with exponents of a million. Each line shows “1 000 000” raised to fractional or whole exponents, with color-coded exponents and word parts: (1 000 000)^0.5 = mille (pink) (1 000 000)^1.0 = milione (prefix “m” in red) (1 000 000)^1.5 = miliardo (prefix “m” in red, “-ardo” in pink) (1 000 000)^2.0 = bilione (prefix “b” in blue) (1 000 000)^2.5 = biliardo (prefix “b” in blue, “-ardo” in pink) (1 000 000)^3.0 = trilione (prefix “tr” in purple).$

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Bill Shillito @solidangles.bsky.social · 25/09/2025

But wait! If we use the "long scale" in British English, we get base million, with a sub-base of a thousand, with a sub-sub-base of ten! And everything makes sense! ...or at least that used to be what British English uses. Now they just use the short scale like Americans. Bummer. (4/5)

$A chart showing the long scale system where powers are based on millions. Each line shows “1 000 000” raised to fractional or whole exponents, with color-coded exponents and names: (1 000 000)^0.5 = thousand (pink) (1 000 000)^1.0 = million (red “m”) (1 000 000)^1.5 = thousand million (pink “thousand,” red “m”) (1 000 000)^2.0 = billion (blue “b”) (1 000 000)^2.5 = thousand billion (pink “thousand,” blue “b”) (1 000 000)^3.0 = trillion (purple “tr”). This illustrates the long scale naming, where each new term is a million times the previous one, unlike the short scale.$

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Bill Shillito @solidangles.bsky.social · 25/09/2025

The new groupings we get are based on powers of a thousand: millions, billions, trillions, and so on. But isn't it weird how the prefixes don't seem to match up with the powers? It's off by one. It would be nice if "trillion" were 1000³, but nope, it's actually 1000⁴. Confusing isn't it? (3/5)

A chart showing powers of a thousand with color-coded exponents and number names.

1000¹ (exponent in red) = thousand

1000² (exponent in blue) = million (prefix “m” in red)

1000³ (exponent in purple) = billion (prefix “b” in blue)

1000⁴ (exponent in yellow) = trillion (prefix “tr” in purple)

1000⁵ (exponent in orange) = quadrillion (prefix “quad” in yellow)

1000⁶ (exponent in green) = quintillion (prefix “quint” in orange).

This highlights how the English naming prefixes don’t match the powers of a thousand, being offset by one.

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Bill Shillito @solidangles.bsky.social · 25/09/2025

For comparison, Japanese is base "man" = ten-thousand (or perhaps we could say base myriad), also with a sub-base of ten. I remember how difficult it was to learn that "million" is "hyaku-man" (百万), which translates to "one hundred myriad". I had to completely reframe how I group numbers. (2/5)

Colorful text showing the number 5,7396,1284 grouped in units of ten-thousand.

First line: “5,7396,1284”.

Second line: “= 5 × 10000² + 7396 × 10000¹ + 1284 × 10000⁰.”

Third line shows the same number in Japanese characters: “五億, 七千三百九十六万, 一千二百八十四,” (go OKU, nana-sen san-byaku kyū-jū roku MAN, is-sen ni-hyaku hachi-jū yon) with each chunk color-matched to the digits above.

The layout highlights that Japanese uses base ten-thousand (“man” units), so numbers are grouped differently than in English.

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Bill Shillito @solidangles.bsky.social · 25/09/2025

Hot take: The English language is not base ten. It's base thousand. Sure, we write numbers using the Hindu-Arabic digits 0 through 9, but the way our language is structured groups numbers in powers of a thousand. If anything, we're sub-base ten. (1/5)

Colorful text showing the number 573,961,284 broken into chunks of thousands.

First line: “573,961,284”.

Second line shows it as “573 × 1000² + 961 × 1000¹ + 284 × 1000⁰.”

Third line spells it out: “Five hundred seventy-three million, nine hundred sixty-one thousand, two hundred eighty-four,” with each chunk matching the colors from the first two lines.

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Bill Shillito @solidangles.bsky.social · 03/09/2025

Today's lesson in COR 314! Students learned how to count in Iñupiaq and write numbers using the Kaktovik numerals (as well as convert between base ten and base twenty). 🧮 Lots of discussion about how the way we're used to thinking isn't the only way to do things! 🙂

A list of number words in Iñupiaq, counting in 1s up through 20 and then counting in 10s up through 120. At the bottom is text that asks "What do you notice? What do you wonder?"

A display of the Kaktovik Inupiaq numerals, along with a conversion of the number 17 : 5 : 13 : 4 (in base twenty) to 138,264 (in base ten).

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Bill Shillito @solidangles.bsky.social · 14/03/2025

Happy #PiDay! To celebrate, here's my favorite representation of π: as a position in the combinatorial game of Stacks.

A infinite stack of bLue and Red chips. Each bLue chip is labeled with an L, and each Red chip is labeled with an R. The stack from bottom to top is labeled LLLLRRRRLRRR... .

Below it, the value of π is given as follows:
π = { 0, 1, 2, 3, 3 1/8, 3 9/64, ... | ... , 3 5/32, 3 3/16, 3 1/4, 3 1/2, 4 }

BRIEF GAME EXPLANATION:
Stacks is a simplified version of Hackenbush. The original concept comes from Carl Lee at University of Kentucky, who calls it "Checker Stacks." The game starts with some number of stacks of bLue and Red chips on the table. On your turn, you can pick any chip of your color and remove it from the board, along with any chips of either color on top of it. If your opponent can't move, you win.

The notation G = { L | R } specifies a game value in terms of the sets L and R of possible values when bLue and Red respectively make a move. If G is a number, then the value of the game will be strictly between the elements of L and R, and hence the sequences listed converge to π from above and below.

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Bill Shillito @solidangles.bsky.social · 01/03/2025

Something seems off about the Euler characteristic of this hush puppy.

A toroidal hush puppy. It looks like an onion ring, but it was, in fact, a hush puppy.

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Bill Shillito @solidangles.bsky.social · 16/01/2025

Apparently this is now what I do for fun when I need a break from work. 😂

A computation showing that if G = {*, ↑|↓*, 0}, then G + G = *, so G is usually called “semi-star”.

The computation involves looking at all moves from G + G and showing that they’re “reversible”, that is, any move by a player has a response that puts their opponent in at least as good a position. In this case, a move to * can always be reversed to 0.

In the process, we get a picture of G’s confusion interval: G is confused with 0 and ↑, but also confused with * and ↓*. This lets us picture where G lives on the “number line” (using scare quotes because none of the games on the board are numbers in the game theoretic sense except for 0).

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Bill Shillito @solidangles.bsky.social · 01/01/2025

This year's deviled eggs for #NewYearsEve! Left: Garlic deviled eggs (using toum!) Right: Ramen deviled eggs (made like onsen eggs but hard boiled)

Left: Garlic deviled eggs (toum [Lebanese garlic sauce], mayo, dijon, green onion, topped with chives and cayenne)

Right: Ramen deviled eggs (miso, mayo, honey, topped with furikake, and the whites have been marinated like onsen eggs)

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Bill Shillito @solidangles.bsky.social · 16/12/2024

About to take my final for Abstract Algebra (group and field theory). If this goes well it will be the *last* final I ever have to take. LET’S GO! 😎

A formula sheet for abstract algebra next to a breakfast sandwich and coffee.

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Bill Shillito @solidangles.bsky.social · 27/11/2024

Presented without comment: The Thermograph of "oof".

Figure 19: The thermograph of "oof". A horizontal line is marked with 0 in the middle, "on" to the left, and "off" to the right. A second line diagonal crosses 0 and continues upward and to the right. The region under the diagonal line and above the horizontal line is shaded, with {0 | off}, the canonical form of "oof", written in that region.

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Bill Shillito @solidangles.bsky.social · 24/11/2024

Optimist: The cup is half full. Pessimist: The cup is half empty. Combinatorial game theorist: The cup is a win for Left, because she can take it to the zero cup. If Right moves, he will take it to the one cup.

A play on the canonical form ½ = { 0 | 1 } in combinatorial game theory. The ½, 0, and 1 have been replaced with glasses that are half cup, empty, and full, respectively.

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Bill Shillito @solidangles.bsky.social · 16/11/2024

tfw you're really bad at a game but you refuse to give up

A figure from Winning Ways for your Mathematical Plays, Volume 1, Chapter 3.

The figure reads: "Figure 1. A Well Advanced Game of Poker-Nim."

Two players are seated at a table playing Poker-Nim. The current piles are 3, 4, and 6. Left is removing one chip from the top of the 3 pile to make the position equal to 2, 4, and 6, which will be a losing position for Right.

Right is looking quite unhappy — he has plenty of chips in reserve that he could add to the pile, but Left will simply take those chips away again immediately, leaving Right again in a poor position. That is, adding chips when in a losing position is a "reversible" move.

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Bill Shillito @solidangles.bsky.social · 16/11/2024

#TodayInMaths we showed how Bayes' Theorem is the math(s) behind both poker bluffing and medical tests! We also talked about the implications of false positives AND false negatives, the importance of non-stigmatizing language, and further application to law. Next week — Texas hold'em!

A slide from today's lesson on "Betting, Bluffing, and Bayes."

The text reads:

Assume that 0.1% of people in a population have HIV. Also assume we know the following about HIV tests:
* If a person is HIV positive, there's a 99% chance they test positive (1% chance of false negative).
* If a person is HIV negative, there's a 98% chance they test negative (2% chance of false positive).

Suppose a person tests positive for HIV. What's the probability they actually are HIV positive?

A slide from today's lesson on "Betting, Bluffing, and Bayes."

The text shows the formula P(A|B) = P(A) · P(B|A) / P(B). Each part is labeled:

P(A|B) — Posterior: The probability that your hypothesis is true, given the observed evidence.
P(A) — Prior: The overall probability of your hypothesis before observing any evidence.
P(B|A) — Likelihood: The probability that your evidence is true, given that your hypothesis is true.
P(B) — Marginal: The overall probability of the new evidence under all possible hypotheses.

A slide from today's lesson on "Betting, Bluffing, and Bayes."

The text reads:

Suppose you're playing a game of poker, and it's down to just you and your friend. Assume there is:

* A 7.7% chance they have a good hand (two pair or better)
* A 42.2% chance they have an okay hand (one pair)
* A 50.1% chance they have a bad hand (just a high card)

During the second betting round, they raise. What is the probability that they're bluffing? (That is, what's the probability they have a bad hand, given that they raised?)

A slide from today's lesson on "Betting, Bluffing, and Bayes."

The relevant events are as follows:
* R: Raise or re-raise
* C: Check or call
* F: Fold
* G: Good hand (two pair or better)
* O: Okay hand (one pair)
* B: Bad hand (high card only)

The following probabilities are given:
* P(R|G) = 0.6; P(C|G) = 0.3; P(F|G) = 0.1
* P(R|O) = 0.2; P(C|O) = 0.4; P(F|O) = 0.4
* P(R|B) = 0.1; P(C|B) = 0.4; P(F|B) = 0.5

The probabilities are used to construct a two-way table, with G, O, B on the rows and R, C, F on the columns. The columns other than R are washed out, so that only the R column is in focus. The probability is then calculated as:

P(B|R) = 5.01% / 18.07% ≈ 27.73%

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Bill Shillito @solidangles.bsky.social · 13/11/2024

Overfull \hbox detected in real life.

A proof on a whiteboard of the fact that a Sylow p-subgroup P of a group G is a normal subgroup if and only if it is unique. The last “s” in the word “subgroups” is written on the metal border of the board.

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Bill Shillito @solidangles.bsky.social · 12/11/2024

For starters, here's the one that starts it all. (No I'm not going to go through every single diagram.) What kind of math book opens up with this? The best kind of math book.

Figure 1: A Blue-Red Hackenbush Picture. The text says "Who's game for an easy pencil-and-paper (or chalk-and-blackboard) game?"

The picture shows a drawing of a tennis player with a ponytail and a skirt, made of blue and red line segments, connected to the ground.

The game of Hackenbush is played by having Blue and Red players (or Left and Right, respectively) take turn cutting segments of their color. Segments disconnected from the ground are removed from play. As with most combinatorial games under normal play, if your opponent can't make a move, you win.

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Bill Shillito @solidangles.bsky.social · 12/11/2024

I need to start interacting here more regularly, but I’m terrible at coming up with what to post… …so until I come up with something better, I’m going to start occasionally posting some of the bizarre and lovable diagrams from Winning Ways. Who’s game?

The covers of all four volumes of “Winning Ways for your Mathematical Plays” by Elwyn R. Berlekamp, John H. Conway, and Richard’s K. Guy. The covers are full of colorful, whimsical drawings of the games explored in the books.

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Bill Shillito @solidangles.bsky.social · 11/11/2024

Oh hey another trinity knot enjoyer! Represent! 😁

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Bill Shillito @solidangles.bsky.social · 11/11/2024

Here's mine, for my liberal arts math class.

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Bill Shillito @solidangles.bsky.social · 31/10/2023

Just taught an entire 105-minute online Statistics lesson like this. 😂 The mouth even moves. Best impulse buy at Spirit Halloween ever. 🐺

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Bill Shillito @solidangles.bsky.social · 19/10/2023

A really handy use case for #ChatGPT as an educator: coming up with problem scenarios. I am *terribly* slow and uncreative at coming up with these, so ChatGPT has been an absolute godsend when it comes to generating practice problems for my Elementary Statistics class.

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