Conjecturing and generalising

problem
Pinned squares

Age

14 to 16

Challenge level

What is the total number of squares that can be made on a 5 by 5 geoboard?
problem
Lower bound

Age

14 to 16

Challenge level

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
problem
Pareq calc

Age

14 to 16

Challenge level

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel lines are 1 unit and 2 units.
problem
Mindreader

Age

11 to 14

Challenge level

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you the last two digits of her answer. Now you can really amaze her by giving the whole answer and the three consecutive numbers used at the start.
problem
How old am I?

Age

14 to 16

Challenge level

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?
problem
Differences

Age

11 to 14

Challenge level

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?
problem
Janine's conjecture

Age

14 to 16

Challenge level

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. Does this always work? Can you prove or disprove this conjecture?
problem
Expenses

Age

14 to 16

Challenge level

What is the largest number which, when divided into these five numbers in turn, leaves the same remainder each time?
problem
Legs eleven

Age

11 to 14

Challenge level

Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?
problem
Dozens

Age

7 to 14

Challenge level

Can you select the missing digit(s) to find the largest multiple?