The Elf Lottery

Santa has 100 reindeer in 100 numbered stalls and 100 elves to boss about. He picks reindeer by the following system:

Initially all the stall doors are closed. Elf 1 opens every door, elf 2 then goes to every other door (i.e. multiples of 2) and changes its state (open to closed or vice versa), elf 3 does this for every multiple of 3 door and so on to elf 100.

On Christmas Eve Santa chooses the reindeer in stalls with open doors. Which reindeer does Santa choose?

Three Line Whip

Construct an equilateral triangle with one vertex on each of 3 parallel lines using just a straight edge and a pair of compasses.

Try the problem on GeoGebraTube: http://tube.geogebra.org/m/1941917

AOB

The points A and B are on the curve *y*=*x*^{2} such that AOB is a right angle. What points A and B will give the smallest possible area for the triangle AOB?

Reciprocating lengths

Given a straight line that intersects the *x* and *y* axes at A and B and the curve *y*=1/*x* at C and D, does AD=BC?

Prime Number Days

The ISO standard for writing dates is YYYY-MM-DD: e.g. the 1^{st} August 2015 is 2015-08-01.

The 21^{st} August 2015 is a *Prime Number Day* because 20150821 is a prime number.

When will other Prime Number Days fall in 2015?

Will there ever be two consecutive Prime Number Days?

A match made in seven

Is it possible to arrange seven points in a plane so that any subset of three points will contain at least one pair that is exactly 1cm apart?

A couple of MEI Conference taster problems

**Problem solving at KS3&4 - 13:45-14:45 Saturday 27th June with Phil Chaffe**

Pattern 1:

Pattern 2:

What would the shaded area of pattern 3 be?

**Florence Nightingale - 09:00-10:00 Friday 26th June with Stella Dudzic**

*“The regulation allowance of raw spirit which a man may obtain at the canteen is no less than 18½ gallons per annum ; which is, I believe, three times the amount per individual which has raised Scotland, in the estimation
of economists, to the rank of being the most spirit-consuming nation in
Europe.”*

Florence Nightingale on the British army in India in *“How people may live and not die in India. London : Emily Faithful, 1863.”*

How does this compare to current drinking habits?

A couple of MEI Conference problems

The following problems appeared in MEI Conference sessions in 2014:

**1×1=2**

You are told that, when rounded to the nearest whole number, *x* and *y* are both 1. What is the probability that, to the nearest whole number, *xy* = 2 ?

**Paper folding a parabola**

Prove that the following steps will produce a parabola:

- Fold a piece of A4 paper in half long-ways (to create the y-axis) and draw this in with a pen.
- Mark a point on the fold a couple of inches from the bottom edge.
- Fold the bottom edge so that it goes through the point and is perpendicular to the vertical fold (to create the x-axis) and mark this with a pen.
- Make repeated folds so that the bottom edge goes through the point – these can be at any angle.

Consecutive Fibonacci Squares

The first few Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34 ... (each number is the sum of the previous two numbers in the sequence and the first two numbers are both 1).

The sums of the squares of some consecutive Fibonacci numbers are given below:

1^{2} + 1^{2} = 2

3^{2} + 5^{2} = 34

13^{2} + 21^{2} = 610

Is the sum of the squares of consecutive Fibonacci numbers always a Fibonacci number?

4 points, 2 lengths

In how many ways can you arrange 4 distinct points in the plane so there are exactly two different distances amongst the 6 pairs?

e.g. if the four points are at the corners of a square then the four sides are the same length and the two diagonals are the same length.

1, 2, 3, 4

Find two quadratic functions f(*x*), g(*x*) so the equation f(g(*x*)) = 0 has the four roots *x* = 1, 2, 3, 4.

Is it possible to find three quadratic functions f(*x*), g(*x*), h(*x*) so the equation f(g(h(*x*))) = 0 has the eight roots *x* = 1, 2, 3, 4, 5, 6, 7, 8?

Happy 2015: A Triple of Triples

2015 is the product of 3 distinct primes: 5×13×31

2014 and 2013 are also the product 3 distinct primes.

Can you find a smaller triple (*n*, *n*+1, *n*+2) where *n*, *n*+1 and *n*+2 are all the product of 3 distinct primes?

Are there any quadruples (*n*, *n*+1, *n*+2, *n*+3) where *n*, *n*+1, *n*+2 and *n*+3 are all the product of 3 distinct primes?