Exploring and noticing

In the 2020 Olympic Games, sport climbing was introduced for the first time, and something very interesting happened with the scoring system. Can you find out what was interesting about it?

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

How many moves does it take to swap over some red and blue frogs? Do you have a method?

Which set of numbers that add to 100 have the largest product?

A cinema has 100 seats. How can ticket sales make £100 for these different combinations of ticket prices?

How many different symmetrical shapes can you make by shading triangles or squares?

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

Draw some isosceles triangles with an area of $9cm^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

Discs are flipped in the air. You win if all the faces show the same colour. What is the probability of winning?