Sprouts

Sprouts is a pencil and paper game with the following rules:

- Mark a number of dots anywhere on a sheet of paper.
- Each player in turn draws a line joining a dot either to itself or to another dot and places a new dot on this line.
- No line may cross either itself or any other line.
- No dot may have more than three lines leaving it.
- The last player able to make a legal move wins the game.

The first two moves from a game of Sprouts with 3 dots are shown below.

Sprouts was devised by by mathematicians John Conway and Michael Paterson. For more details see: en.wikipedia.org/wiki/Sprouts_(game)

Ritangle Competition - Preliminary Question

Given a positive integer *n* we say s(*n*) is the sum of all the factors of *n* not including *n* itself.

Thus s(6) = 1 + 2 + 3 = 6; s(7) = 1; s(8) = 1 + 2 + 4 =7; s(9) = 1 + 3 = 4.

It is easy to find even numbers *n* so the s(*n*) > *n*,

for example s(12) = 1 + 2 + 3 + 4 + 6 = 16.

It's harder to find odd numbers where s(*n*) > *n* but it is possible,

for example, s(1575) = 1649 > 1575.

Find the odd number smaller than 1575 so that s(*n*) > *n*.

This is one of the preliminary questions for the Integral Ritangle competition starting on 9th November. Ritangle is a competition for teams of students of A level Mathematics. For more details about the competition see: integralmaths.org/ritangle/.

Tan lines

Tangents to the graphs of *y*=tan*x* and *y*=cos*x* are drawn at the point of intersection between *x*=0 and *x*=π/2. Parallel lines are drawn to these tangents at the other tangent's point of intersection with the *x*-axis. What is special about the quadrilateral created?

An unexpected fraction?

Start with any parallelogram. Mark the midpoint of each side. Join these midpoints to the vertex two places clockwise around the parallelogram. What fraction of the original parallelogram is the new quadrilateral?

What would happen if you started with any convex quadrilateral?

Square sum of squares

1^{2} + 2^{2} = 5, which is not a square.

1^{2} + 2^{2} + 3^{2} = 14, which is not a square.

What is the smallest positive integer value of *n*, *n*>1, such that 1^{2} + 2^{2} + ... + *n*^{2} is a square number?

Are there any larger possible values of *n*?

Perpendicular parabolas

If the curves *y* = (*x* – *p*)^{2} + *q* and *x* = (*y* – *r*)^{2} + *s* have four points of intersection will these four points always lie on a circle?

MEI Conference taster - Odd and distinct partitions

O(*n*) is the number of ways of writing *n* as the sum of odd positive integers.

e.g. O(6)= 4: {5+1, 3+3, 3+1+1, 1+1+1+1+1+1}

D(*n*) is the number of ways of writing *n*
as the sum of distinct positive integers.

e.g. D(6) = 4: {6, 5+1, 4+2, 3+2+1}

Does O(*n*)=D(*n*) for all natural numbers?

This problem is taken from the 2015 MEI Conference session *Desert Island Mathematics*. To see details of this year's sessions visit the conference website: conference.mei.org.uk

MEI Conference - Sessions about famous mathematicians

The 2015 MEI conference featured a strand of sessions about 12 famous mathematicians. The following problem is from the session about John Conway.

There are only three numbers (>1) that can be written as the sum of fourth powers of their digits:

1634 = 1^{4} + 6^{4} +3^{4} +4^{4}

8208 = 8^{4} + 2^{4} + 0^{4} + 8^{4}

9474 = 9^{4} + 4^{4} + 7^{4} + 4^{4}

Find the smallest number (>1) that can be written as the sum of fifth powers of its digits.

This year the 2016 Conference will feature a strand of sessions about a different set of 12 famous mathematicians. To see details of these, and other sessions, visit the conference website: conference.mei.org.uk

Are you sure?

What’s the area?

Now read the research: Drawing attention to a lack of attention

Cube Slice

A cube is sliced vertically along the line shown in the diagram and the smaller part is thrown away.

The remaining prism is going to be sliced vertically downwards again by a line going through corner D.

Where would the slice have to be to split it into two equal volumes?

This problem is taken from the FMSP GCSE Problem Solving Materials.

50, 60, 70, ... ?

Find the size of the angle α.

Happy 2016

2016 is a triangular number.

The first three triangular numbers are: 1, 3, 6. The first three pentagonal numbers are: 1, 5, 12.

The pentagonal numbers 1, 5 and 12 are all one third of a triangular number.

Are all pentagonal numbers one third of a triangular number?