**Published on** February 20th, 2013 |
*by Simi*

# AustMS Gazette Puzzle Corner 30

Welcome to the Australian Mathematical Society Gazette’s Puzzle Corner No. 30. 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 importance: 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 30 is 28 February 2013. The solutions to Puzzle Corner 30 will appear in Puzzle Corner 32 in the May 2013 issue of the Gazette.

**Peculiar pace**

Jodie jogged for 25 minutes. In any 10-minute period, her average speed was 18 kilometres per hour. How far did she run?

**Rolling roadblocks**

There are 10 cars on an infinitely long, single-lane, one-way road, all travelling at different speeds. When any car catches up to a slower car, it slows down and stays just behind the slower car without overtak- ing. Eventually, the cars form a number of separate blocks. On average, how many blocks do you expect to see?

**Polynomial parity**

*Submitted by Alexander Hanysz*

(i) Let P and Q be complex polynomials with no common factors. Suppose the rational function P/Q is an even function. Prove that P and Q are both even functions.

(ii) Let P , Q and R be complex polynomials. Suppose P Q, P R and QR are all even functions. Prove that either P, Q and R are all even functions, or they are all odd functions.

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**Squaring a pentagon
**

*Submitted by Rory Tarnow-Mordi*

Let ABCDE be a pentagon with AB = BC, CD = DE and ∠B = ∠D = 90◦. Can you cut the pentagon into three pieces and then rearrange them to form a square?

**Team tactics
**

*Submitted by Ross Atkins*In a game show, a team of n girls is standing in a circle. When the game starts, either a blue hat or a red hat is placed on the head of each girl. Due to the set-up of the stage, each girl can only see the hats of the two adjacent girls, but not her own hat nor the hat of anyone else. Without any communication, the girls have to simultaneously guess the colour of their own hats. The team wins if and only if everyone guesses correctly.

Before the show, the girls try to devise a strategy to maximise their probability of winning. What is the maximum probability of winning if

(i) n=3?

(ii) n = 4?

(iii) n = 5?

*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 *AustMS Gazette, *puzzles are created by Ivan Guo unless otherwise stated.

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