# 2014-2015 SoMaS Team Mathematics Challenge

You can browse the problems below or view the download/print the PDF copy of problems.Problem 1

What is the largest prime that you can find with decimal expansion \(2015\cdots\). Your answer should include a verification that your number is indeed prime.

We had three submissions, giving three numbers that were very different in size, but also very different levels of certainty about the primeness of the prime!

At one end, *JD*'s solution claimed that \(2016 \times 10^{1000} - 359\) is prime.
We've managed to check that this is
almost certainly true,
but it's not clear how to be sure of this! Frazer Jarvis
subsequently put all doubts to bed by using
PARI to confirm that the 1000-digit number really is prime.

Then *Hobbit, Hercules and Hulk* claimed that 2015999999999999983 (which is \(2016 \times 10^{15} - 17\)) is prime.
They were able to check this rigorously, but it took them a long
time with a computer to do so.

And, finally, *Noether Berlin vs Borussia Munchengoldbach in the Bundesleibniz* claimed that 201599987 (
which is \(2016 \times 10^5 - 13\)) is prime. They did not merely check it themselves, but found
a way of proving it to us which
requires very little checking.

We reproduce their solution here.

\(n^2\) pegs are arranged in an \(n\times n\) square grid. You have a piece of string which you can wrap around pegs of your choice. How many squares can you form? For example, if \(n=3\) then two such squares are shown below:

There were impeccable solutions from *Hobbit, Hercules and Hulk,
Noether Berlin vs Borussia Munchengoldbach in the Bundesleibniz, JD,
Elastic Pancakes*, and *Chums 2: Chums harder*.

We highlight the *Elastic Pancakes'* solution
because they answered a more general question investigated what happens when you try to count
the number of rectangles. However, the judges felt the questions (particularly 6 and 7) asked
by *Hobbit, Hercules and Hulk* in their solution
made it the stand out solution of the 2015 SoMaS Challenge. We also enjoyed seeing the
youtube video
*Hobbit, Hercules and Hulk* made as part of their submission.

A *cloning move* on an object at the point \( (n,m) \) moves the object to the point
\( (n,m+1)\) and puts a clone on \( (n+1,m) \). Cloning moves are permitted only if \( (n,m+1) \) and \( (n+1,m)\)
are unoccupied. If Dolly the sheep is standing at the origin and
all other points are unoccupied, can she find a sequence of cloning moves that
allow Dolly and her clones to escape from the set
\( \{(0,0), (1,0), (0,1), (2,0),(1,1),(0,2)\}? \)

*Noether Berlin vs Borussia Munchengoldbach in the Bundesleibniz* had a wonderful solution, but unfortunately the
margin of their page was too small to write it down.

*JD* did an exhaustive search for sequences of moves of length 13. Based on this, they suspected the answer was no.
After some internet research they found that this problem is commonly known by the name
``Pebbling a Chessboard''.

*Elastic Pancakes* and *Chums 2: Chums Harder* both gave solutions in line with the classical solution.
We present the Elastic Pancakes' solution.

At the start of the 2013-14 Premier League football season, the
Manchester United manager, David Moyes, complained about his team’s
early fixtures, saying, "I find it hard to believe that's the way the
balls came out of the bag, that's for sure." He also said: "I think
it's the hardest start for 20 years that Manchester United have had."
(He later accepted the scheduler’s assurances that the fixtures were
chosen randomly).

Manchester United’s first five fixtures were Swansea (A), Chelsea (H),
Liverpool (A), Crystal Palace (H), Man City (A). In the 2012-13
season, Chelsea, Liverpool and Man City finished 3rd, 7th and 2nd
respectively. Was there anything ‘surprising’ about the difficulty of
Manchester United’s first five fixtures?

There were two very good submissions for the problem; both of which concluded that Moyes shouldn't have found it hard to believe the way the balls came out.

*Noether Berlin vs Borussia Munchengoldbach in the Bundesleibniz* scored the difficulty of the draw
based on ranks from the previous season, and then compared the 2013-14
draw with draws from the previous 23 Premier League seasons. (An
interesting critique of David Moyes' managerial abilities in big games
was also offered). You can see their solution here.

*Chums 2: Chums Harder took a different approach*,
again scoring the difficulty of the draw based on ranks, but this time
estimating a probability of any particular score using an
approximation based on the normal distribution. You can see
their solution here.

Show that there are arbitrarily large gaps between powers of integers. That is, show that for every \(N\in \mathbb{N}\) there exits an \(n\in \mathbb{N}\) such that no element of \(\{n+1,\dots, n+N\}\) is of the form \(m^k\) with \(k>1\).

No team submitted a complete solution. *Hobbit, Hercules and Hulk* presented some striking
pictures
and *Noether Berlin vs Borussia Munchengoldbach in the Bundesleibniz* left it as an exercise
for the reader and *JD* gave some intuition about why this might be true.

James Cranch suggested a very short solution to *Noether Berlin vs Borussia Munchengoldbach in the Bundesleibniz*'s exercise:
For a set \(\{p_1,\dots,\ p_N\}\) of \(N\) distinct primes, the system of simultaneous congruences
\(x\equiv p_k-k\mod p_k^2\), for \(k=1,\dots,N\) has a solution \(x=n\) by the
Chinese Remainder Theorem.
The prime factorization of \(n+k\) has a single copy of \(p_k\) and is therefore not a power, meaning that
\(n+1,\dots, n+N\) are not powers.

Jayanta Manoharmayum suggested another beautiful solution.

**Judges:** James Cranch,
Jayanta Manoharmayum,
Jeremy Oakley, and
Fionntan Roukema.

**O.B.S. prize:** (overall best submission) *Noether Berlin vs Borussia Munchengoldbach in the Bundesleibniz*
(Dan Graves, Benjamin M Patey).

**M.V.S. prize:** (most valuable solution) *Hobbit, Hercules and Hulk* (Lawrence Beesely-Peck, David Philpott, Robert Nicolaides)
for their solution to question 2.

**Honourable mentions:** *Chums 2: Chums harder* (Sam Baker, James Heseltine), Elastic Pancakes (William Thurley, Sam Watson,
Jordan Williamson), *JD* (Joshua Hawxwell and Daniel Rhodes), Noether Berlin vs Borussia Munchengoldbach in the Bundesleibniz
(Benjamin M Patey, Dan Graves).

**Best team name:** *Noether Berlin vs Borussia Munchengoldbach in the Bundesleibniz* (Dan Graves, Benjamin M Patey).

You can browse past years problems and solutions here.