Exploring and noticing

What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

Start by putting one million (1 000 000) into the display of your calculator. Can you reduce this to 7 using just the 7 key and add, subtract, multiply, divide and equals as many times as you like?

Take any pair of numbers, say 9 and 14. Take the larger number, fourteen, and count up in 14s. Then divide each of those values by the 9, and look at the remainders.

This challenge is to make up YOUR OWN alphanumeric. Each letter represents a digit and where the same letter appears more than once it must represent the same digit each time.

Find another number that is one short of a square number and when you double it and add 1, the result is also a square number.

Using the numbers 1, 2, 3, 4 and 5 once and only once, and the operations x and ÷ once and only once, what is the smallest whole number you can make?

Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?

If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?