UT Math Club Problem of the Month 1 How Many Squares 2

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UT Math Club Problem of the Month

April 1, 2009 1 How Many Squares

Determine the number of squares contained in the sequence

{

1 + 17

n

2

| n ≥

1

}

=

{

18

,

69

,

154

, . . .

}

2 Murdoch Challenge

Australian-born global media mogul Rupert Murdoch writes down the se quence of positive integers from 1 to 10 on a blackboard in RLM 12.104.

Murdoch then challenges American media proprietor Ted Turner to the fol lowing game. The two take turns altering the sequence by erasing a pair of numbers and replacing them with the absolute value of their difference (i.e.

at each turn integers

m

and

n

are erased and the integer if they start with the sequence from 1 to 2009 instead?

| m − n |

is written down). They continue in this fashion until only one number is left written on the blackboard. Ted Turner wins if this number is even. Rupert Murdoch wins if this number is odd. Who can win? Prove your result. What happens Submission Deadline: Wednesday, April 22nd by 11:59 PM 1

3 April Rules

This month the Monthly Problem will include two separate puzzles. One puzzle will be an olympiad style problem and the other will be a brain teaser type of problem. The puzzles are independent. Hence you may submit a solution for either one or both of the puzzles. The winning solution to each puzzle earns a prize of $25. While you (or your team) are encouraged to submit solutions to both puzzles, any given team can win at most $25.

4 Criteria for Evaluation

Submissions will be evaluated based on correctness of the solution, elegance of the argument, and clarity of the exposition, as determined by the particular graduate students over-seeing the Monthly Problem. It will be our goal to select the best written of all the correct solutions submitted for the award.

5 Submission Guidelines

In order to submit a solution for evaluation, e-mail your solution as an attach ment to [email protected]

in one of the following formats: .txt, .pdf, .dvi, .jpg

. Include your full name in the body of the e-mail as well as in the header of your solution. Joint submissions will be accepted, but one member of the group must be chosen as the recipient of the prize in the event that the joint submission receives the award.

2

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