2017-2018 SoMaS Team Mathematics Challenge
You can browse this year's problems below or view the download/print the PDF copy of problems. You can browse other years problems and solutions here.Problem 1
Find all prime numbers of the form \(n^4+4m^4\) for integers \(n\) and \(m\).
Absolutely watertight arguments given by The Prime Suspect and the Algebros, with the Uniformed Unicorns, Ncosamathemajician and Jack Tosh also giving good answers.
Can you find four non-intersecting balls in \(\mathbb{R}^3\) not containing the origin, so that every ray starting at the origin intersects at least one of the four balls?
This problem was found hard, but the Prime Suspect and Jack Tosh were able to see that it was possible to the find four balls.
Write down the largest integer you can on a page. Your integer must be:
- unambiguous
- described using notation which can be understood by the judges
- must not refer to anything not on the page.
This video is a good place to start thinking about the problem. With the exception of a novely sized number 1 submitted by Team Algebros, and the number 11 from I Hope This Submission Isn't Too Late, all our entrants managed to beat the whopping 45,000,000 described in the video. The imaginary Team wrote down the largest number, but it wasn't an interger. The largest integer was given by the Uniformed Unicorns which edged the Prime Suspect's integer.
A force is applied against each edge of the polygon from the outside. The force is applied perpendicularly to each edge, through the centre of that edge, and has magnitude proportional to the length of the edge. What is the result of applying these forces to the polygon?
There were great answers for the case of linear motion from Jack Tosh, the Prime Suspect, Torin and co., and the Uniform Unicorns. A variety of different approaches were used by the different teams.
Consider the following game for two players. Given an integer \(N > 0\) the players take turns in decreasing the integer by changing \emph{only one of its decimal digits}. For instance given \(123\), a valid move can lead to any of the six integers \(23\), \(113\), \(103\), \(120\), \(121\), \(122\). The winner is the player who makes \(N = 0\). A typical game between Alice and Bob with initial value \(N = 123\) may look like this: \(A: 120\), \(B: 110\), \(A: 10\), \(B: 0\), and Bob wins. Investigate the winning strategies for this game, concentrating on starting positions \(1 \le N \le 99\) as well as \(N = 123\), \(N = 1234\), \(N=12345\) and so on. The ultimate question is to find a winning strategy which would work for any initial value \(N\).
It was nicely observed by I Hope This Ssubmission Isn't Too Late that this was a dressed up verion of the game Nim. Really nice answers were submitted by The Prime Suspect, and Ncosamathemajiacian. However, the pick of soltions was Team Algebros, and their effort won them the Most Valuable Solution!
Judges: James Cranch,
Evgeny Shinder,
Jayanta Manoharmayum, and
Fionntan Roukema.
O.B.S. prize: (overall best submission) awarded jointly to Uniformed Unicorns (Ruta Rackaityte, Irina Ichim) and The Prime Suspect (Jonathan Atikinson)
M.V.S. prize: (most valuable solution) Team Algebros (Ethan Freestone, Jonathan Hall) for their answer to Question 5.
Honourable mentions: Imaginary Team (Ryan Tandy), I hope this submission isn't too late (Oliver Feghali), Jack Tosh, nkosamatsemajician (James Mason), The Prime Suspect (Jonathan Atikinson), Torin and Co. (Nicholas Blanchard, Torin Carey, Lilyana N Jankovic), Team Algebros (Ethan Freestone, Jonathan Hall), Uniformed Unicorns (Ruta Rackaityte, Irina Ichim).
Best team name: The Prime Suspect (Jonathan Atikinson)