Reasoning, convincing and proving

problem
Knight defeated

Age

14 to 16

Challenge level

The knight's move on a chess board is 2 steps in one direction and one step in the other direction. Prove that a knight cannot visit every square on the board once and only (a tour) on a 2 by n board for any value of n. How many ways can a knight do this on a 3 by 4 board?
problem
Staircase

Age

16 to 18

Challenge level

Solving the equation x^3 = 3 is easy but what about solving equations with a 'staircase' of powers?
problem
Tetra inequalities

Age

16 to 18

Challenge level

Can you prove that in every tetrahedron there is a vertex where the three edges meeting at that vertex have lengths which could be the sides of a triangle?
problem
Flexi quads

Age

16 to 18

Challenge level

A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?
problem
Diverging

Age

16 to 18

Challenge level

Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.
problem
Latin numbers

Age

14 to 16

Challenge level

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

Age

16 to 18

Challenge level

The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.
problem
Basic rhythms

Age

16 to 18

Challenge level

Explore a number pattern which has the same symmetries in different bases.
problem
Target six

Age

16 to 18

Challenge level

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.
problem
Binary squares

Age

16 to 18

Challenge level

If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?