Solutions to Brain-Twisters V

The solutions too will often involve the use of figures and tables. As in the puzzles page, the figures are in gif format and should be downloadable by text-based browsers like Lynx.

Solution to Puzzle #1

The first student was right, and here is why. From the first statistician's report it follows that there cannot be more than one yellow flower, because if there were two yellows, you could pick two yellows and one blue, thus having a group of three flowers that contained no red. This is contrary to the report that every group of three is bound to contain at least one red flower. Therefore there cannot be more than one yellow flower. Similarly, there cannot be more than one blue flower, because if there were two blues, you could pick two blue flowers and one yellow and again have a group of three that contained no red. And so from the first statistician's report it follows that there is at most one yellow flower and one blue. And it follows from the report of the second statistician that there is at most one red flower, for if there were two reds, you could pick two reds and one blue, thus obtaining a group of three that contained no yellow. It also follows from the second report that there cannot be more than one blue, although we have already deduced this from the first report.

The upshot of all this is that there are only three flowers in the entire garden -- one red, one yellow, and one blue! And so it is of course true that whatever three flowers you pick, one of them must be blue.

Solution to Puzzle #2

If you want to win Prize A, what you should say is: "I will not get Prize B." What can I do? If I give you Prize C, then your statement has turned out to be true -- you didn't get Prize B -- so I have given you the booby prize for making a true statement, which I cannot do. If I give you Prize B, then your statement has turned out to be false, but I can't give you Prize B for having made a false statement. Therefore I am forced to give you Prize A. You have then made a true statement -- you didn't get Prize B -- and have accordingly been awarded one of the two prizes offered for making a true statement.

Of course the statement "I will get either Prize A or Prize C" also works.

Solution to Puzzle #3

To win Prize C, you need merely say: "I will get Prize D." Having worked through the solution of the previous puzzle, the proof of this should be obvious.

Solution to Puzzle #4

Only one of the four quoted statements is true.

(1) Suppose the first statement is true. Then we have two hatters (Mr D and Mr B). So this hypothesis is "out".

(2) Suppose the second statement is true. Then Mr G is d; Mr B is h; Mr H is b. It follows that Mr D is g.

(3) Suppose the third statement is true. Then Mr H is b. So Mr D is neither b, h, nor d, and, once again, must be g.

(4) Suppose the fourth statement is true. Now Mr B is h, Mr G is not h, d, or g, and so is b. Whence as before, Mr D is g.

Hence while we cannot with certainty identify any one of the others, we know that Mr. Draper is the grocer.

Solution to Puzzle #5

He ties the rope round the tree on the shore, and then carries the rope on a walk around the island. As he passes the halfway mark, the rope starts to wrap around the tree on the island, and when he reaches his starting point he ties the other end of the rope to the tree on the shore and pulls himself across on the rope.