Published on December 9th, 2013 | by Daphane Ng0
AustMS Gazette Puzzle Corner 35
Welcome to the Australian Mathematical Society Gazette’s Puzzle Corner number 35. 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 email@example.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 35 is 1 February 2014. The solutions to Puzzle Corner 35 will appear in Puzzle Corner 37 in the May 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.
Row of reciprocals
Harry writes down a strictly increasing sequence of one hundred positive integers. He then writes down the reciprocals of the integers.
(i) Is it possible for the sequence of reciprocals to form an arithmetic progression?
(ii) Apart from the last two reciprocals, is it possi- ble for each reciprocal to be the sum of the next two?
(iii) Would the answers to the previous questions change if Harry had started with an infinite sequence instead?
(i) There are several pebbles placed on an n × n chessboard, such that each pebble is inside a square and no two pebbles share the same square. Perry decides to play the following game. At each turn, he moves one of the pebbles to an empty neighbouring square. After a while, Perry notices that every pebble has passed through every square of the chessboard exactly once and has come back to its original position.
Prove that there was a moment when no pebble was on its original position.
(ii) Peggy aims to place pebbles on an n × n chessboard in the following way. She must place each pebble at the centre of a square and no two pebbles can be in the same square. To keep it interesting, Peggy makes sure that no four pebbles form a non-degenerate parallelogram.
What is the maximum number of pebbles Peggy can place on the chessboard?
Call a nine-digit number flawless if it has all the digits from 1 to 9 in some or- der. An unordered pair of flawless numbers is called harmonious if they sum to 987654321. Note that (a, b) and (b, a) are considered to be the same unordered pair.
Without resorting to an exhaustive search, prove that the number of harmonious pairs is odd.
There are some weights on the two sides of a balance scale. The mass of each weight is an integer number of grams, but no two weights on the same side of the scale share the same mass. At the moment, the scale is perfectly balanced, with each side weighing a total
of W grams. Suppose W is less than the number of weights on the left multiplied by the number of weights on the right.
Is it always true that we can remove some, but not all, of the weights from each side and still keep the two sides balanced?
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.