Christmas (2019) is over

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.


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 (click to enlarge).
If you actually visited this position on the map, you found 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, Blitzen was at, and Rudolph was at
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 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 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 - - "number of ways of factoring n with all factors greater than 1"

Day 4 - - "number of ways to tile a 2 × (n-1) rectangle with 1 × 2 dominoes and 2 × 2 squares"

Day 2 - - "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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