Problemas matemáticos y razonamiento · Operations Research & Combinatorics
Simulador de cruce de ríos y optimización de sartenes
Simulador de algoritmos de optimización: freír 3 tortitas en sartén para 2 y cruce de río
- Cálculo en cliente
- Sin registro
- Derivación en vivo
- Modo proyector
- Hoja de trabajo A4
Ventana de demostración matemática● 60 FPS
Variables de control del experimento
hojas
hojas
min
Derivaciones y métricas en tiempo real
Tiempo total óptimo
--
Total de caras de tortita
--
Uso de la sartén
--
Secuencia de pasos
--
Derivación de fórmulas en tiempo real
Fórmula básica:
--Sustitución de valores:
--Conclusión: --
Ejemplos clásicos y análisis de errores en exámenes
Worked Problem
Problema: Pan holds 2 pancakes; each side takes 3 min. Find minimum time for 5 pancakes and 7 pancakes.
Derivación paso a paso:
- 5 pancakes = pair of 2 (6m) + trio of 3 (9m) = 15 min. Universal rule for N>1: Time = N * 3 min. 5*3 = 15 min; 7*3 = 21 min.
Worked Problem
Problema: Breakfast chores: wash cooker 1m, rinse rice 2m, boil porridge 20m, wash veg 3m, cut 2m, fry 5m. Find minimum time.
Derivación paso a paso:
- Parallel execution! Wash cooker (1m) + rinse rice (2m) + boil (20m) = 23 min. Wash, cut, fry take 10m concurrently during boiling.
Hoja de indagación en clase y tareas
Q1. Variant 1: Pan holds 3 fish, 2 min per side. Minimum time to fry 4 fish? To fry 6 fish?
Answer Key: 4 fish = 8 sides. 3 sides/round => ceil(8/3) = 3 rounds * 2 min = 6 min. 6 fish = 12 sides / 3 = 4 rounds * 2 min = 8 min.
Q2. Variant 2: Cook has 3 orders taking 2m, 3m, 5m. Order of service to minimize total customer wait time?
Answer Key: Shortest job first: 2m, then 3m, then 5m. Waits: 1st=2m, 2nd=5m, 3rd=10m. Total wait = 2 + 5 + 10 = 17 min.
Q3. Variant 3: Drawer has 10 red, 8 white, 6 black balls. Minimum drawn blindly to guarantee a matching pair?
Answer Key: Worst-case pigeonhole: 3 colors. Worst draw is 1 of each color = 3 balls. Next draw guarantees a pair: 3 + 1 = 4 balls.
Preguntas frecuentes y errores conceptuales
Why does optimal frying time for N>1 on a 2-slot pan equal N * side_time exactly?
Because N pancakes require 2N cooked sides. A 2-slot pan cooks 2 sides per interval. Perfect scheduling eliminates empty slots, taking exactly 2N/2 = N intervals.
How long to fry 1 single pancake on a 2-slot pan? Does the N*3 rule apply?
Takes 6 minutes! 1 pancake cannot occupy 2 slots simultaneously, wasting 50% capacity. The N*3 rule requires N ≥ 2.
What is the core game-theoretic principle behind Tian Ji's Horse Racing?
Sacrifice your weakest asset against the opponent's strongest asset to secure decisive 2-to-1 victories on the remaining matches.
How to reliably apply the "Worst-Case Scenario" in pigeonhole problems?
Pretend you are the unluckiest person on Earth! The universe delays your goal as long as mathematically possible. Exhaust all non-matching outcomes, then add 1.