min · The Bridge and Torch
The Bridge and Torch
Four people of different speeds, one torch, a bridge that holds two. What is the minimum time to get everyone across?
The bridge where letting your fastest walker carry the torch backfires.
Think about it first
Make a prediction first. Change one input at a time and compare what happens with what you expected.
Jump in! Get all four walkers (1, 2, 5, 10 minutes) to the far side. The bridge holds two at a time, and crossing needs the 🔥torch. A pair moves at the slower walker's pace.
This banktorch
Far side
Read the solution when you are ready
The bridge is solved by placing the slow feet, not the fast ones.
Why does this question matter?
Four people stand before an old bridge at night. It holds two at a time, and no one can cross without the torch. Their walking times are 1, 2, 5, and 10 minutes. This classic puzzle became famous as a tech-interview question.
Everyone's first answer is the same: 'let the fastest one shuttle the torch.' That takes 19 minutes. But the true answer is 17 — earned not by working the fast walker harder, but by sending the two slow walkers together.
The mathematics of finding the best arrangement is called optimization. Delivery routes, factory schedules, and flight timetables all stand on this question. This bridge is the doorway.
Read the idea from another angle
Pay the expensive cost only once.
The 10-minute crossing is unavoidable. The best strategy hides the 5 inside that 10. Handle other costs while paying the big one, and the small ones vanish.
Optimization lives between the intuitive answer and the best answer.
Nineteen minutes is a plausible answer. But the moment you ask 'can it be less?' two more minutes appear. That one extra question is where optimization begins.
Now read the definitions with their conditions
Write each strategy as a sum and the difference jumps out. Plan A lets the fast walker shuttle; plan B pairs the slow walkers. Only the middle move changes.
Formula anatomy — the five moves of the 17-minute plan
+ + + + = 17
The intuitive answer — fast-foot shuttle
The 1 ferries everyone with the torch. Its weakness: the 5 and the 10 are each paid in full, separately.
The best answer — pairing the slow feet
Send 5 and 10 together and pay only the 10. The return gets pricier (2 instead of 1), but the saved 5 wins.
When pairing wins
With a, b the fast pair and c the third slowest: pairing wins when b's round trip is cheaper than a plus c. Here 4 < 6, so pairing wins.
Make one more distinction
Having the fastest person carry the torch is always optimal.
Send the two slow people together and the 5 rides for free inside the 10. The return trip gets a little pricier, but the saved 5 is bigger. Here 19 minutes drops to 17.
There are so few cases that trial and error finds the best quickly.
Counting return orders, the cases run past several dozen. With five or six people they explode. That is why finding a principle like 'pair the slow ones' beats enumerating.
Where else can this idea help?
Bundled deliveries
A courier carries two orders bound for the same building in one trip. Same strategy as the bridge: pay the long haul only once.
Batch baking
When the oven heats up, you fill it with loaves. Many loaves share the expensive preheat.
Hub airports
Airlines pool passengers onto one long-haul flight, then scatter them on short hops. The sky is engineered to pay the expensive crossing once.
Computer job scheduling
Computers also compute 'when, and with what, should the expensive job run?' The bridge puzzle is scheduling theory in miniature.
Explain it in your own words
Check the idea with practice problems (4)
Questions completed 0 / 4
1. Four people (1, 2, 5, 10 min) cross a two-person bridge. The minimum time is:
2. What is the key move of the 17-minute plan?
3. If the walking times were 1, 2, 6, and 8 minutes, what is the minimum time?
4. The plan where the fastest walker does every torch run is…
What question comes next?
Previous concept
The Monty Hall Problem
Your first betrayal by intuition. On the bridge, 'fastest is best' breaks next.
← The Monty Hall Problem labLeads to next
The Poisoned Wine
You saved time; now save information — ten mice against a thousand bottles.
Go to the The Poisoned Wine lab →Related lab
Optimization
The grown-up version of pairing slow feet — finding the best where the slope hits zero.
Go to the Optimization lab →Related lab
Counting Cases
Guaranteeing the optimum requires counting every case without gaps.
Go to the Counting Cases lab →