๐Ÿงฎ Brain Teaser

Archive

135 problems
#Problemโœ“
135Curse of Dimensionality: Nearest Neighbor Biasโ—‹
134Ridge Regression as Augmented OLSโ—‹
133The Gaussian-Polynomial Integralโ—‹
132Integral of 1/(1+tan^n x)โ—‹
131The Polynomial That Is Always Divisible by n!โ—‹
130The Penny-Passing Game: Who Wins?โ—‹
129Casoratiโ€“Weierstrass: Dense Image Near Essential Singularityโ—‹
128The Secretary Who Knows the Distributionโ—‹
127The Gambler Who Can't Lose... Until He Doesโ—‹
126Quotient of Interval Gives Circleโ—‹
125Fundamental Group of Doubly Punctured Planeโ—‹
124Conditionally Convergent Series of Functions: Pointwise but Not Uniformโ—‹
123The Integral That Knows Its Own Intervalโ—‹
122The Wisdom of Crowds: When Does Averaging Hurt?โ—‹
121The Irrelevant Feature Paradox: When Adding Noise Helps OLSโ—‹
120Integral of ln(sin x) via Symmetryโ—‹
119Wallis-Type Product Integralโ—‹
118A Square Vanishing Function Must Be Identically Zeroโ—‹
117Riemann's Removable Singularity via zf(z)โ†’0โ—‹
116Expected Runs of Headsโ—‹
115The Matching Birthdays: Expected Collisionsโ—‹
114Hawaiian Earring vs Infinite Wedge: Compact vs Notโ—‹
113Suspension Kills Fundamental Groupโ—‹
112Devil's Staircase: FTC Fails Without Absolute Continuityโ—‹
111KDE Bandwidth Bias-Variance Tradeoffโ—‹
110SVM Margin Width: Why 2/โ€–ฮฒโ€–?โ—‹
109The Beta-Flavored Integralโ—‹
108The Sinc Integralโ—‹
107Shanille's Free Throws: Pรณlya Urn Surpriseโ—‹
106Argument of a Product Along a Circleโ—‹
105Schwarzโ€“Pick Lemma: Holomorphic Maps Contract the Diskโ—‹
104The Collector's Rival: Who Finishes First?โ—‹
103The Absent-Minded Secretaryโ—‹
102The Dunce Hat Is Contractibleโ—‹
101โ—‹
100Alternating Series Beyond the M-Testโ—‹
99The Infinite Ensemble That Doesn't Helpโ—‹
98Curse of Dimensionality: Volume Collapseโ—‹
97Integral of sqrt(tan x)โ—‹
96The King's Integralโ—‹
95A Sequence That Always Hits a Perfect Squareโ—‹
94Gaussโ€“Lucas Theoremโ—‹
93Mรถbius Transformation Fixed by Three Pointsโ—‹
92HH vs HT: Waiting Times for Coin Patternsโ—‹
91Second-Best Runner-Upโ—‹
90Fundamental Group of Sยน โˆจ Sยฒโ—‹
89Hawaiian Earring Is Not Semi-Locally Simply Connectedโ—‹
88Riemannโ€“Lebesgue Lemmaโ—‹
87Monotone Convergence Fails for Decreasing Sequences Without Integrabilityโ—‹
86The Kernel Trick: Why Inner Products Are All You Needโ—‹
85Optimism of Training Error and the LOO Shortcutโ—‹
84Feynman's Trick: arctan(x)/x and Catalan's Constantโ—‹
83Functional Equation: From Symmetry to a Differenceโ—‹
82Maximum Modulus & Schwarz Equality Caseโ—‹
81A Holomorphic Function That Is Its Own Derivativeโ—‹
80The Thick Coinโ—‹
79Cycle Lemma: Staying Strictly Aheadโ—‹
78Fundamental Group of Sยน โˆจ Sยน Is Non-Abelianโ—‹
77Fundamental Group of RPยฒโ—‹
76Two Integral Conditions Force Two Zerosโ—‹
75Curse of Dimensionality: Shell Concentrationโ—‹
74The Blessing of Averaging: Boosting Marginsโ—‹
73Integral of x^n ln x and Baselโ—‹
72Fresnel-Flavored Integralโ—‹
71Can Aยฒ + Bยฒ Be Invertible?โ—‹
70A Determinant of Sumsโ—‹
69Winding Number Is Always an Integerโ—‹
68Ballot Problem: Strictly Leading Throughoutโ—‹
67The Collector's Last Couponโ—‹
66Mรถbius Band Has Fundamental Group โ„คโ—‹
65Lebesgue Differentiation: Failure Everywhere?โ—‹
64Zero-Mean Functions: A Closed Hyperplane in C([0,1])โ—‹
63The Double Descent Puzzleโ—‹
62The Variance of Baggingโ—‹
61Integral of ln(1+xยฒ)/(1+xยฒ)โ—‹
60Sum of Squares Under a Linear Constraintโ—‹
59Logarithmic Derivative and Integrality of Windingโ—‹
58Open Mapping + Liouville: Dense Image of Entire Functionsโ—‹
57The First Aceโ—‹
56Broken Bar Triangle Probabilityโ—‹
55Baire Category: Q Is Not a G-deltaโ—‹
54Hairy Ball Theorem via Degree Theoryโ—‹
53Weierstrass Series: Continuity via Abel Summationโ—‹
52Vanishing Integral on All Measurable Setsโ—‹
51Variance of OLS Along Singular Directions and Ridge Shrinkageโ—‹
50The Basel-Flavored Integralโ—‹
49Sophomore's Dream Integralโ—‹
48A Polynomial That Takes the Value 1 Too Oftenโ—‹
47Argument Principle: Zeros in an Annulusโ—‹
46Residue at Infinity and the Sum Ruleโ—‹
45The Prisoner's Dilemmaโ—‹
44The Unfair Coin That Becomes Fairโ—‹
43Fundamental Group of the Torus via Van Kampenโ—‹
42Klein Bottle Is Non-Orientable and Doesn't Embed in Rยณโ—‹
41Dirichlet Function and Lebesgue Measure Zeroโ—‹
40Nowhere-Zero Derivative Paradoxโ—‹
39Optimism of Training Errorโ—‹
38Bias-Variance Decompositionโ—‹
37The Gaussian Integral's Elegant Cousinโ—‹
36Integral of ln(sin x)โ—‹
35Ordered Triples of Sets Covering {1,...,10}โ—‹
34Pigeonhole on an Icosahedronโ—‹
33Rouchรฉ's Theorem: Zeros of zโต+3z+1โ—‹
32Entire Function Avoiding a Disk Must Be Constantโ—‹
31Coupon Collector Problemโ—‹
30Gambler's Ruin: Probability and Expected Durationโ—‹
29Fundamental Group of the Circle Is โ„คโ—‹
28The Comb Space Is Not Locally Connectedโ—‹
27Integral of a Max with a Circleโ—‹
26Dini's Theorem: When Pointwise Becomes Uniformโ—‹
25An Impossible Integer Polynomialโ—‹
24Entire Function with Non-Negative Real Partโ—‹
23The Topologist's Sine Curveโ—‹
22The Cantor Set: Measure Zero yet Uncountableโ—‹
21Why LASSO Gives Sparsity but Ridge Does Notโ—‹
20Continuous but Nowhere Differentiableโ—‹
19Integral of (x-1)ยฒ/(2eหฃ+xยฒ+1)โ—‹
18Consecutive Subsum Divisible by nโ—‹
17Orthogonality to All Monomials Forces Zeroโ—‹
16Stein's Paradoxโ—‹
15Integral of xยฒ sin(x)โ—‹
14No Function with f(f(n)) = n + 2025โ—‹
13Truncated Exponential Has No Repeated Rootsโ—‹
12FTA via Liouville's Theoremโ—‹
11Curvature of an Embedded Torusโ—‹
10The Hat Check Problemโ—‹
9Bertrand's Chord Paradoxโ—‹
8Integral of a Geometric Seriesโ—‹
7Polynomial with No Integer Rootsโ—‹
6Harmonic Series Is Never an Integerโ—‹
5The Schwarz Lemmaโ—‹
4Parallel Transport on a Sphereโ—‹
3Expected Number of Recordsโ—‹
2Bertrand's Ballot Problemโ—‹
1Lp Norm โ†’ Lโˆžโ—‹