Monthly Archives: August 2011

River crossing (3) – couples

There is a third class of river crossing problems, which has the same old origin as those of the farmer crossing the river: the publication of medieval manuscript Propositiones ad Acuendos Juvenes, which is generally attributed to generally attributed to Abbott Alcuin.

The original puzzle was:
1) Three couples want to cross a river. The boat they have available is small and can hold only 2 people.
A complication is that the three men are extremely jealous, and don’t want any man or men to be with their wife on one of the two shores if he is not there himself.
How many trips does it take them to all get across the river?

This puzzle was described as “quite sexist” by David Singmaster in his lecture “The utility of recreational mathematics”. He made the statement during the Eugene Strens memorial conference on recreational mathematics and its history, the proceedings of which were written by Richard K. Guy, Robert E. Woodrow. Part of these proceedings can be read here:
I think it is sexist in the sense that the puzzle makes a difference between men and women, and that the men are depicted as jealous where the women are not described as such. But frankly I don’t see any discrimination in it in the sense that women are regarded as less than the men. Also, the roles can easily be reversed. The only discrimination aspect I could find in it is that the women do not protest against the jealousy of their husbands.

For the solution, see solution 101

There is a variation of this puzzle, usually termed “missionaries and Cannibals”.
2) 3 Missionaries and 3 Cannibals want to cross a river. There boat can hold only 2 people at any time. The missionaries do not have the courage to be a minority on either shore for fear of being eaten.
How many trips does it take them to all get across?
During the same lecture David Singmaster called this one racist. An understandable accusation, though the puzzle does not mention any race of the cannibals. They might as well be the inhabitants of a secret village in mid-Europe. It also does not mention any religion of the missionaries, they might be Christian, as most people probably will imply, but they might as well be Islamic, though the word misionary is usually associated with the first.

For the solution, see solution 91

3) Just two rowers
There is a version with a small additional constraint: Only one of the cannibals and all of the missionaries can row. I will not give the solution here.

4) A boat for three.
Dudeney comes up with an enlarged boat:
During certain local floods five married couples found themselves surrounded by water, and had to escape from their unpleasant position in a boat that would only hold three persons at a time. Every husband was so jealous that he would not allow his wife to be in the boat or on either bank with another man (or with other men) unless he was himself present. Show the quickest way of getting these five men and their wives across into safety.
Call the men A, B, C, D, E, and their respective wives a, b, c, d, e. To go over and return counts as two crossings. No tricks such as ropes, swimming, currents, etc., are permitted.
The problem of larger boats was later fully analyzed by a mathematician, whose name has eluded me.
You can find the solution at solution 111

5) The four elopements
Sam Loyd made a classic expansion, introducing an island. Dudeney came also up with this variation, which might be an indication that they both got it from a third source. Here the puzzle is presented in Dudeneys words:
Colonel B—— was a widower of a very taciturn disposition. His treatment of his four daughters was unusually severe, almost cruel, and they not unnaturally felt disposed to resent it. Being charming girls with every virtue and many accomplishments, it is not surprising that each had a fond admirer. But the father forbade the young men to call at his house, intercepted all letters, and placed his daughters under stricter supervision than ever. But love, which scorns locks and keys and garden walls, was equal to the occasion, and the four youths conspired together and planned a general elopement.
At the foot of the tennis lawn at the bottom of the garden ran the silver Thames, and one night, after the four girls had been safely conducted from a dormitory window to terra firma, they all crept softly down to the bank of the river, where a small boat belonging to the Colonel was moored. With this they proposed to cross to the opposite side and make their way to a lane where conveyances were waiting to carry them in their flight. Alas! here at the water’s brink their difficulties already began.
The young men were so extremely jealous that not one of them would allow his prospective bride to remain at any time in the company of another man, or men, unless he himself were present also. Now, the boat would only hold two persons, though it could, of course, be rowed by one, and it seemed impossible that the four couples would ever get across. But midway in the stream was a small island, and this seemed to present a way out of the difficulty, because a person or persons could be left there while the boat was rowed back or to the opposite shore. If they had been prepared for their difficulty they could have easily worked out a solution to the little poser at any other time. But they were now so hurried and excited in their flight that the confusion they soon got into was exceedingly amusing—or would have been to any one except themselves.
As a consequence they took twice as long and crossed the river twice as often as was really necessary. Meanwhile, the Colonel, who was a very light sleeper, thought he heard a splash of oars. He quickly raised the alarm among his household, and the young ladies were found to be missing. Somebody was sent to the police-station, and a number of officers soon aided in the pursuit of the fugitives, who, in consequence of that delay in crossing the river, were quickly overtaken. The four girls returned sadly to their homes, and afterwards broke off their engagements in disgust.
For a considerable time it was a mystery how the party of eight managed to cross the river in that little boat without any girl being ever left with a man, unless her betrothed was also present. The favourite method is to take eight counters or pieces of cardboard and mark them A, B, C, D, a, b, c, d, to represent the four men and their prospective brides, and carry them from one side of a table to the other in a matchbox (to represent the boat), a penny being placed in the middle of the table as the island.
Readers are now asked to find the quickest method of getting the party across the river. How many passages are necessary from land to land? By “land” is understood either shore or island. Though the boat would not necessarily call at the island every time of crossing, the possibility of its doing so must be provided for. For example, it would not do for a man to be alone in the boat (though it were understood that he intended merely to cross from one bank to the opposite one) if there happened to be a girl alone on the island other than the one to whom he was engaged.

7) Summer Tourists
Sam Loyd, as number 207 of his Cylopedia of Puzzles, comes up with a small variation:

As a preface to a very interesting problem which shows how a company of very quarrelsome picnickers might cross a stream in the same boat without upsetting it, I shall take for granted that all puzzlists, young and old, are familiar with the ingenious tactics of the boatman who had to ferry a fox, a goose and some corn across a river in a small boat “built for two”.
There is a German version of the story which tells of a peasant with a wolf, a goat and, I think a tomato can, which he was to get across the river in a way to circumvent the wolf’s love for goat meat, as well a the natural tendency of the tomato can to telescope into the goat. Eithe of the stories, a familiarly told, possesses interest for the juveniles, and when solved would strengthen a branch of the memory and reasoning powers not generally called into exercise. To a trained puzzlist the problem poses no problem whatsoever, but to one not versed in such matters, if he will just try to run the solution through his mind to test mentally just how many times the boat must cross the river, he will speedily realize what a valuable school it is for learning to concentrate the thoughts.
I wonder, however, if any of our readers who are familiar with both stories have chanced to realize what a funny state of affairs might arise if the two incidents were combined in one? That is a trick I often resort to when I have a couple of easy puzzles which are susceptible of being twisted together into one genuine poser.
Added by the accompanying picture, which explains the situation in a way that words would fail to do, we will tell the story of a party of tourists, who, returning from a picnic were compelled to cross a stream in a small boat, which would hold but two at a time, and none of the ladies could row.
It so happened that Parson Cinch, the popular coloured preacher, has quarreled with the other two gentlemen, and as a result Mrs. Cinch had a falling out with the other ladies.
How is it possible for the gentlemen to conduct them all across the stream in such a way that no two disagreeing parties shall ever cross over together or either remain on either side of the stream at the same time. Another curious feature of the strained relations being that no one gentlemen should remain on either side with two ladies.
The puzzle is merely to show how many times the little two seated boat must cross the stream, to ferry the entire party over; but I take occasion to say that not one person out of a thousand is endowed with a headpiece which would figure it out mentally, without recourse to pencil and paper, although the faculty of doing so may readily be acquired.

In the unlikely case that you are hopelessly lost, peek here.

8) The four elopements

Of course all good puzzlists are familiar with the time honoured problem of the countryman who had to ferry a fox, goose and some corn across a river in a boat which would carry but two at a time. The story of the four elopements, equally old, is built upon similar lines, but presents so many complications that the best or shortest answer seems to have been overlooked by mathematicians and writers upon the subject.
It is told that four men eloped with their sweethearts, but in carrying out their plan they were compelled to cross a stream in a boat which would hold only two at a time. It appears that the young men were so extremely jealous that not one of them would permit his prospective bride to remain at any time in the company of any other man or men unless he was also present.
Nor was any men to get in the boat alone, when there happened to be a girl alone or on the island or shore, other than the one to whom he was engaged. This feature of the condition looks as if the girls were also jealous and feared that their fellows would run off with the wrong girl if they got a chance. Well, be that as it may, the problem is to guess the quickest way to get the whole party across the river according to the conditions imposed. let us suppose the river is two hundred yards wide, with the island in the middle. How many trips would the boat make to get the four couples safely across in accordance with the stipulations?
Don’t spoil puzzle fun to quick by peeking at the solution

9) Four jealous couples and a picnic

It occurred to me that Sam Loyds puzzle with four couple and an island, described fully above, can be taken slightly further. Again we have four couples, a boat for 2 persons to cross the river, and an island.
Again the boys have that awful stroke of jealousy, which forbids them to let one of their friends to be with their girlfriend if he is not present himself.
But in addition, the girls have prepared a huge bag with food for the pick-nick. The bag is so heavy that it wont fit into the boat if it already holds two persons, it has to be ferried across by one person. The food smells so good that girls don’t allow the boys to be with the bag with food if not at least one girl is there to keep watch on the food.

You can find the solution at number 141

UPDATE 2016, December 17
Peter Rowlett ran an interesting discussion on the sexist nature of the three couples puzzle. One reformulation of the puzzle, which maintains the personal relationship aspect, is:
Three actors and their three agents want to cross a river in a boat that is capable of holding only two people at a time, with the constraint that no actor can be in the presence of another agent unless their own agent is also present, because each agent is worried their rivals will poach their client. How should they cross the river with the least amount of rowing?
The variation was suggested by James Summer.