Can you solve it? Do you cut cake correctly?

And....cut! Photograph: Alamy

Hi guzzlers

To celebrate my birthday this week I thought I’d serve up three cake-based puzzles. Ent-icing! No soggy cerebellums please. Ready. Steady. THINK

1.You have a square cake, and four friends. How do you divide the cake into five slices of equal size? Each slice must be slice-like, meaning that the knife cuts vertically through the cake and the tip of each slice is at the centre of the cake. You have no ruler or tape measure, but you can use the horizontal grid here.

2. You have a rectangular cake, and two friends. One of your friends (the annoying one) just cut herself a rectangular slice, as below. With a single cut, show how to divide what’s left into two portions of equal size.

3. You have a circular cake. You sprinkle exactly 100 hundreds and thousands on the cake. These are mathematical hundreds and thousands, meaning that each one is a single point with no length or width. Show how it is always possible to slice the cake into two with a single cut of the knife so that each slice has exactly 50 hundreds and thousands on it. You can assume that the surface of the cake is perfectly horizontal.

Oh, and if you have any space for an extra mouthful, here’s the puzzle from the cover of my new puzzle book reinterpreted as a patisserie window display.

Which is the odd one out and why?

I’ll be back at 5pm GMT with the answers.

In the comments, please try to avoid spoilers, and instead discuss the many interesting aspects of the mathematics of cake. Or maybe give hints as to how to solve question 3, as it is harder than the others.

I set a puzzle here every two weeks on a Monday. If you would like to suggest a puzzle email me.

My latest book Can You Solve My Problems? A Casebook of Ingenious, Perplexing and Totally Satisfying Puzzles is just out. It contains my favourite puzzles from the last 2000 years, along with historical, biographical and mathematical background. Simon Singh reviewed it here. Available from the Guardian Bookshop and other retailers.

You can check me out on Twitter, Facebook, Google+, my personal website or my Guardian maths blog.