193 lines
4.1 KiB
Typst
193 lines
4.1 KiB
Typst
#import "@local/handout:0.1.0": *
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#import "@preview/cetz:0.3.1"
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#show: handout.with(
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title: [Warm-Up: Passing Balls],
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by: "Mark",
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)
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#problem()
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Twelve people are standing in a circle. Each is assigned a number between 1 and 12. \
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Participants numbered 1, 2, 3, and 4 hold red, green, yellow, and black balls, respectively. \
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Everyone else is empty-handed.
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#v(2mm)
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Each participant can pass their ball to any student that is exactly 5 positions away. \
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Balls cannot be passed to someone who has one in hand.
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#v(2mm)
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After a number of passes, the first four participants again hold all the balls. \
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Participant 1 has a black ball. Which balls are held by participants 2, 3, and 4?
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#solution([
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The graph of possible moves is isomorphic to a circle (since 5 and 12 are coprime), \
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so the order of the balls cannot be changed as they are passed around.
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#v(2mm)
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Therefore, the balls will stay in their initial (cyclic) order:
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#v(2mm)
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#{
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let s = 0.7 // scale
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let t = 12pt * s // text size
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let radius = 0.35
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let pts = (
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(0 * s, 3 * s),
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(1 * s, 2 * s),
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(2 * s, 1 * s),
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(3 * s, 0 * s),
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(2 * s, -1 * s),
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(1 * s, -2 * s),
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(0 * s, -3 * s),
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(-1 * s, -2 * s),
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(-2 * s, -1 * s),
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(-3 * s, 0 * s),
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(-2 * s, 1 * s),
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(-1 * s, 2 * s),
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)
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let pts_shuf = (
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(0 * s, 3 * s), // 1
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(1 * s, -2 * s), // 6
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(-2 * s, 1 * s), // 11
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(3 * s, 0 * s), // 4
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(-2 * s, -1 * s), // 9
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(1 * s, 2 * s), // 2
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(0 * s, -3 * s), // 7
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(-1 * s, 2 * s), // 12
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(2 * s, -1 * s), // 5
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(-3 * s, 0 * s), // 10
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(2 * s, 1 * s), // 3
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(-1 * s, -2 * s), // 8
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)
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table(
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stroke: none,
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align: center,
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columns: (1fr, 1fr, 1fr),
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cetz.canvas({
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import cetz.draw: *
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set-style(stroke: (thickness: 0.4mm, paint: black))
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line(..pts_shuf, close: true)
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let i = 1
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for p in pts {
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circle(
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p,
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radius: radius * s,
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fill: if i == 1 {
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ored
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} else if i == 2 {
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ogreen
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} else if i == 3 {
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oorange
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} else if i == 4 {
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oblue
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} else { white },
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)
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content(
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p,
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text(
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fill: if i <= 4 {
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white
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} else {
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black
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},
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size: t,
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[*#i*],
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),
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)
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i = i + 1
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}
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}),
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cetz.canvas({
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import cetz.draw: *
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set-style(stroke: (thickness: 0.4mm, paint: black))
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line(..pts, close: true)
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let i = 1
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for p in pts {
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let l = calc.rem(((i - 1) * 5), 12) + 1
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circle(
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p,
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radius: radius * s,
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fill: if l == 1 {
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ored
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} else if l == 2 {
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ogreen
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} else if l == 3 {
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oorange
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} else if l == 4 {
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oblue
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} else { white },
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)
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content(
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p,
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text(
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fill: if l <= 4 {
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white
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} else {
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black
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},
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size: t,
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[*#l*],
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),
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)
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i = i + 1
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}
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}),
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cetz.canvas({
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import cetz.draw: *
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set-style(stroke: (thickness: 0.4mm, paint: black))
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line(..pts, close: true)
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let i = 1
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for p in pts {
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let l = calc.rem(((i - 1) * 5), 12) + 1
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circle(
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p,
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radius: radius * s,
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fill: if l == 1 {
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oblue
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} else if l == 2 {
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oorange
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} else if l == 3 {
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ored
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} else if l == 4 {
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ogreen
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} else { white },
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)
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content(
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p,
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text(
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fill: if l <= 4 {
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white
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} else {
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black
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},
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size: t,
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[*#l*],
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),
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)
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i = i + 1
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}
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}),
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)
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}
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])
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