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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.

Show answer


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?

Show answer


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.

Show answer


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 numbermean 7range 2

Show answer

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:
1Valentin Valciu
2Tom Anderson
3Alex Bolton
4Kevin Fray
5Jack Kiuttu
6Ben Baker
7Joe Gage
8Michael Thomas
9Martin Holtham
10Beth 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.

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Comments

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"
Adam Abrams
   ×1              Reply
Never mind, I found them, they were your example ASC. Very clever!
Michael T
   ×1              Reply
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...

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Comments

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
                 Reply
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
                 Reply
@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   ×1   ×1   ×1     Reply
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
                 Reply
It was fun.
Rishabh
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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!
As you solve the puzzles, your answers will be stored. To share your stored answers between multiple devices, enter your email address below the calendar and you will be emailed a magic link to visit on your other devices.
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!

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Comments

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              Reply
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)
                 Reply
@(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
                 Reply
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)
                 Reply
Thanks for the nice puzzles!
Gert-Jan
                 Reply
@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.

Show answer


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.

Show answer


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. Sometime 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.

 

Show answer


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).

Show answer

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 LueyFred MetcalfeJo RangerKip UrplesMeg Reeny
0:00–1:21
Billiards
0:00–2:52
Maths puzzles
0:00–2:33
Maths puzzles
0:00–1:21
Billiards
0:00–1:10
Ice skating
1:10–2:33
Skiing
1:21–2:52
Ice skating
1:21–2:52
Stealing presents
2:33–4:45
Billiards
2:33–4:45
Billiards
2:52–3:30
Lunch
2:52–3:30
Lunch
2:52–3:30
Lunch
3:30–4:45
Climbing
3:30–4:45
Climbing
3:30–4:45
Climbing
4:45–5:42
Curling
4:45–5:42
Curling
4:45–5:42
Curling
4:45–5:42
Curling
4:45–5:42
Lunch
5:42–7:30
Maths puzzles
5:42–7:30
Ice skating
5:42–7:30
Chess
5:42–7:30
Chess
5:42–7:30
Maths puzzles
7:30–10:00
Skiing
7:30–9:45
Chess
7:30–8:45
Skiing
7:30–10:00
Maths puzzles
7:30–9:45
Chess
8:45–9:45
Lunch
9:45–10:00
Table tennis
9:45–10:00
Table tennis
9:45–10:00
Table 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:
1Sarah Brook
2Mihai Zsisku
3Bhavik Mehta
4Peter Byrne
5Martin Harris
6Gert-Jan de Vries
7Lyra
8James O'Driscoll
9Harry Poole
10Albert 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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 2018-12-23 
Recently, I made myself a new Christmas decoration:
Loyal readers may recognise the Platonic solid presents from last year's Christmas card, last year's Advent calendar medals, or this year's Advent calendar medals.
If you'd like to make your own Platonic solids Christmas cross stitch, you can find the instructions below. I'm also currently putting together some prototype Platonic solids cross stitch kits (that may be available to buy at some point in the future), and will be adding these to the piles of prizes for this year's Advent calendar.

You will need

Instructions

Cross stitch thread is made up of 6 strands twisted together. When doing your cross stitch, it's best to use two strands at a time: so start by cutting a sensible length of thread and splitting this into pairs of strands. Thread one pair of strands into the needle.
Following the patterns below, cross stitch rows of red and non-red crosses. To stitch a row, first go along the row sewing diagonals in one direction, then go back along the row sewing the other diagonals. When you've finished the row, move the the row above and repeat. This is shown in the animation below.
When doing the first row, make sure the stiches on the back cover the loose end of thread to hold it in place. Looking at the back of the aida, it should look something like this:
When you are running out of thread, or you have finished a colour, finish a stitch so the needle is at the back.
Then pass the needle through some of the stitches on the back so that the loose end will be held in place.

Patterns

The patterns for each solid are shown below. For each shape, the left pattern shows which colours you should cross in each square (/ means red, o means non-red); and the right pattern shows where to add black lines after the crosses are all done.
Pattern for a tetrahedron (click to enlarge)
Pattern for a cube (click to enlarge)
Pattern for a octahedron (click to enlarge)
Pattern for a dodecahedron (click to enlarge)
Pattern for a icosahedron (click to enlarge)

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Mar 2020

Log-scaled axes

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PhD thesis, chapter ∞
PhD thesis, chapter 5
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Inverting a matrix
PhD thesis, chapter 2

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PhD thesis, chapter 1
Gaussian elimination
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