![Obviously?](/sites/default/files/styles/medium/public/thumbnails/content-98-08-15plus1-icon.jpg?itok=aO3S6H7Z)
Mathematical induction
![Obviously?](/sites/default/files/styles/medium/public/thumbnails/content-98-08-15plus1-icon.jpg?itok=aO3S6H7Z)
![Dirisibly Yours](/sites/default/files/styles/medium/public/thumbnails/content-98-10-15plus1-icon.jpg?itok=81QLSGRF)
problem
Dirisibly Yours
Find and explain a short and neat proof that 5^(2n+1) + 11^(2n+1) +
17^(2n+1) is divisible by 33 for every non negative integer n.
![Counting Binary Ops](/sites/default/files/styles/medium/public/thumbnails/content-03-10-15plus2-icon.jpg?itok=-uUfGxNQ)
problem
Counting Binary Ops
How many ways can the terms in an ordered list be combined by
repeating a single binary operation. Show that for 4 terms there
are 5 cases and find the number of cases for 5 terms and 6 terms.
![Walkabout](/sites/default/files/styles/medium/public/thumbnails/content-00-09-six2-icon.gif?itok=bA1H7_aH)
problem
Walkabout
A walk is made up of diagonal steps from left to right, starting at
the origin and ending on the x-axis. How many paths are there for 4
steps, for 6 steps, for 8 steps?
![One Basket or Group Photo](/sites/default/files/styles/medium/public/thumbnails/content-00-06-six1-icon.jpg?itok=e5Gvn7qE)
problem
One Basket or Group Photo
Libby Jared helped to set up NRICH and this is one of her favourite
problems. It's a problem suitable for a wide age range and best
tackled practically.
![Golden Powers](/sites/default/files/styles/medium/public/thumbnails/content-03-02-15plus3-icon.gif?itok=sI3O98yL)
problem
Golden Powers
You add 1 to the golden ratio to get its square. How do you find higher powers?
![Binary Squares](/sites/default/files/styles/medium/public/thumbnails/content-02-05-15plus2-icon.jpg?itok=UvvW0ax0)
problem
Binary Squares
If a number N is expressed in binary by using only 'ones,' what can
you say about its square (in binary)?
![Growing](/sites/default/files/styles/medium/public/thumbnails/content-00-07-15plus1-icon.jpg?itok=Jry1mts7)
problem
Growing
Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)
![Overarch 2](/sites/default/files/styles/medium/public/thumbnails/content-00-01-15plus5-icon.gif?itok=SaemCGM-)
problem
Overarch 2
Bricks are 20cm long and 10cm high. How high could an arch be built
without mortar on a flat horizontal surface, to overhang by 1
metre? How big an overhang is it possible to make like this?
![OK! Now prove it](/sites/default/files/styles/medium/public/thumbnails/content-00-01-15plus3-icon.jpg?itok=iazXtkSp)
problem
OK! Now prove it
Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?