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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

Matthew

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

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2020-12-03
In November, I spent some time designing this year's Chalkdust puzzle Christmas card.
The card looks boring at first glance, but contains 9 puzzles. By splitting the answers into two digit numbers, then colouring the regions labelled with each number (eg if an answer to a question in the red section is 201304, colour the regions labelled 20, 13 and 4 red), you will reveal a Christmas themed picture.
If you want to try the card yourself, you can download this pdf. Alternatively, you can find the puzzles below and type the answers in the boxes. The answers will be automatically be split into two digit numbers, and the regions will be coloured...

### Similar posts

 Christmas card 2019 Christmas card 2018 Christmas card 2017 Christmas card 2016

Comments in green were written by me. Comments in blue were not written by me.
@JDev: lots of the card will still be brown once you're done, but you should see a nice picture. Perhaps one of your answers is wrong, making a mess of the picture?
Matthew

I finished all of the puzzles but the picture is far from colored in. Am I missing something?

These puzzles have been a blast!
JDev

Thanks a lot, Matthew. It was great fun!
Gert-Jan
×1
@Tara: I initially made the same mistake. Maybe you didn't take into account that 6 is not one of the available digits in question 1?
Sean
×1   ×2
@Tara: Yes, looks like you may have got an incorrect answer for one of the black puzzles
Matthew
×1   ×2   ×1   ×1   ×1

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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
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@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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2019-12-08
Just like last year, the year before and the year before, TD and I spent some time in November this year designing a Chalkdust puzzle Christmas card.
The card looks boring at first glance, but contains 9 puzzles. By splitting the answers into two digit numbers, then drawing lines labelled with each number (eg if an answer is 201304, draw the lines labelled 20, 13 and 4), you will reveal a Christmas themed picture. Colouring the regions of the card containing circles red, the regions containing squares green, and the regions containing stars white or yellow will make this picture even nicer.
If you want to try the card yourself, you can download this pdf. Alternatively, you can find the puzzles below and type the answers in the boxes. The answers will be automatically be split into two digit numbers, the lines will be drawn, and the regions will be coloured...

### Similar posts

 Christmas card 2020 Christmas card 2018 Christmas card 2017 Christmas card 2016

Comments in green were written by me. Comments in blue were not written by me.
Rishabh, what do you see that I do not?
Rob Glencairn

Thanks for the feedback. (I now understand the need for redaction). My son sent me your link as a Xmas present. I must think of an appropriate retaliation. (What is a PDF?)Think I've fixed 1,6 and 9....8 eludes me, for the moment.
Rob

@Rob: It looks to me like you've made mistakes in questions 1, 6, 8, and 9. The hints from the back of the pdf might help:

1. How many numbers between 1 and 10,000 have 1 as their final digit? How many have 1 as their penultimate digit?

6. How many ways can you write 1? 2? 3? 4? 5? What's the pattern?

8. How many zeros does 10! end in? How many zeros does 20! end in? How many zeros does 30! end in?

9. Carol’s sum is odd. What does this tell you about the 5- and 6-digit numbers?
Matthew
×1   ×1   ×2   ×1   ×1
I'm 71, with one good eye left. What am I missing?
1. 400001
2. 1849
3. 2002
4. 130405
5. 120306
6. 53?
7. 171175
8. 59?
9. 313525
Rob

It was fun.
Rishabh
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