Top Tweets for #QEDEventually
Problem 12: THE HARMONIC STAIRCASE.
Step n has height H_n = 1 + 1/2 + ... + 1/n and weight 1/n³.
Weigh the whole infinite staircase: what is the exact value of the sum of H_n / n³?
It is about 1.3529.
Solution 10 is live.
#QEDEventually #EulerSums #InfiniteSeries
Problem 11: THE LOCKSMITH'S CIRCLE.
A two-button keypad opens when the last n presses match the code.
A shortest cyclic sequence of 2^n presses tries every code once.
How many exist, up to rotation?
Solution 9 is live.
#QEDEventually #deBruijn #BESTTheorem #Combinatorics
Problem 11: THE LOCKSMITH'S CIRCLE.
A two-button keypad opens when the last n presses match the code.
A shortest cyclic sequence of 2^n presses tries every code once.
How many exist, up to rotation?
Solution 9 is live.
#QEDEventually #deBruijn #BESTTheorem #Combinatorics
Problem 10: THE WARDEN'S LOCKERS.
2n prisoners, 2n lockers, n openings each.
Open the locker with your own number, then follow the numbers you find.
What is the chance all succeed, and its limit as n grows?
Solution 8 is live.
#QEDEventually #Probability #Permutations
Problem 10: THE WARDEN'S LOCKERS.
2n prisoners, 2n lockers, n openings each.
Open the locker with your own number, then follow the numbers you find.
What is the chance all succeed, and its limit as n grows?
Solution 8 is live.
#QEDEventually #Probability #Permutations
Problem 9: THE LIE OF POWERS OF TWO.
n points on a circle, every chord drawn, no three chords meeting inside.
Regions: 1, 2, 4, 8, 16... Does it keep doubling?
Find the formula, with proof.
Solution 7 is live.
#QEDEventually #Geometry #Combinatorics
Problem 9: THE LIE OF POWERS OF TWO.
n points on a circle, every chord drawn, no three chords meeting inside.
Regions: 1, 2, 4, 8, 16... Does it keep doubling?
Find the formula, with proof.
Solution 7 is live.
#QEDEventually #Geometry #Combinatorics
Problem 8: THE CAPITAL'S ROADS.
Connect n labeled towns by n-1 loop-free roads, chosen uniformly among all n^(n-2) ways. How many roads does the capital get?
Find the exact law and its limit as n grows.
Solution 6 is live.
#QEDEventually #RandomTrees #CayleyFormula
Problem 8: THE CAPITAL'S ROADS.
Connect n labeled towns by n-1 loop-free roads, chosen uniformly among all n^(n-2) ways. How many roads does the capital get?
Find the exact law and its limit as n grows.
Solution 6 is live.
#QEDEventually #RandomTrees #CayleyFormula
Problem 7: THE WATCHMAN'S CORRIDOR.
2n steps, forward or back, from the door and back to it.
Count only the forward steps taken on the north side. Show every count from 0 to n is equally likely.
Solution 5 is live.
#QEDEventually #Combinatorics #GeneratingFunctions
Problem 7: THE WATCHMAN'S CORRIDOR.
2n steps, forward or back, from the door and back to it.
Count only the forward steps taken on the north side. Show every count from 0 to n is equally likely.
Solution 5 is live.
#QEDEventually #Combinatorics #GeneratingFunctions
Problem 6: THE MERCHANT'S COINS.
Coins worth a, a+d, a+2d, ..., a+kd with gcd(a, d) = 1.
Some totals can never be paid exactly. What is the largest one? Find a formula, with proof.
Solution 4 is live.
#QEDEventually #FrobeniusCoinProblem #NumberTheory #ArithmeticProgression
Problem 6: THE MERCHANT'S COINS.
Coins worth a, a+d, a+2d, ..., a+kd with gcd(a, d) = 1.
Some totals can never be paid exactly. What is the largest one? Find a formula, with proof.
Solution 4 is live.
#QEDEventually #FrobeniusCoinProblem #NumberTheory #ArithmeticProgression
Problem 5: THE SQUARE THAT WAITED.
26 sits between 25 = 5² and 27 = 3³. Fermat said it is the only such number.
Find every integer solution of y² + 2 = x³.
Solution 3 is live.
#QEDEventually #NumberTheory #MathPuzzle
Problem 5: THE SQUARE THAT WAITED.
26 sits between 25 = 5² and 27 = 3³. Fermat said it is the only such number.
Find every integer solution of y² + 2 = x³.
Solution 3 is live.
#QEDEventually #NumberTheory #MathPuzzle
Problem 4: TWO MONKS, ONE QUEEN.
One chess queen, two monks.
She moves only left, down or diagonally down-left.
Whoever lands her on the corner wins.
Which squares lose for the player to move?
Solution 2 is live.
#QEDEventually #GameTheory #WythoffsGame
Problem 4: TWO MONKS, ONE QUEEN.
One chess queen, two monks.
She moves only left, down or diagonally down-left.
Whoever lands her on the corner wins.
Which squares lose for the player to move?
Solution 2 is live.
#QEDEventually #GameTheory #WythoffsGame
Problem 3: BERNOULLI'S DREAM.
In 1697, Bernoulli saw an integral and an infinite sum share every digit.
Prove it, then find the sibling series for the integral of x^x.
Tonight: Solution 1.
Tomorrow: Solution 2.
#QEDEventually #CalculusPuzzle #InfiniteSeries
Problem 3: BERNOULLI'S DREAM.
In 1697, Bernoulli saw an integral and an infinite sum share every digit.
Prove it, then find the sibling series for the integral of x^x.
Tonight: Solution 1.
Tomorrow: Solution 2.
#QEDEventually #CalculusPuzzle #InfiniteSeries
Problem 2: THE LIGHTHOUSE RING.
A lighthouse, n harbor towns in a ring, 2n cables.
Keep exactly n so everything stays connected with no loops.
How many ways? Find the closed form.
Hint 1 is live.
Solution 1 tomorrow.
#QEDEventually #GraphTheory #DiscreteMath
Problem 2: THE LIGHTHOUSE RING.
A lighthouse, n harbor towns in a ring, 2n cables.
Keep exactly n so everything stays connected with no loops.
How many ways? Find the closed form.
Hint 1 is live.
Solution 1 tomorrow.
#QEDEventually #GraphTheory #DiscreteMath
Problem 1: THE VISIBLE FOREST
A surveyor, an endless orchard, a tree at every lattice point.
Pick one at random: what is the chance its line of sight holds a perfect-square number of trees?
Hint tomorrow.
Proof the night after.
#QEDEventually #MathPuzzle #NumberTheory
Problem 1: THE VISIBLE FOREST
A surveyor, an endless orchard, a tree at every lattice point.
Pick one at random: what is the chance its line of sight holds a perfect-square number of trees?
Hint tomorrow.
Proof the night after.
#QEDEventually #MathPuzzle #NumberTheory
Are you ready for the challenge?
A story and a problem tonight.
A hint tomorrow.
The proof the night after.
21 problems, each inspired by a classic, each with a solution checked by hand and by computer.
Problem 1 arrives next.
#QEDEventually #MathPuzzle #BrainTeaser
Are you ready for the challenge?
A story and a problem tonight.
A hint tomorrow.
The proof the night after.
21 problems, each inspired by a classic, each with a solution checked by hand and by computer.
Problem 1 arrives next.
#QEDEventually #MathPuzzle #BrainTeaser
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