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

Earlier today I solved these three problems for you 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 provided. Nevertheless, it is possible to deduce the answer. If (d) is true then (b) is true and therefore (d) is false. Since (d) is false, (a) is false. If (c) is true, then (e) is true since we know that (d) is not, and so (c) is actually false. Since (c) is false, all of the following are false: that is, (e) is false. Therefore (b) is true.

Now we need to make sure 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. Pin blocks

Pins are arranged in a rectangular grid on a board, and rubber bands can be placed around the pins to form geometric shapes. The figure above shows how to construct squares of areas 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 points

Alexander, David, Esther, Jacinda and Simon all got different marks in the maths test which was held unexpectedly last week. In the following dialogue, students are either true or false, and those students who made correct statements always scored higher than those who made incorrect statements.

Simon: Alexander and Esther took the top two spots.

Jacinda: No, what Simon just said is wrong.

David: I was ranked between Simon and Jacinda.

Alexander: Jacinda came second.

Jacinda: I scored less 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 Simon’s statement is correct. Then Alexander came higher on the list than Simon and therefore must also have spoken correctly. But this is impossible as it would mean that Alexander, Esther and Jacinda each occupy one of the top two places. So Simon must have been wrong. This means that Jacinda’s first statement is true, and so her second statement must also be true. So Esther came before Jacinda. To summarize what we know so far, (part of) the sequence of points

…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 comments, one is true and one is false. If David was correct and Alexander wrong, then David came under Jacinda, and so did Alexander (since his statement was false); so Jacinda came second and Alexander’s statement was true after all. So 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 out Parabolic problems, a compilation of more than 300 of the magazine’s best mysteries from the past 60 years. Worth reading!

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

I’ve been doing a puzzle here on alternate Mondays since 2015. I’m always on the lookout for great puzzles. If you want to suggest one, email me.