Mathematicaphiles more than sum of parts
Fancy yourself a bit of a maths whizz. How’s the below for a bit of a calculus quandary to ponder then?
There is a tower of identical double-spouted cups of volume one cubic unit. This tower has infinitely many rows, with each row having one more cup than the one above it. The cups are arranged in a triangular pattern with the spouts facing left and right (Figure 1). At the top of the tower, a leaky tap starts to drip into the top cup. When a cup is full, the overflowing liquid begins to drip into the two cups directly below it. At first, the tap drips at a rate of 0.004 cubic units per second.
After two full days, how many cups have been filled? What is the volume of liquid in each cup at the bottom-most row of non-empty cups?
Let Vi,j(t) be the volume of liquid at time t in a cup positioned at the ith row and jth column where i, j ≥ 1.
Is it possible to find a formula for Vi,j(t), where t is the number of seconds since the tap started leaking? If so, modify and test your formula for a different flow rate. If not, explain why.
To extend this problem, consider a new situation. Before the tap starts to drip, one cup in the tower is randomly replaced with a special siphon cup that is otherwise externally similar to the regular cups. When the liquid inside the siphon cup reaches 0.7 cubic units, it will start to drain into the cup directly below it (i.e. two rows down) at a rate of 0.5 cubic units per second. Whenever the siphon cup is drained completely, it will start to fill up again (until it reaches 0.7 cubic units, when it will start to drain again).
How do your responses to the above change? What if you replaced more of the original cups with siphon cups?
This brain breaker was just one of the mathematical and statistical conundrums posed to some of Newington’s 15 sharpest maths minds in this year’s Junior paper (Years 7 to 10) in the UNSW School of Mathematics Competition.
Running annually since 1962, the UNSW School of Mathematics Competition is open to secondary school students throughout New South Wales and the Australian Capital Territory and is staged in two divisions: Junior, from Year 7 up to and including Year 10, and Senior, Years 11 and 12.
Mr Luke Dudman, Assistant Head of Mathematics, says the competition is designed to assess mathematical insight and ingenuity rather than efficiency in tackling routine examples, and is acknowledged as ‘an extremely difficult competition’.
Each year, about 700 students participate with approximately half in each division, with prizes and/or certificates are awarded to about 60 students in each division.
This year, Year 9 Newington students Hugo N and Paul H and Year 10 students Rhys Hand Benjamin M excelled in the Junior Division with Hugo taking out overall second place. In the Senior Division, Eamon J, Sebastian W and Preston Z also excelled with Eamon coming first outright across the entire competition. Eamon was congratulated by UNSW School of Mathematics & Statistics Head of School, Professor Andrew Francis at a prize presentation ceremony at UNSW’s Sydney campus recently (as seen in this photograph)
Huge congratulations to all our Newington mathematics virtuosos below who also took part and received awards for their mind crunching mathematical abilities.
Year 8: Matthew M, Gabriel Y, Erain S, Jacob L, Daniel C
Year 9: Kent S, Hamish T, George W, Henry W
Year 10: Nam L
Year 11: Michael Z, Michael K, Jacob H, Marcus K, Logan G, Aiden L