Sled-packing with Santa

Santa has packed up all his presents into 16 boxes. Each box is a cube: the first is 1m×1m×1m, the second 2m×2m×2m, the third 3m×3m×3m, ... up to the largest which is 16m×16m×16m.

He wants to fit all the boxes onto two sleds but is bound by the Elf Safety Regulations. These state that on each sled he must have the same number of boxes. In addition to this on each sled the sum of the side-lengths of the boxes must be the same, the sum of the areas of the bases of the boxes must be the same and the sum of the volumes of the boxes must be the same. Is this possible?

Right on the Curve

On *y* = *x*^{2} plot a point on the curve, A, and draw the tangent to the curve at A. Plot the other tangent to the curve that is perpendicular to this tangent. Label the point at which this touches the curve B.

What are the coordinates of the point of intersection of the two tangents (shown as C on the diagram)?

What are the coordinates of the point of intersection of the line AB with the *y*-axis (labelled as D on the diagram).

Repeat this for different values of A - what do you notice?

Would this work for other parabolas?

MathsJam 2011

The annual MathsJam event is a chance for like-minded self-confessed maths enthusiasts to get together and share stuff they like. Puzzles, games, problems - or just anything they think that is cool or interesting. MathsJam 2011 will take place at Wychwood Park, Crewe on the weekend of 12-13th November.

A couple of sample puzzles from MathsJam 2010:

Dissecting a Circle

It is possible to dissect a square into congruent pieces that don't all touch the centre. Is it possible to do this for a circle?

Euler's Infinite Tetration

What is the value of *x* in:

For more information about MathsJam 2011 please visit the MathsJam website.

Venn diagrams with areas proportional to probabilities

It is possible to construct Venn diagrams in a unit square in such a way that the areas of the regions are proportional to the probabilities and the area of the overlapping areas is the probability of the combined event.

If this is done so that P(A) is a rectangle with base 1 and P(B) is a square* then the events will be independent if and only if the overlapping area is a similar rectangle to the rectangle representing P(B). The diagonal on the diagram helps visualise when the rectangles are similar. An alternative way of looking at this is to identify if the fraction of the square for P(B) that is shaded twice is the same fraction of the whole square that is shaded for PA).

A dynamic version of the diagram is available.

The Geogebra file can be downloaded.

*If A and B are independent it will be possible to draw P(B) as a square; however, for some other values of P(A∩B) it may not be possible to satisfy all the conditions and draw P(B) as a square.

and is a Pythagorean triple.

Add the reciprocals of any two consecutive odd numbers. Will the resulting fraction, , always generate an integer Pythagorean triple, ?

**Some sums**

What numbers can be made from the sum of some (i.e. at least two) consecutive positive whole numbers?

e.g.

1 + 2 = 3

5 + 6 + 7 = 18

**Circles in a triangle**

In an equilateral triangle with side length 1 what is the largest area that can be covered by three non-overlapping circles?

**Frequency of numbers in Pascal's triangle**

In Pascal's triangle the number 1 appears infinitely many times. All other numbers will appear a finite amount of times*.

The number 2 appears just once.

The number 3 appears twice.

What is the first number that appears exactly 3 times?

What is the first number that appears exactly 4 times?

Can you find a number that appears exactly 5 times?

Can you find a number that appears exactly 6 times?

Can you find a number that appears more than 6 times?

*Can you prove this?

**Not all triangles are perfect, but...**

The first two perfect numbers are 6 and 28

A number is perfect if it is equal to the sum of its factors other than itself: e.g. 6 = 1 + 2 + 3 and 28 = 1+ 2 + 4 + 7 + 14.

The first four triangular numbers are 1, 3, 6, 10. Both 6 and 28 are triangular numbers. Are all perfect numbers triangular?

The first four hexagonal numbers are 1, 6, 15, 28. Both 6 and 28 are hexagonal numbers. Are all perfect numbers hexagonal?

**11 x 11 Square**

The five rectangles in the diagram below, which cover the 11 by 11 square, have dimensions 1,2,3,4,5,6,7,8,9,10, each length used exactly once.

Can you find a different way of doing this (rotations and reflections not allowed)?

**Circle Square**

The diagram shows four circles of radius 1/4 in a unit square

A circle can fit in the gap in the middle.

What is the radius of the circle in the middle? What proportion of the original square is covered by the circle in the middle?

If you placed eight spheres of radius 1/4 in a unit cube a sphere can fit in the middle.What is the radius of the sphere in the middle? What proportion of the original cube is covered by the sphere in the middle?

What happens in 4 dimensions?

And in higher dimensions how big does the diameter of the hypersphere in the gap get and what proportion of the hypercube is covered?

**What Shape is an egg?**

The shape of an egg can be modelled using the equation:

The "egg" created using k = 0.25 and a = 1.6 is shown below:

An alternative method is to find the locus of the points of intersection of two circles whose centres are distance d apart as their radii vary.

e.g. If two circles' centres are 3 units apart and the radii are t/3 and 5-t/2 then the locus of the points of intersection as t varies is:

A dynamic version is available.

Are these two models equivalent? I.e. are there values of k and a in the first model that will give the same shape?

These ideas were developed from the comprehension paper "What shape is an egg?". There are many good ideas for starting points for mathematical investigations in the comprehension papers for Core 4. These can be found by downloading the comprehension papers from resources section of the website.