Bill Shillito

@solidangles.bsky.social

1121 Followers 1059 Following

Math instructor at Oglethorpe University. Views my own. Talk to me about anything combinatorial game theory related! He/him. Pronounced SHILL-lit-toe. Websites: https://www.solidangl.es, https://1dividedby0.com

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
A video of the following subtraction problem: 416852 - 278393. The digit subtractions are performed left to right: 4 - 2 = 2, 1 - 7 = -6, 6 - 8 = -2, 8 - 3 = 5, 5 - 9 = -4, 2 - 3 = -1. The numbers are written on the third line if they're positive or the fourth line if they're negative, and then zeros are filled in for the remaining spots, so the problem now becomes 200500 - 62041. Then the remaining problem is broken into two quick subtraction problems: 200 - 62 = 138, and 500 - 41 = 459. So the final answer is 138459.
Left: Everything bagel deviled eggs, with smoked salmon and chives. Right: Dua Belibis deviled eggs, with green onion, cilantro, and chopped peanuts.
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.
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.”
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!"
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
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.
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.”
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 45⅓ (variance*). An upward arrow labeled “√( )” connects 45⅓ 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 √62 ≈ 5.568.”
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.
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.”
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.”
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.
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.
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.”
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.
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.
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).”
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).
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.
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.
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.
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.
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 picture of Le Poisson Steve (an orange fish with arms and legs). Above his head is the probability mass function for the Poisson distribution: P(X = k) = λ^k e^-λ / k!.
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.
A toroidal hush puppy. It looks like an onion ring, but it was, in fact, a hush puppy.
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).
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)
A formula sheet for abstract algebra next to a breakfast sandwich and coffee.
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.
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.
A poke bowl made with romaine lettuce, brown rice, and topped with salmon "noodles." Arranged around the edges clockwise from the top are seaweed salad banchan, spicy squid banchan, hard-boiled egg slices, grape tomatoes, green onion, spicy octopus banchan, mashed avocado, and red onion. The top is drizzled with sriracha mayo and gochujang and sprinkled with furikake. To the left is a pair of disposable chopsticks propped up on a holder made from their wrapper, and on the right is a glass of mixed berry juice with ice.
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.
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 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.
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.
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.