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 2016-01-01 
The deadline for entering the 2015 Advent calendar competition has now passed. You can see all the puzzles and their answers here.
Thank-you to everyone who took part in the competition! In total, 23 people submitted answers to the puzzles. Every one of these had correct answers to at least 18 of the puzzles. The winners are as follows:
#NameDetails
1ScottAll correct at 5:00:18 GMT on Christmas Day
2Louis de MendoncaAll correct at 5:00:32 GMT on Christmas Day
3Alex BoltonAll correct at 5:00:34 GMT on Christmas Day
4Martin HarrisAll correct at 6:15 GMT on Christmas Day
5Linus HamiltonAll correct at 14:12 GMT on Christmas Day
6ZephiAll correct at 20:40 GMT on Christmas Day
7Daniel ChivertonAll but one (5 December) correct at 5:00:24 GMT on Christmas Day
8Jon PalinAll but one (12 December) correct at 5:00:34 GMT on Christmas Day
9Kathryn CoffinAll but one (5 December) correct at 6:28 GMT on Christmas Day
10Félix BretonAll but one (15 December) correct at 9:05 GMT on Christmas Day
I will be in touch will all the entrants in the next few days and I will post pictures of prizes here once they are on their way!
I have already started working on puzzles for next year's (in fact this year's) calendar, so make sure you're back here in December...

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 2016-07-14 
@stephan: It looks like the code I wrote to check the solutions were unique contained errors. This might explain why first question was very difficult to solve with logic alone.
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Matthew
 2016-01-16 
Are you sure http://www.mscroggs.co.uk/puzzles/126 have unique solutions. See e.g. http://cryptarithms.awardspace.us/solver.html which gives lots of solutions
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stephan
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 2015-11-24 
This year, the front page of mscroggs.co.uk will feature an advent calendar. Behind each door, there will be a puzzle with a three digit solution.
As you solve the puzzles, your answers will be stored in a cookie. Behind the door on Christmas Day, there will be a form allowing you to submit your answers. The first person to submit the correct answers will win this array of prizes:
The prizes include an mscroggs.co.uk t-shirt, a DVD of Full Frontal Nerdity and a beautiful mscroggs.co.uk 2015 winner's medal:
I will be adding to the pile of prizes throughout December.
The next nine people to submit the correct answers will win a winner's medal plus a smaller goody bag of prizes. If less than ten correct entries are submitted, prizes will be given to those with one incorrect answer, then those with two incorrect answers and so on.
Each day's puzzle (and the entry form on Christmas Day) will be available from 5:00am GMT.
To win a prize, you must submit your entry before the start of 2016. Only one entry will be accepted per person. Once ten correct entries have been submitted, I will add a note here and below the calendar. If you have any questions, ask them in the comments below or on Twitter.
So once December is here, get solving! Good luck and have a very merry Christmas!
Edit: The winners and answers can now be found here.

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Comments

Comments in green were written by me. Comments in blue were not written by me.
 2015-12-28 
@Anonymous2: Other winners and answers will be available from 1st January
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Matthew
 2015-12-28 
How many wrong answers do I have ? And when will you publish the solutions ?
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Anonymous2
 2015-12-25 
@Anonymous1: You are (now) correct and on the leaderboard. Well done!
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Matthew
 2015-12-25 
I've submitted my answer! On Christmas too! It's just that I sorta crammed all of the answers today :/

Are you allowed to tell me if I am correct?
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Anonymous1
 2015-12-25 
I'm so glad for OEIS.

hint hint
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Anonymous
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 2013-12-23 
Here is a collection of Christmas relates mathematical activities.

Flexagons

I first encountered flexagons sometime around October 2012. Soon after, we used this template to make them at school with year 11 classes who had just taken GCSE papers as a fun but mathematical activity. The students loved them. This lead me to adapt the template for Christmas:
And here is an uncoloured version of the template on that site if you'd like to colour it yourself and a blank one if you'd like to make your own patterns:
The excitement of flexagons does not end there. There are templates around for six faced flexagons and while writing this piece, I found this page with templates for a great number of flexagons. In addition, there is a fantastic article by Martin Gardner and a two part video by Vi Hart.

Fröbel stars

I discovered the Fröbel star while searching for a picture to be the Wikipedia Maths Portal picture of the month for December 2013. I quickly found these very good instructions for making the star, although it proved very fiddly to make with paper I had cut myself. I bought some 5mm quilling paper which made their construction much easier. With a piece of thread through the middle, Fröbel starts make brilliant tree decorations.

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