PRH | Educ | 6.5 • Blur-as-a-Method: Small, Hands-On Problems (Elementary → Undergraduate)

Blur-as-a-Method: Small, Hands-On Problems (Elementary → Undergraduate) A friendly note for students & teachers (simplified K–12 wording) Quick words for students Blur = take a tiny, fair average so noisy details calm down. Safety margin = how far off your blurred answer could be. We say it out loud. Plan = (1) pick a small blur, (2) do a few easy calculations, (3) compare with the goal, (4) write the safety margin. Teacher guide (1 minute) • Name the blur & the safety margin in every task (how we average, how big the possible miss is). •Keep it small: each task is designed for ≤30 simple calculations. • Ablate one knob: try again with less blur and see what breaks—this reveals which knob mattered. 1 Elementary school (ages ∼ 10–13) : same problems, simpler language Each fits on a single page. Use a ruler, pencil, and simple arithmetic. 1. Circle perimeter by blur (two polygons). Draw a circle of radius r = 5 cm. Mark 12 equally spaced points. Make two 12-gons: one inside the circle, one outside. Let Pblur =inside perimeter +outside perimeter 2. Goal: compare Pblur with 2πr (use a calculator). Safety margin: half the gap between the two perimeters. Why blur helps: one is a bit too small, one a bit too big—average is close. 2. Smoothing a noisy count (marbles). 1) Prepare a table with 20 boxes. Each time you draw a marble and put it back, write 1 if it was red and 0if it was blue. □ □ □ ··· □ | {z } 20 boxes 1 2) Compute the raw red rate: add all 20 numbers and divide by 20. Raw rate =x1+x2+···+x20 20 . 3) Make a 3-number moving average list (this is the blur): for each middle position i= 2,3,...,19, MAi=xi−1+xi+xi+1 3. (You will get 18 blurred values: MA2,MA3,...,MA19.) 4) Look at the numbers (or a quick line plot): the blurred list MAi should wiggle less than the raw xi. This makes it easier to see the typical level. 5) Compare two summaries: Blurred rate =MA2+···+MA19 18 vs. Raw rate =x1+···+x20 20 . Which one seems more stable from class to class? 6) Safety margin (simple): find the biggest jump between neighboring blurred values max2≤i≤18  MAi+1 −MAi . Use ±half of that as a quick margin for the blurred rate. Margin ≈1 2max 2≤i≤18  MAi+1 −MAi . 7) Tiny worked example (first few steps): suppose your first five draws were 1 , 0 , 1 , 1 , 0. Then MA2 = (1+0+1) / 3 = 2 / 3, MA3 = (0+1+1) / 3 = 2 / 3, MA4 = (1+1+0) / 3 = 2 / 3. Already smoother than 1,0,1,1,0. 3. Pair-and-average a number list (perfect blur). 1) Write the first 12 multiples of 7: 7,14,21,28,35,42,49,56,63,70,77,84. 2) Make 6 neighbor pairs: (7,14),(21,28),(35,42),(49,56),(63,70),(77,84). 3) Find the average of each pair: 7 + 14 2= 10.5,21 + 28 2= 24.5,35 + 42 2= 38.5, 49 + 56 2= 52.5,63 + 70 2= 66.5,77 + 84 2= 80.5. 4) Add these 6 averages: Savg = 10.5 + 24.5 + 38.5 + 52.5 + 66.5 + 80.5. 5) Double that sum to get 2 Savg . This equals the sum of all 12 numbers (the “perfect blur” keeps totals exact). 6) Why it always works (one line): for any neighboring pair (a, b), 2×a+b 2=a+b. Adding this over all 6 pairs gives exactly the total of all 12 numbers. 2 7) Quick check (optional): the sum of the 12 numbers is also 7 (1 + 2 + ···+ 12) = 7 ×78 = 546. If you compute Savg above, you will get Savg = 273,so2Savg = 546, which matches. 4. Collatz-lite with simple counting (no logs). Pick some odd numbers under 50. For each: •Do 3n+ 1, •divide by 2again and again until it’s odd, •count how many times you divided by 2. Now sort by the remainder of n when divided by 8(1 , 3 , 5 , 7). For each remainder, average the “how many times” counts. Look: the “ × 3 + 1” step makes numbers bigger, but dividing by 2several times makes them smaller. If on average you divide by 2at least twice, the shrinking usually wins. That’s the whole idea of blur here. 2 Secondary school (ages ∼14–18) A bit more structure, still paper-and-pencil. 1. Circle & area with two blurs. For r = 1, find areas of regular n -gons for n = 6 , 8 , 12 both inside and outside the circle. Let Ablur =1 2(Ain +Aout). Claim: Ablur gets close to πas ngrows. Safety margin: Aout −Ain 2. 2. Prime gaps with a tiny “bell” weight. List primes up to 97; make the table of gaps g=pk+1 −pk. Give each gap a weight w ( g ) = exp ( −g2/ 8) (this is a small bell-shaped number). Add up w(g)separately for gaps 2,4,6,8. Question: Which gaps survive this small blur best? (Notice how gap 2behaves.) 3. AM–GM as “fair averaging.” For a, b > 0, show a+b 2≥√ab using (√a−√b)2≥0. Read it as blur: the ordinary average (blur) can’t go below a fair baseline. For three numbers, average in two steps. 4. Right triangles by color classes (mod 4). Squares are 0or 1modulo 4. In a2 + b2 = c2 you can’t have 1+1≡0 (mod 4). Blur idea: We don’t follow exact numbers—just their color class (0or 1). That already rules things out. 5. Cubes modulo 9(a coarse filter). Make the table of x3 ( mod 9) for x = 0 ,..., 8; the only results are 0,±1. Task: Check there are no solutions to x3 + y3≡z3 ( mod 9) when x, y, z are all not divisible by 3. Blur idea: one small modulus blocks many impossible triples at once. 3 6. “Blurred” slope (symmetric difference). Take f ( x ) = x2 at x = 2. With h = 0 . 5, compute D+=f(2+h)−f(2) h, D−=f(2) −f(2 −h) h, Dblur =D++D− 2. Repeat with h= 0.25. Compare each to f′(2) = 4. Why blur helps: the forward and backward errors cancel; Dblur is closer. 3 Undergraduates (unchanged core tasks) Compact exercises with explicit safety margins. 1. Twin-prime constant via prime-side blur. Compute C2(y) = 1 2Y 3≤p≤y p(p−2) (p−1)2for y= 13 (p= 3,5,7,11,13). Tail margin: 0 ≤C2−C2 ( y ) ≤1 2Pp>y 1 (p−1)2≤1 2(y−1) . Give a numeric interval for C2 and 2C2. 2. Li-type toy positivity (small angle blur). With θ∈ { 0 . 4 , 0 . 9 , 1 . 6 } , set λn = 2 Pθ (1 − cos(nθ)) for n= 1,2,3,4. Blur each angle by ± 0 . 05 and average the λn over all perturbations. Nonnegativity is stable; margin = max change across the 23cases. 3. Beurling–Nyman on a grid (measurement blur). N = 6, ak∈ { 1 / 2 , 2 / 3 , 3 / 4 , 4 / 5 , 5 / 6 , 6 / 7 } . Least-squares fit f(x)=1by g(x) = Pck({ak/x}−ak)on grid xj=j/20. Blur is the finite grid; margin is the ℓ2residual. Larger grid ⇒smaller margin. 4. Collatz drift by residue blur (hand-size). For odd residues r∈ { 1 , 3 , 5 , 7 , 9 , 11 , 13 , 15 } modulo 16, compute v2(3r+ 1) and a(r) = log 3 −v2(3r+ 1) log 2 (use three-decimal logs). Average ¯a=1 8Pa(r); check ¯a < 0(negative on average). 5. Cesàro blur of Leibniz partial sums. π 4 = Pn≥0 (−1)n 2n+1 . Compute partial sums S1, . . . , S7 and their blurred means Ck=1 kPk j=1 Sj. Compare Skvs Ckto π/4. Margin: oscillation shrinks under Cesàro blur. One-page classroom summary • Blur first: pick a small, fair average (inside/outside mean, moving average, symmetric difference, small modulus). •Name a safety margin: half-gap, tail bound, grid error, or “max wiggle” after blur. •Decide: if blurred value already meets the goal with margin to spare, you’re done. • Learn by ablation: redo with less blur; watch which conclusions fail—those knobs matter most. 4