๐Ÿงฎ Brain Teaser

Archive

361 problems
#Problemโœ“
361Injective Entire Functions Are Linearโ—‹
360Winning an Unfair Gameโ—‹
359Random Quadratic Equations: Real Roots Almost Surely?โ—‹
358Maps from RPยฒ to the Circle (and Borsukโ€“Ulam)โ—‹
357The Plane Cannot Be Sliced into Circlesโ—‹
356Fekete's Subadditive Lemmaโ—‹
355The Factorial That Unmasks eโ—‹
354The Bootstrap That Fools Itself: 1-NN on Pure Noiseโ—‹
353The Cross-Validation That Peeked: Screening Before the Foldsโ—‹
352The Quartic That Folds onto Itselfโ—‹
351Poisson's Log-Cosine Integralโ—‹
350The Fraction That Eventually Stops Simplifyingโ—‹
349A Finite Set Closed Under Cross Productโ—‹
348The Circle Has Only One Entire Parametrizationโ—‹
347The Function Polynomials Cannot Reachโ—‹
346Doubling Your Accuracyโ—‹
345Curing the Compulsive Gamblerโ—‹
344A Compact Space Has No Room for a Proper Isometric Copy of Itselfโ—‹
343Exactly Twice Is Impossibleโ—‹
342How Fast Does Iterated Sine Die?โ—‹
341Olivier's Theorem: Decreasing Terms Are o(1/n)โ—‹
340Why the Wavelet Threshold Is ฯƒโˆš(2 log N)โ—‹
339Nearest Neighbor Is Never More Than Twice as Bad as Bayesโ—‹
338The Integral That Only Cares About the Ratioโ—‹
337Lobachevsky's Folding Trickโ—‹
336Four Random Points on a Sphere: Tetrahedron Contains the Centerโ—‹
335Zeros Hidden Before the Beginningโ—‹
334A Polynomial That Lives on the Unit Circleโ—‹
333Positive Real Part of the Derivative Forces Injectivityโ—‹
332The Cliff-Hangerโ—‹
331Buffon's Needle and the Noodle Trickโ—‹
330A Connected Metric Space Cannot Be Countableโ—‹
329Furstenberg's Topological Proof of Infinitely Many Primesโ—‹
328The Universal Chord Theoremโ—‹
327The Derivative That Drags Its Function Alongโ—‹
326Why ICA Cannot Unmix Gaussiansโ—‹
325Ridge Regression with Cloned Featuresโ—‹
324Why ฯ€ Is Less Than 22/7โ—‹
323Two Integrals, One Curveโ—‹
322Twelve Lengths and an Acute Triangleโ—‹
321Three-Fold Closure Forces Two-Fold Closureโ—‹
320The Function That Cannot Explode at the Boundaryโ—‹
319The Monic Polynomial That Cannot Hide Below 1โ—‹
318The Ballot Box Surprise: Early Leader Locks Inโ—‹
317Successive Wins: Father or Champion in the Middle?โ—‹
316Cofinite Topology: Tโ‚ but Not Hausdorffโ—‹
315The Wedge That Knows Its Loopsโ—‹
314The Function That Is Its Own Averageโ—‹
313The Continuous Function That Cannot Be Monotone on Any Intervalโ—‹
312Precision Matrix and Conditional Independenceโ—‹
311Gradient Boosting Secret: Why Each Tree Fits Residualsโ—‹
310The Integral That Sums to an Arcsineโ—‹
309The Integral That Absorbs a Parameterโ—‹
308The Polynomial That Stays Small on Integersโ—‹
307The Conformal Map That Unfolds a Wedgeโ—‹
306โ—‹
305The Duelist's Dilemma: Who Wins When A Shoots First?โ—‹
304The Flippant Juror: Three Jurors and a Coinโ—‹
303Cofinite Topology: Compact but Not Closedโ—‹
302Torus Minus Disk: Free Group Appearsโ—‹
301The Sequence That Must Have a Convergent Subsequence... or Does It?โ—‹
300Vanishing Fourier Coefficients Force Zeroโ—‹
299The EM Algorithm That Overshoots: Why Hard Assignment Failsโ—‹
298Cross-Entropy Decomposition: Entropy + KL Divergenceโ—‹
297The Integral That Knows About Primesโ—‹
296Frullani-Type Log Integral via Feynmanโ—‹
295The Integers That Sum to Their Productโ—‹
294The Holomorphic Function That Knows Its Zeros Are Realโ—‹
293The Gambler's Ruin on a Circleโ—‹
292The Flippant Juror's Cousin: Four Jurors and a Coinโ—‹
291Disk Boundary Reflection Gives Sphereโ—‹
290The Integral That Knows Its Own Symmetryโ—‹
289The Equicontinuous Sequence That Must Convergeโ—‹
288Fisher Information Curvature Identityโ—‹
287Precision Matrix and Conditional Independenceโ—‹
286The Integral That Sums Geometric Anglesโ—‹
285The Polynomial That Takes Integer Values at Half-Integersโ—‹
284The Holomorphic Function That Integrates to Zero on Every Circleโ—‹
283Natural Boundary via Factorial Lacunary Seriesโ—‹
282The Collector Who Stops Too Earlyโ—‹
281The Ballot Box Twist: Runs of the Same Colorโ—‹
280Cofinite Topology: Compact but Not Hausdorffโ—‹
279Punctured RPยฒ Has Fundamental Group โ„คโ—‹
278Stoneโ€“Weierstrass: Even vs Odd Power Subalgebrasโ—‹
277โ—‹
276PCR Bias-Variance per Principal Componentโ—‹
275The Integral That Knows Euler's Constantโ—‹
274Sophomore's Dream: Integral of x^xโ—‹
273Expected Fixed Points of a Random Permutationโ—‹
272The Conformal Map That Doubles Its Angleโ—‹
271The Entire Function That Grows Too Slowlyโ—‹
270Expected Number of Matchesโ—‹
269The Prisoner's Pardon: Three Boxes, One Clueโ—‹
268Connected Sum of Two RPยฒs and Its Fundamental Groupโ—‹
267Torus Knot Curve: Simple Closed Curve and Annular Complementโ—‹
266Orthogonality to Monomials: Missing n=0โ—‹
265Naive Bayes Beats Its Own Assumptionsโ—‹
264The Gaussian That Knows Its Own Varianceโ—‹
263The Integral That Squares Its Sineโ—‹
262Integral of arctan(x)/(1+xยฒ)โ—‹
261The Coloring That Cannot Avoid Arithmeticโ—‹
260The Expression That Only Skips Perfect Cubesโ—‹
259The Entire Function That Squares to Anotherโ—‹
258The Lazy Gambler's Last Dollarโ—‹
257Ties in Matching Penniesโ—‹
256The Quotient Square That Becomes a Diskโ—‹
255The Covering Space That Knows Its Sheetsโ—‹
254Surjective Map on Interval Has Two Fixed Pointsโ—‹
253Uniform Limit of Derivatives via Integrationโ—‹
252The Posterior That Forgets the Priorโ—‹
251The Calibration Trap: Perfect Accuracy vs Probability Estimationโ—‹
250The Integral That Loops Back Aroundโ—‹
249Integral of sinยณx/(sinยณx+cosยณx)โ—‹
248The Polynomial That Divides Its Own Compositionโ—‹
247The Checkerboard Walk: Can the Knight Return Home?โ—‹
246The Holomorphic Function That Misses Two Pointsโ—‹
245The Conformal Map That Straightens a Half-Stripโ—‹
244Twin Knights: Meeting in the Tournamentโ—‹
243The Three-Door Switcheroo: Generalized Monty Hallโ—‹
242The Covering Space That Wraps Around Twiceโ—‹
241The Lebesgue Integral That Measures Its Own Level Setsโ—‹
240The Absolutely Continuous Function That Forgets Its Derivativeโ—‹
239LOO Risk That Knows Its Smootherโ—‹
238The Softmax That Forgets Its Past: Invariance to Label Permutation vs. Feature Permutationโ—‹
237The Integral That Completes the Square in the Exponentโ—‹
236The Integral That Loves a Log-Sine Cousinโ—‹
235Determinant Tic-Tac-Toeโ—‹
234The Entire Function Bounded on a Lineโ—‹
233Gambler's Duration via Martingaleโ—‹
232The Torus That Forgets a Diskโ—‹
231Fundamental Group of RPยฒโ—‹
230The Continuous Function That Must Change Signโ—‹
229The Measurable Set That Fills Every Intervalโ—‹
228EM Algorithm's Hidden Monotonicityโ—‹
227The Logistic Regression Coefficient That Goes to Infinityโ—‹
226The Integral That Reflects Twiceโ—‹
225The Integral That Walks Down Stairsโ—‹
224Central Binomial Coefficient mod Primeโ—‹
223The Polynomial Pigeonholeโ—‹
222Weighted Argument Principle: Summing Zeros and Polesโ—‹
221Growth Bound Forces Singularity Typeโ—‹
220The Absent Ticket Inspectorโ—‹
219The Three-Cornered Duelโ—‹
218Closed Disk Minus a Boundary Arc: Contractible?โ—‹
217The Sphere Is Simply Connected: A Covering Space Argumentโ—‹
216The Integral That Knows ฮ“โ—‹
215The Sinc Integral: Conditional but Not Absoluteโ—‹
214โ—‹
213LDA Is Secretly Logistic Regressionโ—‹
212Integral of 1/(1+tan^โˆš2 x)โ—‹
211The Integral That Hides a Telescoping Heartโ—‹
210The Derivative That Divides 2016โ—‹
209The Holomorphic Map That Fixes Too Many Pointsโ—‹
208The Mรถbius Transformation That Sends the Real Line to Itselfโ—‹
207The Clumsy Chemist's Cousin: Breaking at Two Random Pointsโ—‹
206The Envelope Paradoxโ—‹
205Torus Minus a Point Deformation Retracts onto Wedgeโ—‹
204Punctured Torus Has Free Fundamental Groupโ—‹
203The Nowhere-Monotone Continuous Functionโ—‹
202Integral of the Cantor Function via Symmetryโ—‹
201The Variance That Vanishes: Why PCA Features Are Uncorrelatedโ—‹
200The Ghost Feature: Useless Predictor Inflates OLS Varianceโ—‹
199The Integral That Wants to Be a Probabilityโ—‹
198Integral of 1/(1+tan^ฮฑ x)โ—‹
197Integer-Valued Polynomials and the Binomial Basisโ—‹
196Weierstrass Product Convergence and Logarithmic Derivativeโ—‹
195The Sock Drawer Surpriseโ—‹
194Gambler's Ruin: Martingale and Win Probabilityโ—‹
193Which Spheres Admit a Topological Group Structure?โ—‹
192The Torus and the Annulus: A Quotient Surpriseโ—‹
191The Function That Equalizes Its Own Averagesโ—‹
190Polynomials Dense: Orthogonality Forces Zeroโ—‹
189Naive Bayes Independence Violation: Wrong Probabilities, Right Decisionsโ—‹
188PCA Reconstruction Error and Discarded Eigenvaluesโ—‹
187The Integral That Swaps Its Limitsโ—‹
186Integral of ln(x)/sqrt(x(1-x))โ—‹
185Sums That Know Their Partsโ—‹
184Two Laurent Series for One Functionโ—‹
183The Conformal Map That Squares the Diskโ—‹
182The Coin That Remembers Its Pastโ—‹
181The Lazy Drunk: Random Walk Returns in 2D vs 3Dโ—‹
180Open Identity Forces Discrete Topological Groupโ—‹
179Covering Space of Wedge of Circles Has Larger Fundamental Groupโ—‹
178The Monotone Function That Isn'tโ—‹
177The Integrable Function Whose Integral Vanishes on Every Intervalโ—‹
176The Vanishing Gradient Plateau: Why Sigmoid Networks Saturateโ—‹
175The SVM That Saw Only Dot Productsโ—‹
174The Gaussian Meets Its Reflectionโ—‹
173Integral of arctan(x)/x and Catalan's Constantโ—‹
172A Polynomial Taking Only Non-Negative Integer Valuesโ—‹
171Infinite Product Telescoping to 2โ—‹
170The Cauchy Integral That Evaluates Itselfโ—‹
169Evening Sales: Poisson Parity via e^m + e^{-m}โ—‹
168The Craps Shooter's Oddsโ—‹
167Collapsing Square Boundary Gives Sphereโ—‹
166The Quotient of a Torus by an Involutionโ—‹
165The Integral That Oscillates Into Submissionโ—‹
164Dini's Theorem via Compactnessโ—‹
163Curse of Dimensionality: Nearest Neighbor Becomes Globalโ—‹
162Temperature Scaling and Softmax Saturationโ—‹
161Integral of ln(1+x)/(1+xยฒ) via Feynmanโ—‹
160Integral of ln(1+x)/(1+xยฒ) via Feynman Self-Referenceโ—‹
159A Polynomial with Three Prescribed Valuesโ—‹
158A Polynomial Vanishing on All Lattice Points in a Stripโ—‹
157The Holomorphic Function Determined by Its Real Partโ—‹
156Schwarz Reflection Principle via Moreraโ—‹
155The Pepysโ€“Newton Problemโ—‹
154The Shared Birthday Secretaryโ—‹
153Fundamental Group of the Klein Bottleโ—‹
152Open Subgroups of Topological Groups Are Closedโ—‹
151Frullani Integral: (e^{-x}-e^{-2x})/xโ—‹
150Monotone Function with Countably Many Discontinuitiesโ—‹
149AdaBoost Exponential Training Error Boundโ—‹
148Precision Matrix and Partial Correlationsโ—‹
147Integral of 1/(1+sinยฒx)โ—‹
146The Dirichlet Beta Integralโ—‹
145A Polynomial Evaluated at Consecutive Integersโ—‹
144The Tournament Bracket Paradoxโ—‹
143The Zeros That Refuse to Accumulateโ—‹
142The Harmonic Function That Knows Its Boundaryโ—‹
141The Lazy Prisoner: A Probability Paradoxโ—‹
140The Ballot Box Surprise: Two Candidates, One Mysteryโ—‹
139Collapsing Disk Boundary Gives Sphereโ—‹
138The Closed Subgroup of โ„โ—‹
137Generic Continuous Function Is Nowhere Differentiableโ—‹
136Pointwise Limits and Meagre Discontinuity Setsโ—‹
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—‹