Updated weekly
One mathematical puzzle, every Tuesday. Send in your solution by email, drop by a meeting, or just enjoy it on your own. Past puzzles and solutions are below.
This week
Find the smallest positive integer n such that 1 + 1/2 + 1/3 + … + 1/n exceeds 5. Closed-form arguments welcome; clever bounds even more so.
The harmonic numbers grow like ln(n) + γ. Use this to estimate n, then check carefully around your estimate.
A new puzzle appears here each Tuesday. They're chosen to be approachable with pen and paper, no special background required. Try the hint first; the solution gets posted with the next week's puzzle.
Email us your solution before the deadline, or bring it to the next meeting. We love elegant solutions, but partial credit is real, and so is "here's where I got stuck."
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Tap a card to reveal the hint or the full solution. Solvers are listed with permission.
In how many ways can you tile a 2 × 10 board using 1 × 2 dominoes (placed either horizontally or vertically)?
Let T(n) be the number of tilings of a 2 × n board. Find a recurrence by considering the leftmost piece.
T(n) satisfies T(n) = T(n − 1) + T(n − 2) with T(1) = 1, T(2) = 2, which gives the Fibonacci sequence shifted by one. So T(10) = F(11) = 89.
Solvers: Anna L., Marcus W., Priya S.
Find all positive integers n such that n + 1 divides n² + 1.
Polynomial long division. What is n² + 1 modulo n + 1?
Since n² + 1 = (n + 1)(n − 1) + 2, we have (n² + 1) mod (n + 1) = 2. So (n + 1) | (n² + 1) iff (n + 1) | 2, which gives n + 1 ∈ {1, 2}, hence n = 1.
Solvers: Devi T., Jonas K.