# September 2002, All Stages

## Problems

##### Age 5 to 7 Challenge Level:

Can you find 2 butterflies to go on each flower so that the numbers on each pair of butterflies adds to the same number as the one on the flower?

##### Age 7 to 11 Challenge Level:

Here are the prices for 1st and 2nd class mail within the UK. You have an unlimited number of each of these stamps. Which stamps would you need to post a parcel weighing 825g?

##### Age 7 to 11 Challenge Level:

The Scot, John Napier, invented these strips about 400 years ago to
help calculate multiplication and division. Can you work out how to
use Napier's bones to find the answer to these multiplications?

##### Age 7 to 11 Challenge Level:

This multiplication uses each of the digits 0 - 9 once and once only. Using the information given, can you replace the stars in the calculation with figures?

##### Age 7 to 11 Challenge Level:

What do the digits in the number fifteen add up to? How many other
numbers have digits with the same total but no zeros?

##### Age 7 to 14 Challenge Level:

Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.

##### Age 7 to 16 Challenge Level:

The challenge is to produce elegant solutions. Elegance here implies simplicity. The focus is on rhombi, in particular those formed by jointing two equilateral triangles along an edge.

##### Age 11 to 14 Challenge Level:

A triangle ABC resting on a horizontal line is "rolled" along the
line. Describe the paths of each of the vertices and the
relationships between them and the original triangle.

##### Age 11 to 14 Challenge Level:

Choose two digits and arrange them to make two double-digit
numbers. Now add your double-digit numbers. Now add your single
digit numbers. Divide your double-digit answer by your single-digit
answer. . . .

##### Age 11 to 14 Challenge Level:

There are two forms of counting on Vuvv - Zios count in base 3 and
Zepts count in base 7. One day four of these creatures, two Zios
and two Zepts, sat on the summit of a hill to count the legs of. . . .

##### Age 14 to 16 Challenge Level:

Find the five distinct digits N, R, I, C and H in the following
nomogram

##### Age 14 to 16 Challenge Level:

How many different ways can you arrange the officers in a square?

##### Age 14 to 16 Challenge Level:

An equilateral triangle is constructed on BC. A line QD is drawn,
where Q is the midpoint of AC. Prove that AB // QD.

##### Age 14 to 16 Challenge Level:

Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".

##### Age 14 to 16 Challenge Level:

Can you create a Latin Square from multiples of a six digit number?

##### Age 14 to 16 Challenge Level:

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

##### Age 16 to 18 Challenge Level:

Prove that k.k! = (k+1)! - k! and sum the series 1.1! + 2.2! + 3.3!
+...+n.n!

##### Age 16 to 18 Challenge Level:

Prove that the graph of f(x) = x^3 - 6x^2 +9x +1 has rotational
symmetry. Do graphs of all cubics have rotational symmetry?

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