Daily Math · 2026-05-05

Question 1

An airplane row has 3 seats: window, middle, aisle. You're seating 3 people: Alice prefers the window, Bob prefers the aisle, Carol has no preference. In how many of the $3! = 6$ possible orderings is everyone happy (i.e., neither Alice nor Bob ends up away from their preferred seat)?

Accepted Answer

Alice must take the window seat (1 way). Bob must take the aisle seat (1 way). Carol takes the middle (1 way). Only $1 	imes 1 	imes 1 = 1$ ordering satisfies both preferences.

More generally — when each person's preference is for a distinct seat, only the permutation that matches preferences exactly is 'fully happy'.

Answer

$1$

Answer

$2$

Answer

$3$

Answer

$6$

Question 2

How long, in years, does it take for an investment to double at 7% annual interest, compounded annually? (Use the closest answer.)

Accepted Answer

We need $t$ such that $(1.07)^t = 2$, i.e. $t = \dfrac{\ln 2}{\ln 1.07} \approx \dfrac{0.693}{0.0677} \approx 10.24$ years.

The banker's shortcut: the **Rule of 72** — divide 72 by the interest rate. $72 / 7 \approx 10.3$. Quick, accurate enough.

Answer

About 7 years

Answer

About 10 years

Answer

About 14 years

Answer

About 70 years

Question 3

Solve for $x$: $\quad 2^{x+1} = 3^{x}$.

Accepted Answer

Take $\ln$ of both sides: $(x+1) \ln 2 = x \ln 3$.

Expand: $x \ln 2 + \ln 2 = x \ln 3$.

Collect $x$: $\ln 2 = x(\ln 3 - \ln 2)$.

So $x = \dfrac{\ln 2}{\ln 3 - \ln 2} \approx \dfrac{0.693}{0.405} \approx 1.71$.

Answer

$x = \log_3 2$

Answer

$x = \dfrac{\ln 2}{\ln 3 - \ln 2}$

Answer

$x = \log_2 3$

Answer

$x = \dfrac{\ln 2}{\ln 3}$

Problems & worked solutions

Problem 1 — Combinatorics

Problem 2 — Compound Interest

Problem 3 — Exponential Functions