# Somerset Sport

##### By Candice McKenna - Director of Sport

Well done to students and staff on Season 2 of APS sport. The effort contributed by all to weekly practise, and the energy committed to Friday sport is appreciated.

Andrew Wrigley, Senior School Teacher

Term One, 2017

2016 was a wonderful year mathematically, as 16 is both a square number (4^{2}) and quite neatly a fourth power (2^{4}). This threw up a number of interesting dates; there was Square Root Day on 4 April (4/4/16), ANZAC Day was a Perfect Square Day (25/4/16) while 4 August was Exponent Day 2^{2}/2^{3}/2^{4}.

The 9 October (9/10/16) was a Rotational Symmetry Day where a 180^{o} rotation of the numbers around the zero gets the same ones back. We also had a number of Happy Days where the day and the month multiply together to make the year (16/1/16, 8/2/16, 4/4/16, 2/8/16).

Sadly, there will be only one Happy Day in 2017 (17 January) as 17 is a prime number (it has no factors other than 1 and itself). Interestingly, some species of cicadas in North America have a life cycle of 17 years, spending most of their life underground as a nymph and only a few weeks as an adult. They use this property of prime numbers in order to deter predators from wiping them out. Predators with a 1 year life cycle would only coincide with the adult cicadas every 17 years. A 2 year life cycle predator would coincide every 34 years, while one with a 3 year life cycle would be 3 x 17 = 51 years.

2017 is also a prime number and is the sum of two squares 2017 = 44^{2} + 9^{2} and as 9 is a square number 2017, it is also the sum of a square and a fourth power - 2017 = 44^{2} + 3^{4}. Furthermore, 2017 is the sum of three cubes: 2017 = 11³ + 7³ + 7³. It is found in a Pythagorean Triple 792² + 1855² = 2017² and indeed 15 August this year (15/8/17) will be a Pythagorean Triple Day as 15^{2} + 8^{2} = 17^{2}.

2017 is a member of what is known as The Lazy Caterer’s Sequence which can be formulated as follows. Imagine a pancake being cut once - it produces two pieces, 2 cuts will produce 4 pieces, 3 cuts = 7 pieces (see Figure One) and so on up to 63 cuts (if possible) to give 2017 pieces.

*Figure one: Pancake with 3 cuts gives 7 pieces*

The sequence actually starts at 1 (zero cuts) and develops as follows, 1, 2, 4, 7, 11, 16, 22, 29, 37, ... 2017

Or alternatively as a series, 1 + 1 + 2 + 3 + 4 + 5 + 6 ... + 63 = 2017

For those who like to see it written as an equation we could write it; P = ^{(n^2+n+2)}/_{2} where p is the maximum number of pieces and n is the number of cuts.

So for 63 cuts;
P = (63^2+63+2)/2

P = 2017

On the New Year’s Eve countdown to 2017, I enjoyed this sequence from 10 to 0 where each number used all the digits of 2017.

10 = 2+0+1+7

9 = 2+0+1x7

8 = 2^0+1x7

7 = 2x0x1+7

6 = -(2+0-1-7)

5 = -(2-0x1-7)

4 = -(2+0+1-7)

3 = 2+0+17

2 = 2+0x1x7

1 = 2^(0^(1^7))

0 = 2x0x1x7

… but wait a minute … an extra leap second was added to the clock at the end of 2016 to keep our planet’s rotation in line with atomic clocks, so the countdown should have started at 11 (= 2 + 0! + 1 +7).

And here is a pandigital representation of 2017, 2017 = (10 x 9 x 8 x 7 x 6 )/((5+4+3+2+1) ) + 0!

Oh! Happy New Year!

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