## Le Monde puzzle [#1078]

**R**ecalling Le Monde mathematical puzzle first competition problem

Given yay/nay answers to the three following questions about the integer 13≤n≤1300 (i) is the integer n less than 500? (ii) is n a perfect square? (iii) is n a perfect cube? n cannot be determined, but it is certain that any answer to the fourth question (iv) are all digits of n distinct? allows to identify n. What is n if the answer provided for (ii) was false.

When looking at perfect squares less than 1300 (33) and perfect cubes less than 1300 (8), there exists one single common integer less than 500 (64) and one single above (729). Hence, it is not possible that answers to (ii) and (iii) are both positive, since the final (iv) would then be unnecessary. If the answer to (ii) is negative and the answer to (iii) is positive, it would mean that the value of n is either 512 or 10³ depending on the answer to (i), excluding numbers below 500 since there is no unicity even after (iv). When switching to a positive answer to (ii), this produces 729 as the puzzle solution.

Incidentally, while Amic, Robin, and I finished among the 25 ex-aequos of the competition, none of us reached the subsidiary maximal number of points to become the overall winner. It may be that I will attend the reward ceremony at Musée des Arts et Métiers next Sunday.

November 30, 2018 at 9:54 am

Do you have to buy the paper edition to get the puzzles ?

ps: you should add some contact information. It is not easy to contact you from R-bloggers posts.

November 30, 2018 at 10:34 am

Most of the time, the puzzles appear the week after on the website. Most of the time.

November 29, 2018 at 7:38 pm

But sqrt(729)=27 and the puzzle said n is not a perfect square…

November 30, 2018 at 8:34 am

Thank you, Gustavo, the way I understand it, the puzzle states that with genuine answers to all questions, the answer to (ii) must be negative and the values of n are either 512 or 1000 depending on the answer to (iv). But the final step is that the answer to (ii) was wrong, thus that n is a perfect square as well. Does this sound a reasonable explanation?