Did you solve it? Can you outwit the wizards of Oz? | Mathematics

Earlier today I set you these three problems from Parabola, a wonderful magazine from Australia that was first published 60 years ago this month. Here they are again with solutions.

1. The question with no question

(a) All of the following.

(b) None of the following.

(c) Some of the following.

(d) All of the above.

(e) None of the above.

[Just to reassure you, nothing has been omitted here.]

Solution

This is a multiple choice question with no information given. Nevertheless it is possible to deduce the answer. If (d) is true then (b) is true and so (d) is false after all. Since (d) is false, (a) is false. If (c) is true, then (e) is true since we know that (d) is not, and hence (c) is actually false. Since (c) is false, all those following are false: that is, (e) is false. Hence, (b) is true.

Now we have to check that all the true and false specifications are consistent; otherwise, there will be no solution to the problem. This is easily done, and so the answer is (b).

2. Peg squares

Pegs are arranged in a rectangular grid on a board, and rubber bands can be placed around the pegs to form geometrical shapes. The figure above shows how to construct squares of area 1 and 5 using pegs and rubber bands.

Show how to construct squares of areas 8 and 10.

[Knowledge of the Pythagorean theorem may be useful.]

Solution

3. Groucho marks

Alexander, David, Esther, Jacinda and Simon all received different marks in the maths test which was held unexpectedly last week. In the following dialogue, either students are truthful or not, and those students who made correct statements invariably had obtained higher marks than those who made incorrect statements.

Simon: Alexander and Esther gained the top two places.

Jacinda: No, what Simon just said is wrong.

David: I was ranked in between Simon and Jacinda.

Alexander: Jacinda came second.

Jacinda: I scored fewer marks than Esther.

Esther: Exactly three of the previous five statements are correct.

Find the order in which the students finished.

Solution Esther, Jacinda, Alexander, Simon, David

Suppose that Simon’s statement is correct. Then Alexander came higher up the list than Simon and therefore must also have spoken correctly. But this is impossible since it would mean that Alexander, Esther and Jacinda each occupy one of the top two places. Thus, Simon must have been wrong. This means that Jacinda’s first statement is true, and so her second statement must be true too. Thus, Esther came ahead of Jacinda. To summarise what we know so far, (part of) the order of marks is

…Esther…Jacinda…Simon…

and Jacinda made two correct statements, Simon one incorrect statement. Since three of the first five statements are true, we see that of Alexander’s and David’s remarks one is true and one false. If David was correct and Alexander incorrect, then David came below Jacinda, and so did Alexander (since his statement was false); thus, Jacinda came second and Alexander’s statement was true after all. Thus, David must have made a false statement and finished last, while Alexander made a true statement and came third.

Thanks to Parabola for these puzzles. They are taken from Parabolic Problems, a compilation of more than 300 of the magazine’s best puzzles from the last 60 years. Well worth a read!

The first issue of Parabola came out in July 1964, published by the University of New South Wales, Sydney. It continues to exist online as a free resource. Aimed at sixth form students (ages 16-18), teachers and enthusiasts, it has the format of a scholarly journal, with fascinating papers on a wide variety of mathematical fields.

I’ve been setting a puzzle here on alternate Mondays since 2015. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.