Puzzles

Published on March 20th, 2013 | by Simi

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# AustMS Gazette Puzzle Corner 31

Welcome to the Australian Mathematical Society Gazette’s Puzzle Corner num- ber 31. Each puzzle corner includes a handful of fun, yet intriguing, puzzles for adventurous readers to try. They cover a range of difficulties, come from a variety of topics, and require a minimum of mathematical prerequisites for their solution. Should you happen to be ingenious enough to solve one of them, then you should send your solution to us.

For each puzzle corner, the reader with the best submission will receive a book voucher to the value of \$50, not to mention fame, glory and unlimited bragging rights! Entries are judged on the following criteria, in decreasing order of impor- tance: accuracy, elegance, difficulty, and the number of correct solutions submitted. Please note that the judge’s decision — that is, my decision — is absolutely final. Please email solutions to Ivan Guo.

The deadline for submission of solutions for Puzzle Corner 31 is 1 May 2013.

Rolling in riches

Place four \$1 coins as shown in the diagram below. Now roll the shaded coin anti- clockwise around the other three, touching them the entire time, until it returns to the original position. How much has the shaded coin rotated relative to its centre?

An ancient scroll is kept in a chest, which is locked by a number of padlocks. All padlocks must be unlocked in order to open the chest. Copies of the keys to the padlocks are distributed amongst 12 knights, such that any group of 7 or more knights can open the chest should they choose to do so, but any group of less than 7 cannot. What is the minimal number of padlocks required to achieve this?

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Stargazing

An astronomer observed 20 stars with his telescope. When he added up all the pairwise distances between the stars, the result was X. Suddenly a cloud obscured 10 of the stars. Prove that the sum of the pairwise distances between the 10 re- maining stars is less than 1/2 X.

Bonus: Can you improve the bound? What is the smallest real number r such that the new sum is always less than rX, regardless of the configuration of the stars?

Golden creatures
Submitted by Joe Kupka

At the beginning of time, in a galaxy far, far away, the Queen of Heaven gives birth to 40 golden creatures. On the last day of each year the King of Heaven sacrifices a randomly chosen creature to his own glory. After every 20 sacrifices, the Queen gives birth to 20 new creatures. Every creature lives until it is sacrificed by the King. Any creature who reaches 100 years of age receives a congratulatory letter from the Queen.

(i) What is the probability that a creature will receive a congratulatory letter?
(ii) How many congratulatory letters, on average, will the Queen write in the first 1000 years?
(iii) One of the first 40 creatures is named Adam. One of the 20 creatures born after 40 sacrifices is named Eve. What is the probability that Adam will outlive Eve?

Uncovered construction

Can you construct a set of 100 rectangles, with the property that not one of the rectangles can be completely covered by the the other 99?

Ivan Guo
Ivan is a PhD student in the School of Mathematics and Statis- tics at The University of Sydney. His current research involves a mixture of multi-person game theory and option pricing. Ivan spends much of his spare time playing with puzzles of all flavours, as well as Olympiad Mathematics.

The Puzzle Corner is a regular feature in the puzzles are created by Ivan Guo unless otherwise stated.

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