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There Are More Brainteasers About Crossing Rivers Than You Ever Imagined

Perhaps you’ve heard the classic puzzle about the fox, the goose and the grain?

Contributor

December 3, 2012

Perhaps you’ve heard the classic puzzle about the fox, the goose and the grain? It goes like this. A farmer needs to get a fox, a goose and a bag of grain across a river using a boat. This boat is small, and it can only hold one additional item alongside the farmer. The fox cannot be left alone with the goose, because he will eat it. The goose cannot be left alone with the grain for the same reason. How can the farmer get the three items across in one piece?

There are a number of surprising variations of this problem. In one version, there are three married couples trying to cross that same river in that same two person boat. The catch is that in this case, the husbands are jealous, so no married woman can cross the river with another man unless her husband is present. In another version there is an entirely dysfunctional group made up of a father, a mother, two sons, two daughters, a guard and a prisoner. The father cannot be left along with any of the daughters without the mother, the mother cannot be left with any of the sons without the father, the criminal can’t be with any family member without the guard and only the mother, father and guard know how to steer a boat.

Alternatively, what if you had a man and a woman of equal weight, along with two children who weighed half that? The boat can only carry the weight of one adult at a time. How do all four get across?

In another version, there’s a bridge rather than a boat. Four people get to this bridge at night, but the bridge can only hold two people and there’s only one torch. The added complexity here is that each person takes different amounts of time to cross—Person A takes one minute, B takes two, C takes five and D takes eight. When two people cross, the slow-poke holds them up, so they can only travel as fast as the slowest crosser.

Perhaps you prefer missionaries and cannibals? Retronaut sums up this version:

Three cannibals and three missionaries arrive at the bank of a river which they must somehow cross. There is but one boat. This boat will carry but two people. Of the missionary group all three can row, but only one of the cannibals can row. In no case can there be a greater number of cannibals than missionaries left on either bank of the river. The number of missionaries in all cases must equal of exceed the number of cannibals.

The Physicist Karen Lingel wrote a poem about the problem involving four hungry men:

Four men start out to cross the sea
And yet they all walk different speeds!
The first, a sprinter, he goes fast
He leaves the others in the past!
The second takes a bit more time

The third’s a somewhat pokey man
He strolls along, sees what he can.
The last one is so very slow
You’d think he had no place to go!

So now they come upon a bridge
And on the other side — a fridge!
Well — you know men — they’ve gotta see
What’s inside the fridge to eat!
One flashlight is the light they’ve got
To guide them to the eating spot.
The batteries will only last
Seventeen minutes — that’s a fact.
The bridge, alas, — and here’s the trap –
Is apparently a piece of crap.

So only two men at a time
can cross the bridge — or they’ll sink in brine!
How can they all then make the trip?
And use the light so no one slips?
Send the fast guys first across
The fastest returns with little loss.
The pokey ones are next to go
While Fast Guy waits (they sure are slow)
Then send the other fast guy back
To get his friend and complete the pack.

Here are even more versions of the puzzle, from the University of Bielefeld Mathematics department.

The answers to all these puzzles can easily be found online, so we won’t ruin them here for you. But these classic logic puzzles are useful not just to keep you busy for a while trying to figure them out, but also to programmers. In fact, Microsoft apparently asked a variation of this question to potential employees:

I must warn you, you can really get caught up trying to solve this problem. Reportedly, one guy solved it by writing a C program, although that took him 37 minutes to develop (compiled and ran on the 1st try though). Another guy solved it in three minutes. A group of 50, at Motorola, couldn’t figure it out at all. See how long it takes you.

They asked:

U2 has a concert that starts in 17 minutes and they must all cross a bridge to get there. All four men begin on the same side of the bridge. You must help them across to the other side. It is night. There is one flashlight. A maximum of two people can cross at one time. Any party who crosses, either 1 or 2 people, must have the flashlight with them. The flashlight must be walked back and forth, it cannot be thrown, etc. Each band member walks at a different speed. A pair must walk together at the rate of the slower man’s pace:

Bono:- 1 minute to cross

Edge:- 2 minutes to cross

Adam:- 5 minutes to cross

Larry:- 10 minutes to cross

For example: if Bono and Larry walk across first, 10 minutes have elapsed when they get to the other side of the bridge. If Larry then returns with the flashlight, a total of 20 minutes have passed and you have failed the mission.

How fast can you solve these?