Threesomes

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I know?

Have You Got It?

Can you explain the strategy for winning this game with any target?

Tilted Squares

Age 11 to 14 Challenge Level:

It's easy to work out the areas of squares drawn on a grid if they are oriented in the usual way:

Can you find a quick and easy method to work out the areas of tilted squares?

Here are some squares with a tilt of 1:
See the hint for suggested ways to calculate their areas.

Notice anything special about their areas?
Can you predict the areas of other squares with a tilt of 1?

What about squares with a tilt of 2? Or 3? Or 4? Or...?
Notice anything interesting?

Can you make any conjectures about the areas of tilted squares?