15 hikers arrive at a river they need to cross. Only 12 boats are available.

Question

15 hikers arrive at a river they need to cross. Only 12 boats are available. Which of the following statements must be true/false? Explain your answers for each statement.

a. Some boat will be used by 2 hikers.
b. Some boat will be used by two or more hikers.
c. Three boats will be used by two or more hikers.
d. Every boat will be used.

Answer 1

If you think about it, people could go all 3 to a boat or 1 to a boat, so a does not have to be true. It could be, but not necessarily.

 

b is true, because there is no way to have all the hikers cross together without someone doubling up in some configuration.

 

c is true if they put the 3 extras into separate boats, but it is not necessarily true.

 

You would think d would be true, but maybe they will just use 5 boats with 3 people apiece. No one said how big the boats are.

 

So the only one you can actually say anything for certain about is b, which is true. (you can add a caveat that if they take multiple trips you could avoid b being true, if there are other people also manning the boats, or someone goes back and pulls empty boats with them. But this is just to cover your bases.)

Answer 2

In all logic questions asking whether something "must be" true or false, it is usually easiest to try to figure out any exceptions to the statement.  In other words, if something "must be true," then to disprove it, we only need one example of its being false.  This is usually much easier than trying to prove that the statement is true.  Since your question requires that statements are either 100% true or false, you must look for exceptions to either case:

 

a. Is it 100% true? No, because they could just have all 15 hikers on one boat.  (Caveat: If you read the statement as "Some boat will be used by 2 hikers, but there may also be other hikers on the boat," then this is 100% true (see b. below); If you read the statement as "Some boat will be used by exactly 2 hikers," then this could be false.).

   Is it 100% false? No, two hikers could definitely be on one boat: 2 hikers on three boats; 1 hiker on the other 9 boats.

 

b. Is it 100% true? Yes, because there are more hikers than boats.  (Note: this is true even if you can make more than one trip across, unless the boats can return to the first side by themselves.)

   Is it 100% false? No, two or more hikers could definitely be on one boat: all 15 on one boat.

 

c. Is it 100% true? No, 15 hikers on one boat.
   Is it 100% false? No, 2 hikers on three boats; 1 hiker on the other 9 boats.

  

d. Is it 100% true? No, 15 hikers on one boat.
   Is it 100% false? No, 2 hikers on three boats; 1 hiker on the other 9 boats.

 

Some amount of proof will sometimes be necessary (see b. above) and even easy, but it is very often easier to disprove a 100%/must be/always statement.

 

Cheers,

Ashley

 

PS - Look up "scientific method" for more on this "only one exception" principle.

 

Answer 3

A. True. (Pigeonhole principle: http://en.wikipedia.org/wiki/Pigeonhole_principle)

B. True. B is a consequence of A. In other words, if two people have to use a boat, then two or more are using that boat.

C. False. Although the best way to distribute the hikers would be evenly (first 12 all get their own boat, then the remaining three hop on three different occupied boats), The problem doesn't say how many hikers each boat can hold. They could all be on the same boat for instance. In this case only one boat has two or more hikers.

D. False. Same reasoning as above.

 

If you are wondering, this is an introduction into functions and their domain, co-domain, and range. The hikers representing a domain with 15 element, and the boats representing a co-domain with 12 elements. (The boats that are actually used represent the range, a subset of the co-domain).

A and B are asking, could the function be injective (1-1). D is asking, could the function be surjective (unto).