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# Blog

2021-01-03
It's 2021, and the Advent calendar has disappeared, so it's time to reveal the answers and annouce the winners. But first, some good news: with your help, Santa received all his letters and Christmas was saved!
Now that the competition is over, the questions and all the answers can be found here. Before announcing the winners, I'm going to go through some of my favourite puzzles from the calendar, reveal the solution and a couple of other interesting bits and pieces.

### Highlights

My first highlight is this puzzle from 5 December, that I think doesn't seem obvious that there's even a unique answer until you spot some nice properties of dice.

### 5 December

Carol rolled a large handful of six-sided dice. The total of all the numbers Carol got was 521. After some calculating, Carol worked out that the probability that of her total being 521 was the same as the probability that her total being 200. How many dice did Carol roll?

My next highlight is the puzzle from 15 December. I had a lot of fun trying to write this one. At the end, I tell you that T represents 4: this puzzle still has a unique answer without this, but it was too hard to get started.

### 15 December

When talking to someone about this Advent calendar, you told them that the combination of XMAS and MATHS is GREAT. They were American, so asked you if the combination of XMAS and MATH is great; you said SURE. You asked them their name; they said SAM.
Each of the letters E, X, M, A, T, H, S, R, U, and G stands for a different digit 0 to 9. The following sums are correct:
Today's number is SAM. To help you get started, the letter T represents 4.

My next highlight is the puzzle from 16 December. I had a lot of fun writing this one too. At least a few people seem to have enjoyed solving it, as indicated by this comment.

### 16 December

Solve the crossnumber to find today's number. No number starts with 0.

My final highlight is the puzzle from 21 December. My method for solving this one was quite complicated. Let me know in the comments or on Twitter if you have a better way of solving it.

### 21 December

There are 3 ways to order the numbers 1 to 3 so that no number immediately follows the number one less that itself:
• 3, 2, 1
• 1, 3, 2
• 2, 1, 3
Today's number is the number of ways to order the numbers 1 to 6 so that no number immediately follows the number one less that itself.

### Hardest and easiest puzzles

Once you've entered 24 answers, the calendar checks these and tells you how many are correct. I logged the answers that were sent for checking and have looked at these to see which puzzles were the most and least commonly incorrect. The bar chart below shows the total number of incorrect attempts at each question.
You can see that the most difficult puzzles were those on 6, 14, 19, 21, and 24 December; and the easiest puzzles were on 3, 9, 10, 16, and 17 December.

### The solutions

The solutions to all the individual puzzles can be found here. If you want to read some alternative solutions to the puzzles, you can find Alira's solutions here, including a very nice explanation of how the final clues fit together to find the directions to Santa's house.
Using the daily clues, it was possible to work out that Santa's house could be found by following the directions 3,7,1,1,4,3,3,4,9,3,3,9 on this map.
Directions to Santa's house.
Due to the way the Advent compass worked, there are actually a few different ways to get to Santa's house. For example, 7,5,7,1,1,4,3,3,4,9,3,3 also gets you there:
Alternative directions to Santa's house.
I'll leave you to work out how many different ways there are to get there...
Here's the full collection of tiles that you could find by walking around the map:
The map tiles.

### The winners

And finally (and maybe most importantly), on to the winners: 223 people found Santa's house entered the competition. This is quite a lot more than in previous years:
From the correct answers, the following 10 winners were selected:
 1 Mars He 2 Ben Baker 3 Amelia Taylor 4 Alex Ayres 5 Hannah Charman 6 Diane Keimel 7 Alex Burlton 8 Rob March 9 Mahmood Hikmet 10 Guillermo Heras Prieto
Congratulations! Your prizes will be on their way shortly.
The prizes this year include 2020 Advent calendar T-shirts. If you didn't win one, but would like one of these, I've made them available to buy on Teespring alongside the T-shirts from previous years.
Additionally, well done to 4nder, Ashley Frazer Jarvis, Aaron Stiff, Alan Buck, Alejandro Villarreal, Alex Bolton, Alex Davis, Alex Klapheke, Alexander Rakitin, Anders Kaseorg, Andrew, Andrew P Turner, Andy Ennaco, Annabel, Anthony Della Pella, Artie Smith, Athena, Austin, Austin Shapiro, Beau Ferguson, Becky Russell, Ben Buchwald, Ben Jones, Ben Reiniger, Benjamin Tozer, Bennet, Beth Jensen, Blake, Brennan Dolson, Brian Carnes, Brian Wellington, Burak Kadron, Caleb Likely, Callum Hobbis, Carl Westerlund, Carmen Günther, Cath Simpson, Cathy Hooper, Chris Hellings, Christy Hales, Colin, Colin Brockley, Connie, Corbin Groothuis, Cory Peters, Cosmo Viola, Dan Colestock, Dan DiMillo, Dan McIntyre, Dan Whitman, Dana Sussman, Daniel De la Paz, David, David Ault, David Berardo, David Fox, David Kendel, David Mitchell, David Parmenter, David and Ivy Walbert, Dean Thomas, Deborah Tayler, Don Anderson, Donagh Collins, Elizabeth Blackwell, Emilie Heidenreich, Emily Troyer, Eric Kolbusz, Eric Skoglund, Erik Eklund, Eva, Evan Louis Robinson, Fionn Woodcock, Frances Haigh, Frank Kasell, Franklin Ta, Fred Verheul, Georges, Gabriella Pinter, Gert-Jan de Vries, Graham Greve, Gregory Loges, Gwendal Collet, Harry Allen, Heerpal Sahota, Helen, Helen Matthews, Herschel Pecker, Håkon Balteskard, Ivan Arribas, Jacob Young, James Demers, James Laver, Jan, Jason Demers, Jason Kass, Jean-Sébastien Turcotte, Jeff Rubin, Jefff Michael, Jen Huang, Jessica Marsh, Jim Rogers, Joe Gage, Johan Asplund, John Alasdair Warwicker, Jon Foster, Jon Lipscombe, Jon Palin, Jonathan Chaffer, Jonathan Forsythe, Jonathan Winfield, Jordan Susskind, Jorge del Castillo, Kai Lam, Karen Climis, Keith Pound, Ken Cheung, Killeen, Kristen Koenigs, Kristin Gramza, Kurt Engleman, Laura Midgley, Lauren Woolsey, Lewis Dyer, Linus Banghart-Linn, Liz Madisetti, Louis de Mendonca, Magnus Eklund, Mark Stambaugh, Martijn O., Martin Harris, Martin Holtham, Martine Nome, Matt Askins, Matt Hutton, Matthew Reynier, Matthew Riggle, Matthew Schulz, Matthew Scroggs, Matthias Planitzer, Maxime Rivet, Maximilian Pfister, Michael DeLyser, Michael Prescott, Mihai Zsisku, Mike Hands, Moritz Stocker, Nadine Chaurand, Nathan Chun, Nick C, Nick Keith, Nick Mohr, Norvin Richards, Olov Wilander, Pamela Docherty, Pau Batlle, Paul Livesey, Pranshu Gaba, Ray Arndorfer, Rea, Reuben Cheung, Riccardo Lani, Richard Pemberton, Rileigh Luczak, Road White, Roni, Rosie Paterson, Russ Collins, Ruth Franklin, Ryan Coomer, Ryan Seldon, Ryan Wise, Sam Dreilinger, Sam Hartburn, Sammie Buzzard, Sara Van Hoy, Sarah Brook, Saul Freedman, Scott, Sean, Sean Carmody, Sean Henderson, Sean McDonald, Sebastian, Selina Glauert, Seth Cohen, Shivanshi Adlakha, Simon Schneider, Sophie Maclean, Spencer B., Stephen Cappella, Stephen Dainty, Stephen Jasina, Steve Blay, Steve Paget, Steven Richardson, Susana Early, Tarim, Tim Holman-Wilkens, Tim Lewis, Todd Geldon, Tom Fryers, Tony Mann, Tristan Shephard, Valentin Valciu, Vinesh, Yasha, Yuliya Nesterova, Yurie Ito, and Zachary Polansky who all also found Santa's house but were too unlucky to win prizes this time.
See you all next December, when the Advent calendar will return.

### Similar posts

 Christmas (2020) is coming! Christmas (2019) is over Christmas (2019) is coming! Christmas (2018) is over

Comments in green were written by me. Comments in blue were not written by me.
Dec 15th was my favorite. I kept making a logical error and had to restart, so it took me way to long, but I really enjoyed it.

The 16th was just cheeky, after I spent way to much time on the 15th it was nice to have something like that!

It's been 16+ years since I did any probability or combinations and permutations, so it was nice to brush off that part of my brain, not that I did any of them well, but it should serve me well when my kids start doing them and ask me for help.
Dave
×1
You can solve the Dec 21 puzzle using the principle of inclusion/exclusion:

-There are 6! total ways of arranging 6 numbers.
-Now we have to exclude the ones that don't fit. How many ways have 2 following 1? You can think of 12 as a pair, so you're arranging 12/3/4/5/6 in any order, so there are 5! ways to do this. And there are (5 choose 1)=5 total pairs that might exist, so there are 5*5! ways that have either 12, 23, 34, 45, or 56.
-Of course, we've double counted some that have more than one pair. (This is where inclusion/exclusion comes in, we have to include them back in). So how many have, say, 12 and 45? Well now we're arranging 12/3/45/6, so there are 4! ways to do so. There are (5 choose 2)=10 different pairs, so the double counting was 10*4!.
-We continue this on, and inclusion/exclusion says we keep alternating adding and subtracting as we add more pairs, so the answer is:
6!
- (5 choose 1) * 5!
+ (5 choose 2) * 4!
- (5 choose 3) * 3!
+ (5 choose 4) * 2!
- (5 choose 5) * 1!
= 309
Todd
×3   ×1
There seems to be a missing diagram for the answer to the Dec 2 puzzle.
Kai

@(anonymous): Thanks, links corrected
Matthew

Messed up HTML. What I mean is sequence A000255 on OEIS.
(anonymous)
×2

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2020-11-22
This year, the front page of mscroggs.co.uk will once again feature an Advent calendar, just like in the five previous Decembers. Behind each door, there will be a puzzle with a three digit solution. The solution to each day's puzzle forms part of a logic puzzle:
It's nearly Christmas and something terrible has happened: you've just landed in a town in the Arctic circle with a massive bag of letters for Santa, but you've lost to instructions for how to get to Santa's house near the north pole. You need to work out where he lives and deliver the letters to him before Christmas is ruined for everyone.
Due to magnetic compasses being hard to use near the north pole, you brought with you a special Advent compass. This compass has nine numbered directions. Santa has given the residents of the town clues about a sequence of directions that will lead to his house; but in order to keep his location secret from present thieves, he gave each resident two clues: one clue is true, and one clue is false.
The residents' clues will reveal to you a seqeunce of compass directions to follow. You can try out your sequences on this map.
Behind each day (except Christmas Day), there is a puzzle with a three-digit answer. Each of these answers forms part of a resident's clue. You must use these clues to work out how to find Santa's house.
Ten randomly selected people who solve all the puzzles, find Santa's house, and fill in the entry form behind the door on the 25th will win prizes!
The winners will be randomly chosen from all those who submit the entry form before the end of 2020. Each day's puzzle (and the entry form on Christmas Day) will be available from 5:00am GMT. But as the winners will be selected randomly, there's no need to get up at 5am on Christmas Day to enter!
To win a prize, you must submit your entry before the end of 2020. Only one entry will be accepted per person. 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!

### Similar posts

 Christmas (2020) is over Christmas (2019) is over Christmas (2019) is coming! Christmas (2018) is over

Comments in green were written by me. Comments in blue were not written by me.
Nice one today (16 December) :)
Did he?... did he really?... starts to look like it... yes he did! :D
Gert-Jan
×3   ×6   ×3   ×1   ×2
@A: Aha, I should have thought of this! Puzzles completed :-)
(anonymous)
×1
@Dean: you can also go through each answer one at a time and change a digit; if the number of wrong answers goes up then your answer for that question was correct
A
×1   ×1
@Dean: Yes, I'm planning to change how that bit works. Check back tomorrow or the next day for a more fun finish!
Matthew
×1
A bit harsh that we can’t tell exactly which answers are wrong! I won’t have time to revisit every puzzle - and its kind of less fun redoing something that is already correct... :-(
Dean

@Marty: Yes, I'm in the middle of correcting the clues page to add these details back
Matthew
×1

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2020-01-02
It's 2020, and the Advent calendar has disappeared, so it's time to reveal the answers and annouce the winners. But first, some good news: with your help, Santa and his two reindeer were found and Christmas was saved!
Now that the competition is over, the questions and all the answers can be found here. Before announcing the winners, I'm going to go through some of my favourite puzzles from the calendar, reveal the solution and a couple of notes and Easter eggs.

### Highlights

My first highlight is this puzzle from 10 December that seems difficult to get started on, but plotting the two quadratics reveals a very useful piece of information that can be used.

### 10 December

For all values of $$x$$, the function $$f(x)=ax+b$$ satisfies
$$8x-8-x^2\leqslant f(x)\leqslant x^2.$$
What is $$f(65)$$?
Edit: The left-hand quadratic originally said $$8-8x-x^2$$. This was a typo and has now been corrected.

My next highlight is the puzzle from 12 December, which was election day in the UK. Although the puzzle isn't that difficult or interesting to calculate, the answer is surprising enough to make this one of my favourites.

### 12 December

For a general election, the Advent isles are split into 650 constituencies. In each constituency, exactly 99 people vote: everyone votes for one of the two main parties: the Rum party or the Land party. The party that receives the most votes in each constituency gets an MAP (Member of Advent Parliament) elected to parliament to represent that constituency.
In this year's election, exactly half of the 64350 total voters voted for the Rum party. What is the largest number of MAPs that the Rum party could have?

My next highlight is the puzzle from 13 December. If you enjoyed this one, then you can find a puzzle based on a similar idea on the puzzles pages of issue 10 of Chalkdust.

### 13 December

Each clue in this crossnumber (except 5A) gives a property of that answer that is true of no other answer. For example: 7A is a multiple of 13; but 1A, 3A, 5A, 1D, 2D, 4D, and 6D are all not multiples of 13. No number starts with 0.

My final highlight is the puzzle from 16 December. I always include a few of these, as they can be designed to give any answer so are useful for making the final logic puzzle work. But I was particularly happy with this one.

### 16 December

Arrange the digits 1-9 in a 3×3 square so that: the median number in the first row is 6; the median number in the second row is 3; the mean of the numbers in the third row is 4; the mean of the numbers in the second column is 7; the range of the numbers in the third column is 2, The 3-digit number in the first column is today's number.
 median 6 median 3 mean 4 today's number mean 7 range 2

### Notes and Easter eggs

A few of you may have noticed the relevance of the colours assigned to each island: Rum (Red Rum), Land (Greenland), Moon (Blue Moon), and County (Orange County).
Once you've entered 24 answers, the calendar checks these and tells you how many are correct. I logged the answers that were sent for checking and have looked at these to see which puzzles were the most and least commonly incorrect. The bar chart below shows the total number of incorrect attempts at each question.
You can see that the most difficult puzzles were those on 4 and 15 December; and the easiest puzzles were on 3, 11, 17, and 24 December.
This year, the final logic puzzle revealed the positions of Santa, Rudolph and Blitzen, then you had to find them on this map. The map has 6 levels, with 81 positions on each level, so the total size of the map is $$81^6=282\,429\,536\,481$$ squares. This is a lot; one Advent solver even wondered how large a cross stitch of the whole thing would be. I obviously didn't draw 282 billion tiles: the whole map was generated using the following tiles, which were reused across the map.
The map tiles.
I also snuck a small Easter egg into the map. Below the Advent calendar, this example was shown:
The ASC coordinates of this pair of flowers are 12.52.12.13.84.55 (click to enlarge).
If you actually visited this position on the map, you found Wally.
Wally.
At least one Advent solver appears to have found Wally, as they left this cryptic comment under the name Dr Matrix (an excellent Martin Gardner reference).

### The solution

The solutions to all the individual puzzles can be found here.
Using the daily clues, it was possible to work out that Santa was at 36.11.19.79.79.43, Blitzen was at 23.12.23.11.23.11, and Rudolph was at 16.64.16.16.16.64.
Santa (left), Blitzen (centre) and Rudolph (right).

### The winners

And finally (and maybe most importantly), on to the winners: 126 people found Santa and both reindeer and entered the competition. Their (very) approximate locations are shown on this map:
From the correct answers, the following 10 winners were selected:
 1 Valentin Valciu 2 Tom Anderson 3 Alex Bolton 4 Kevin Fray 5 Jack Kiuttu 6 Ben Baker 7 Joe Gage 8 Michael Thomas 9 Martin Holtham 10 Beth Jensen
Congratulations! Your prizes will be on their way shortly.
The prizes this year include 2019 Advent calendar T-shirts. If you didn't win one, but would like one of these, you can buy one at mscroggs.co.uk/tshirt until 7 January (when I will be sending them for printing).
The design on the T-shirt.
Additionally, well done to Adam Abrams, Adam NH, Adam Vellender, Alan Buck, Alex Ayres, Alex Burlton, Alexander Ignatenkov, Andrew Ennaco, Andrew Tindall, Artie Smith, Ashley Jarvis, Austin Antoniou, Becky Russell, Ben Jones, Ben Reiniger, Brennan Dolson, Brian Carnes, Carl Westerlund, Carmen Guenther, Clare Wallace, Colin Beveridge, Connie, Corbin Groothuis, Cory Peters, Dan Colestock, Dan DiMillo, Dan Whitman, David, David Ault, David Fox, Diane Keimel, Duncan Ramage, Emilie Heidenreich, Emily Troyer, Eric, Eric Kolbusz, Erik Eklund, Evan Louis Robinson, Frances Haigh, Franklin Ta, Fred Verheul, Félix Breton, Gabriella Pinter, Gautam Webb, Gert-Jan de Vries, Hart Andrin, Heerpal Sahota, Herschel Pecker, Jacob Juillerat, Jan, Jean-Noël Monette, Jen Shackley, Jeremiah Southwick, Jessica Marsh, Johan Asplund, John Warwicker, Jon Foster, Jon Palin, Jonathan Chaffer, Jonathan Winfield, Jose, Kai, Karen Climis, Karen Kendel, Katja Labermeyer, Laura, Lewis Dyer, Louis de Mendonca, M Oostrom, Magnus Eklund, Mahmood Hikmet, Marian Clegg, Mark Stambaugh, Martin Harris, Martine Vijn Nome, Matt Hutton, Matthew Askins, Matthew Schulz, Maximilian Pfister, Melissa Lucas, Mels, Michael DeLyser, Michael Gustin, Michael Horst, Michael Prescott, Mihai Zsisku, Mike, Mikko, Moreno Gennari, Nadine Chaurand, Naomi Bowler, Nathan C, Pat Ashforth, Paul Livesey, Pranshu Gaba, Raymond Arndorfer, Riccardo Lani, Rosie Paterson, Rupinder Matharu, Russ Collins, S A Paget, SShaikh, Sam Butler, Sam Hartburn, Scott, Seth Cohen, Shivanshi Adlakha, Simon Schneider, Stephen Cappella, Stephen Dainty, Steve Blay, Thomas Tu, Tom Anderson, Tony Mann, Yasha, and Yuliya Nesterova, who all also submitted the correct answer but were too unlucky to win prizes this time.
See you all next December, when the Advent calendar will return.

### Similar posts

 Christmas (2020) is over Christmas (2020) is coming! Christmas (2019) is coming! Christmas (2018) is over

Comments in green were written by me. Comments in blue were not written by me.
It's interesting that the three puzzles with the most incorrect attempts can all be looked up on OEIS.

Day 15 - https://oeis.org/A001055 - "number of ways of factoring n with all factors greater than 1"

Day 4 - https://oeis.org/A001045 - "number of ways to tile a 2 × (n-1) rectangle with 1 × 2 dominoes and 2 × 2 squares"

Day 2 - https://oeis.org/A002623 - "number of nondegenerate triangles that can be made from rods of length 1,2,3,4,...,n"
×1
Never mind, I found them, they were your example ASC. Very clever!
Michael T
×1
Where did the coordinates for Wally come from? Are they meaningful, or are they just some random coordinates thrown together?
Michael T

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2019-11-24
This year, the front page of mscroggs.co.uk will once again feature an Advent calendar, just like last year, the year before, the year before and the year before. Behind each door, there will be a puzzle with a three digit solution. The solution to each day's puzzle forms part of a logic puzzle:
It's nearly Christmas and something terrible has happened: while out on a test flight, Santa's sled was damaged and Santa, Rudolph and Blitzen fell to the ground over the Advent Isles. You need to find Santa and his reindeer before Christmas is ruined for everyone.
You have gathered one inhabitant of the four largest Advent Isles—Rum, Land, Moon and County—and they are going to give you a series of clues about where Santa and his reindeer landed. However, one or more of the islanders you have gathered may have been involved in damaging Santa's sled and causing it to crash: any islander involved in this will lie to you to attempt to stop you from finding Santa and his reindeer. Once you are ready to search for Santa, Rudolph and Blitzen, you can find the map by following this link.
Each of the clues will be about Santa's, Rudolph's or Blitzen's positions in Advent Standard Coordinates (ASC): ASC are given by six two-digit numbers with dots inbetween, for example 12.52.12.13.84.55. For this example coordinate, the islanders will refer to (the first) 12 as the first coordinate, 52 as the second coordinate, (the second) 12 as the third coordinate, 13 as the fourth coordinate, 84 as the fifth coordinate, and 55 as the sixth coordinate.
To find a point's ASC coordinates, split a map of the islands into a 9×9 grid, then number the rows and columns 1 to 9: the first two digits of ASC give the vertical then horizontal position of a square in this grid. The next two digits then give a smaller square when this square is then itself split into a 9×9 grid, and so on. An example is show below.
The ASC coordinates of this pair of flowers are 12.52.12.13.84.55 (click to enlarge).
Behind each day (except Christmas Day), there is a puzzle with a three-digit answer. Each of these answers forms part of a fact that one of the islanders tells you. You must use these clues to find Santa and his two reindeer.
Ten randomly selected people who solve all the puzzles, find Santa and his reindeer, and fill in the entry form behind the door on the 25th will win prizes!
The winners will be randomly chosen from all those who submit the entry form before the end of 2019. Each day's puzzle (and the entry form on Christmas Day) will be available from 5:00am GMT. But as the winners will be selected randomly, there's no need to get up at 5am on Christmas Day to enter!
To win a prize, you must submit your entry before the end of 2019. Only one entry will be accepted per person. 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!

### Similar posts

 Christmas (2020) is over Christmas (2020) is coming! Christmas (2019) is over Christmas (2018) is over

Comments in green were written by me. Comments in blue were not written by me.
This search space is so large, it's way harder than trying to find Wally like in the children's books. Best to read everything carefully looking for any clues, and then solve all the puzzles and logic before searching randomly!
Did you know Wally is called Waldo in the US, Walter in Germany, and Charlie in France?
Dr. Matrix
×3   ×1
Thank you, I was worried. Thanks also for the puzzles, makes a change from my usual sudokus. I especially liked 3, 9, 11, 16, 21 and 23.
(anonymous)

@(anonymous): You don't get a confirmation email but if you hit submit you'll be entered. (I'll add confirmation emails before next year...)
Matthew

Do we get a confirmation email after submission of the entry form? I never received one, so I'm not sure if I am entered.
(anonymous)

Thanks for the nice puzzles!
Gert-Jan

@Seth: If you find them on the map then that's all you need to do. (And if you didn't find them yet then the entry form won't appear so you definably did everything you need to do.)
Matthew

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2019-01-01
It's 2019, and the Advent calendar has disappeared, so it's time to reveal the answers and annouce the winners. But first, some good news: with your help, Santa was able to work out who had stolen the presents and save Christmas:
Now that the competition is over, the questions and all the answers can be found here. Before announcing the winners, I'm going to go through some of my favourite puzzles from the calendar, reveal the solution and a couple of notes and Easter eggs.

### Highlights

My first highlight is the first puzzle in the calendar. This is one of my favourites as it has a pleasingly neat solution involving a surprise appearance of a very famous sequence.

### 1 December

There are 5 ways to write 4 as the sum of 1s and 2s:
• 1+1+1+1
• 2+1+1
• 1+2+1
• 1+1+2
• 2+2
Today's number is the number of ways you can write 12 as the sum of 1s and 2s.

My next highlight is a puzzle that I was particularly proud of cooking up: again, this puzzle at first glance seems like it'll take a lot of brute force to solve, but has a surprisingly neat solution.

### 10 December

The equation $$x^2+1512x+414720=0$$ has two integer solutions.
Today's number is the number of (positive or negative) integers $$b$$ such that $$x^2+bx+414720=0$$ has two integer solutions.

My next highlight is a geometry problem that appears to be about polygons, but actually it's about a secret circle.

### 12 December

There are 2600 different ways to pick three vertices of a regular 26-sided shape. Sometimes the three vertices you pick form a right angled triangle.
These three vertices form a right angled triangle.
Today's number is the number of different ways to pick three vertices of a regular 26-sided shape so that the three vertices make a right angled triangle.

My final highlight is a puzzle about the expansion of a fraction in different bases.

### 22 December

In base 2, 1/24 is 0.0000101010101010101010101010...
In base 3, 1/24 is 0.0010101010101010101010101010...
In base 4, 1/24 is 0.0022222222222222222222222222...
In base 5, 1/24 is 0.0101010101010101010101010101...
In base 6, 1/24 is 0.013.
Therefore base 6 is the lowest base in which 1/24 has a finite number of digits.
Today's number is the smallest base in which 1/10890 has a finite number of digits.
Note: 1/24 always represents 1 divided by twenty-four (ie the 24 is written in decimal).

### Notes and Easter eggs

I had a lot of fun this year coming up with the names for the possible theives. In order to sensibly colour code each suspect's clues, there is a name of a colour hidden within each name: Fred Metcalfe, Jo Ranger, Bob Luey, Meg Reeny, and Kip Urples. Fred Metcalfe's colour is contained entirely within his forename, so you may be wondering where his surname came from. His surname is shared with Paul Metcalfe—the real name of a captain whose codename was a certain shade of red.
On 20 December, Elijah Kuhn emailed me to point out that it was possible to solve the final puzzle a few days early: although he could not yet work out the full details of everyone's timetable, he had enough information to correctly work out who the culprit was and between which times the theft had taken place.
Once you've entered 24 answers, the calendar checks these and tells you how many are correct. This year, I logged the answers that were sent for checking and have looked at these to see which puzzles were the most and least commonly incorrect. The bar chart below shows the total number of incorrect attempts at each question.
You can see that the most difficult puzzles were those on 13, 24, and 10 December; and the easiest puzzles were on 5, 23, 11, and 15 December.
I also snuck a small Easter egg into the door arrangement: the doors were arranged to make a magic square, with each row and column, plus the two diagonals, adding to 55.

### The solution

The solutions to all the individual puzzles can be found here. Using the clues, you can work out that everyone's seven activities formed the following timetable.
 Bob Luey Fred Metcalfe Jo Ranger Kip Urples Meg Reeny 0:00–1:21Billiards 0:00–2:52Maths puzzles 0:00–2:33Maths puzzles 0:00–1:21Billiards 0:00–1:10Ice skating 1:10–2:33Skiing 1:21–2:52Ice skating 1:21–2:52Stealing presents 2:33–4:45Billiards 2:33–4:45Billiards 2:52–3:30Lunch 2:52–3:30Lunch 2:52–3:30Lunch 3:30–4:45Climbing 3:30–4:45Climbing 3:30–4:45Climbing 4:45–5:42Curling 4:45–5:42Curling 4:45–5:42Curling 4:45–5:42Curling 4:45–5:42Lunch 5:42–7:30Maths puzzles 5:42–7:30Ice skating 5:42–7:30Chess 5:42–7:30Chess 5:42–7:30Maths puzzles 7:30–10:00Skiing 7:30–9:45Chess 7:30–8:45Skiing 7:30–10:00Maths puzzles 7:30–9:45Chess 8:45–9:45Lunch 9:45–10:00Table tennis 9:45–10:00Table tennis 9:45–10:00Table tennis
Following your investigation, Santa found all the presents hidden under Kip Urples's bed, fired Kip and sucessfully delivered all the presents on Christmas Eve.

### The winners

And finally (and maybe most importantly), on to the winners: 73 people submitted answers to the final logic puzzle. Their (very) approximate locations are shown on this map:
From the correct answers, the following 10 winners were selected:
 1 Sarah Brook 2 Mihai Zsisku 3 Bhavik Mehta 4 Peter Byrne 5 Martin Harris 6 Gert-Jan de Vries 7 Lyra 8 James O'Driscoll 9 Harry Poole 10 Albert Wood
Congratulations! Your prizes will be on their way shortly. Additionally, well done to Alan Buck, Alex Ayres, Alex Bolton, Alex Lam, Alexander Ignatenkov, Alexandra Seceleanu, Andrew Turner, Ashwin Agarwal, Becky Russell, Ben Reiniger, Brennan Dolson, Carl Westerlund, Cheng Wai Koo, Christopher Embrey, Corbin Groothuis, Dan Whitman, David, David Ault, David Kendel, Dennis Oltmanns, Elijah Kuhn, Eric, Eric Kolbusz, Evan Louis Robinson, Felix Breton, Fred Verheul, Gregory Loges, Hannah, Jean-Noël Monette, Jessica Marsh, Joe Gage, Jon Palin, Jonathan Winfield, Kai Lam, Louis de Mendonca, M Oostrom, Martine Vijn Nome, Matt Hutton, Matthew S, Matthew Wales, Michael DeLyser, MikeKim, Naomi Bowler, Pranshu Gaba, Rachel Bentley, Raymond Arndorfer, Rick Simineo, Roni, Rosie Paterson, Sam Hartburn, Scott, Sheby, Shivanshi, Stephen Cappella, Steve Paget, Thomas Smith, Tony Mann, Valentin Vălciu, Yasha Ayyari, Zack Wolske, and Zoe Griffiths, who all also submitted the correct answer but were too unlucky to win prizes this time.
See you all next December, when the Advent calendar will return.

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Comments in green were written by me. Comments in blue were not written by me.