Biscuit Decorations Problem

Andrew decorated 20 biscuits to take to a party.

He lined them up and put icing on every second biscuit.

Then he put a cherry on every third biscuit.

Then he put a chocolate button on every fourth biscuit.

So there was nothing on the first biscuit.

How many other biscuits had no decoration? Did any biscuits get all three decorations?

Printable Sheet

https://nrich.maths.org/content/99/07/letme1/Biscuit%20Decorations%20printable%20sheet.pdf

Notes for Adults

This problem fits in well with counting and skip-counting (counting by twos etc.) and can be solved by physically modelling the biscuits and decorations with whatever objects are convenient. It is a good opportunity for children to choose the way they represent the problem in order to solve it. It may also be appropriate to introduce vocabulary such as "multiple".

What is a multiple?

https://www.bbc.co.uk/bitesize/articles/zgbpnbk

Invite your child  to work on the problem using whatever they find most helpful - have paper, pens, pencils, cubes, counters etc. easily available.

You may like to stop them part way through to share some different ways they could represent the biscuit decorations.  Some children might make  models with differently-coloured cubes for the decorations, some may draw pictures, some may use symbols.

 You may find that your child adopts a different representation following the discussion and it would be interesting to know why this is.

For those children who are more mathematically experienced, consider linking this problem with the idea of common multiples through the multiplication tables and the hundred square.

Key questions

Which other biscuits have icing on?

Which biscuits have a cherry on them as well as the third one?

What about the biscuits with a chocolate button on them? Which ones are they?

Tell me about the biscuits that have no decorations on them.