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Senior boys get UCAS sorted - the City of London School

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Friday October 15 2009<br />

Maths puzzle<br />

Daniel Hu 5H Maths Puzzler<br />

Quadratic Residues<br />

You are given that (a) squared plus (b) squared equals (c) squared.<br />

Prove that one <strong>of</strong> a, b, and c is a multiple <strong>of</strong> 3 using <strong>the</strong> solution to last week’s puzzle.<br />

Notice that if you square any integer, you can only <strong>get</strong> an integer which ei<strong>the</strong>r is a multiple<br />

<strong>of</strong> 3, or is 1 more than a multiple <strong>of</strong> 3 i.e. (3n) or (3n+1). You cannot <strong>get</strong> (3n+2).<br />

Solution to last week’s puzzle<br />

1) What are <strong>the</strong> next three square numbers after 1, 4, 9? 16, 25, 36<br />

2) What are <strong>the</strong>ir remainders when you divide by 3? 1, 1, 0<br />

3) Do you notice a pattern? Can you prove it?<br />

The sequence goes 1, 1, 0, 1, 1, 0 etc.<br />

(3n+1) squared is 3(3n^2+2n)+1 so it is 1 more than a multiple <strong>of</strong> 3 and has remainder 1<br />

when you divide by 3<br />

Therefore if you square any number 1 more than a multiple <strong>of</strong> 3, you <strong>get</strong> a number 1<br />

more than a multiple <strong>of</strong> 3 which has remainder 1 when you divide by 3.<br />

Try to do <strong>the</strong> same with (3n+2) and (3n). You should <strong>get</strong> remainder 1 and 0 respectively.<br />

Standing at 2,228 metres (7,310 ft) above sea level, Mount Kusciuszko was last<br />

week’s mountain and is <strong>the</strong> tallest mountain on <strong>the</strong> Australian mainland, although<br />

<strong>the</strong>re is a taller one on <strong>the</strong> island <strong>of</strong> Tasmania. It was named by a Polish explorer,<br />

after General Kusciuszko who featured in <strong>the</strong> American Civil war. Despite <strong>the</strong> height,<br />

it is a relatively easy mountain to climb, or <strong>the</strong>re is a road to <strong>the</strong> summit for <strong>the</strong> less<br />

active.<br />

Which is this mountain?<br />

1. This mountain was successfully climbed <strong>the</strong> day before <strong>the</strong> coronation <strong>of</strong> Queen<br />

Elizabeth II.<br />

2. The Olympic torch passed over <strong>the</strong> summit <strong>of</strong> this mountain in 2008.<br />

3. It is <strong>the</strong> home <strong>of</strong> <strong>the</strong> black jumping spider, thought to be <strong>the</strong> highest altitude<br />

macroscopic animal on <strong>the</strong> planet. It survives by eating frozen insects that have been<br />

blown upslope.<br />

4. Its height has been downgraded several times over <strong>the</strong> years due to continual<br />

overestimation.<br />

PSM<br />

Classics<br />

Challenge<br />

Page 11<br />

Puzzle Page<br />

The answers to <strong>the</strong> Greek mythology Challenge<br />

are: (a) Hercules (b) Prome<strong>the</strong>us (c)<br />

Daedalus (d) Sisyphus (e) Jocasta.<br />

Congratulations to <strong>the</strong> <strong>Senior</strong> and Junior<br />

winners, Patrick Benjamin in 5H and Ariel<br />

Barton in OGM. £10 Bookshop tokens are<br />

on <strong>the</strong>ir way (real not mythical tokens).<br />

This week’s <strong>the</strong>me is ano<strong>the</strong>r perennial favourite:<br />

<strong>the</strong> Trojan War. So reach for your<br />

Iliad and try to answer <strong>the</strong> following:<br />

(a) Who shoots but does not kill Paris with<br />

an arrow?<br />

(b) One <strong>of</strong> <strong>the</strong> rivers <strong>of</strong> Troy is Scamander.<br />

Name <strong>the</strong> o<strong>the</strong>r.<br />

(c) When Agamemnon gives back Chryseis,<br />

who does he take instead?<br />

(d) Who kills Patroclus? (Hint: <strong>the</strong> answer<br />

is not Apollo!)<br />

(e) What sort <strong>of</strong> tree do Achilles and Hector<br />

run past in <strong>the</strong>ir fateful encounter?<br />

Answers on paper to any member <strong>of</strong> <strong>the</strong><br />

Classics Department by <strong>the</strong> end <strong>of</strong> next<br />

week (i.e. Friday 22 October). Correct<br />

answers will go into a hat, and <strong>the</strong> winner<br />

will be chosen by lot. Prize: a £10 Bookshop<br />

voucher.<br />

CITI-DOKU

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