(Above: JP McCarthy, PhD student from UCC assists budding DCS mathematicians.)
Since September, Douglas Community School has been involved in running one of the first Maths Circles in Cork. These are after-school mathematics clubs for First Year students who have shown a particular interest in maths, and are run in conjunction with staff in UCC and CIT. Under the scheme, students from UCC are sent to any school interested in setting up a Maths Circle to help run the first number of classes. We were lucky enough to have J.P. McCarthy, currently studying for his PhD, who ran the first five sessions, in conjunction with teachers in the school. Since then, the Maths Circle has been run by Mr Ciarán Ó Conaill.
The classes are generally centred around problem-solving type questions, with students often working in groups of two or three rather than singly to solve puzzles. The aim of the classes is to continue to engage the more able students, and to challenge them in a subject they may find very easy in school. Given the enthusiastic responses from those involved, we seem to be achieving this aim!
All of the notes from the classes, as well as some supplementary material, are available online at oconailldcs.wikidot.com/mathscircles. We continue to be supported by UCC in the form of a dedicated website, www.mathscircles.com, which contains an ever-growing number of puzzles and resources for possible use in Maths Circles around the country, although it has to be said that DCS is one of the main contributors to the site in terms of adding puzzles and problems! We also look forward to having more students from UCC in the new year bring their enthusiasm and innovation to the students of Douglas Community School.
Ciarán Ó Conaill
Some Sample Problems:
- In a knock-out tournament with 50 teams, how many games will be played?
- Cork has about 270,000 inhabitants. How do you know for sure that at least two non-bald people in Cork have the same number of hairs on their head?
- How many squares are on a chessboard?
- A cable is made that runs all the way around the equator. After a few days, it’s decided that, instead of running along the ground, the cable needs to be 1 metre off the ground, all the way around the earth. How much extra cable is needed?
- Add up all the numbers from 1 to 100 without using a calculator.
- Make the number 1,000 using the number 8 eight times. You can use any operation (e.g. adding, subtracting, multiplying, etc).
- Each team has to lose 1 match. You need 49 losers, so you need 49 games.
- The most number of hairs a person can have is less than 200,000. So, the worst-case scenario is that you have 1 person with no hairs, 1 person with 1 hair, 1 person wih 2 hairs, etc, all the way to 1 person with 200,000 hairs. Now the next person must have some number of hairs between 0 and 200,000, so they have the same number of hairs as someone already counted.
- There’s one 8×8 square, four 7×7 squares, nine 6×6 squares, etc, up to sixty-four 1×1 squares. That’s a total of 204 squares.
- The circumference of a circle is , and the radius of the earth is roughly 6,400km, 6,400,000 m. Assuming that the equator is a circle, that gives a cable length of 12,800,000 m. To get the cable 1 m off the ground means using a radius of 6,400,001 m, which gives a length of 12,800, 002 m. That’s an extra length of 2 m, or 6.28 m. It’s a bit strange that this is so small, but what’s really odd is that the answer is the same, no matter what the initial radius. So the same would be true for the moon.
- This is an old problem. The trick is to pair off the first and last numbers, so you look at 1+100, 2+99, 3+98, etc. There are 50 of these pairs (the last one will be 50+51), and each pair adds to 101, so that’s a total of 50×101 = 5050.
- This requires a little, well, cheating. We need to use what’s called concatenation – joining numbers together. So one solution is: 888 + 88 + 8 + 8 + 8 = 1000.