**Published on** September 11th, 2013 |
*by Daphane Ng*

# AustMS Gazette Puzzle Corner 34

Welcome to the Australian Mathematical Society Gazette’s Puzzle Corner number 34. 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 ivanguo1986@gmail.com or send paper entries to: Gazette of the Australian Mathematical Society, School of Science, Information Technology & Engineering, University of Ballarat, PO Box 663, Ballarat, Vic. 3353, Australia.

The deadline for submission of solutions for Puzzle Corner 34 is 1 November 2013. The solutions to Puzzle Corner 34 will appear in Puzzle Corner 36 in the March 2014 issue of the Gazette.

Notice: If you have heard of, read, or created any interesting mathematical puzzles that you feel are worthy of being included in the Puzzle Corner, I would love to hear from you! They don’t have to be difficult or sophisticated. Your submissions may very well be featured in a future Puzzle Corner, testing the wits of other avid readers.

**Crowded square**

There are four points inside an 8 metres by 8 metres square. Prove that two of those points are at most √65 metres apart.

**Fraction practice 2**

Franny is practising her fractions again. She begins with the numbers written on the board.

At each turn, Franny may erase two numbers a, b and replace them with a single number f(a,b). This is repeated until only one number remains.

(i) If f (a, b) = ab/(a + b), what are the possible values of the final number?

(ii) If f (a, b) = ab + a + b, what are the possible values of the final number?

**Prickly pair**

I am thinking of a pair of positive integers. To help you work out what they are, I will give you some clues. Their difference is a prime, their product is a perfect square, and the last digit of their sum is 3. What can they possibly be?

**Tessellation test
**

Tess is trying to draw an n-sided convex polygon which can be tessellated by a finite number of parallelograms. For which n will Tess be able to succeed?

**Diminishing differences
**

Begin with n integers x1, . . . , xn around a circle. At each turn, simultaneously replace all of them by the absolute differences

Repeat this process until every number is 0, then stop. Prove that this process always terminates if and only if n is a power of 2.

**About the author**

Ivan is a PhD student in the School of Mathematics and Statistics at The University of Sydney. His cur- rent 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 *AustMS Gazette, *puzzles are created by Ivan Guo unless otherwise stated.

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