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Advent puzzles II

Behind today’s door… a puzzle!

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This post was part of the Chalkdust 2016 Advent Calendar.

Welcome to the sixth day of the 2016 Chalkdust Advent Calendar. Today, we have another two puzzles for you to enjoy, plus the answer to the puzzle from the 02 December.

If you’ve spent time browsing the internet recently, you will have noticed that we’re not the only site running an advent calendar. Today’s puzzles are taken from two of our rival calendars, run by Matthew Scroggs (who?) and nrich.

First a puzzle from my own advent calendar.

Digital sums

Source: mscroggs.co.uk Advent calendar, day 6
When you add up the digits of a number, the result is called the digital sum.

How many different digital sums do the numbers from 1 to 1091 have?

Second, a puzzle from the excellent nrich Advent calendar.

Wipeout

Source: nrich Secondary Advent calendar, day 10
You are given the numbers 1,2,3,4,5,6 and are allowed to erase one. If you erase 5, the mean of the remaining numbers will be 3.2. Is it possible to erase a number so that the mean of the remaining number is an integer?

If you are given the numbers 1,2,3,4,…,N, can you erase one number so that the mean of the remaining numbers is an integer?

I’ll be back with answers and more puzzles later in Advent.

Back on 02 December, I posted a longer version of the following puzzle:

Decorations

You love big equilateral triangles but hate small equilateral triangles. Can you arrange ten red and blue baubles in a triangle so that no three baubles of the same colour form the vertices of an equilateral triangle?
bauble-puzzle-tri

This is not possible. To see this, first pick a colour for the central bauble. I’ve picked red.
bauble-puzzle-sol1
Now we try to colour the rest without making a triangle. One of the three baubles on the following triangles must be red (otherwise there is a blue triangle). Pick one of them to make red. If a different one is red, rotate the triangle to make this one red.

bauble-puzzle-sol2bauble-puzzle-sol3
The baubles must be coloured as follows. In each step, the colour is chosen to avoid a triangle.

bauble-puzzle-sol4bauble-puzzle-sol5bauble-puzzle-sol6
Now, the bauble shown in green below cannot by either colour, as in each case it makes a triangle.
bauble-puzzle-sol7
Hence, it is impossible to find a triangle without a smaller triangle.

Matthew Scroggs is a PhD student at UCL working on finite and boundary element methods. His website, mscroggs.co.uk, is full of maths and now features a video of him completing a level of Pac-Man optimally.
Twitter  @mscroggs    Website  mscroggs.co.uk    + More articles by Matthew

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